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1. 900 0 05 So you should not type 550 1 1 0 05 30 170 05 but rather 550 1 1 0 05 7 30 0 05 where the brackets have become parentheses Some very old math books use braces and as a sort of auxiliary also to the parentheses and brackets These too if they are for grouping in a formula must become parentheses As it turns out SAGE as well as modern mathematical books use the braces and to denote sets By the way the above formula was not artificial It is the value of a loan at 5 compounded annually for 30 years with an annual payment of 550 The formula is called the present value of an annuity Three Mistakes that are 90 of My Errors in SAGE There are three mistakes that I make a lot when using SAGE For example if you want to say 11 000x 1200 then you have to type 11000 x 1200 The first error that I sometimes make is that I leave out the asterisk between 11000 and x In SAGE you must include that asterisk as that is the symbol of multiplication The second error that Pll often add a comma inside the 11000 but it is not acceptable in SAGE to write 11 000 for 11000 The third error is that Pll have mismatched parentheses Any SAGE expression should have the same number of s as it has of s no more and no less 1 2 Using SAGE with Common Functions Now Pl discuss how SAGE works with square roots logarithms exponentials and so forth Square Roots The standard high school
2. Business Calculating the optimal array of vehicles in a corporation s fleet for transporting gasoline Biology Modeling the interactions between two species one a predator the other prey The examples for physics and finance as well as microeconomics are fully written and are included here I have used the examples for engineering and business in class as projects but need to convert those PDFs into LaTeX for inclusion in the book The pure mathematics example is available as a Sage worksheet but needs conversion into LaTeX The ecology example was done in a lecture of mine so it might require a bit of time to convert into a project but not much The astronomy example was an exam question of mine so that might require a bit more time The earth sciences medicine and biology examples have not been written yet 2 1 The Pure Math Example Quintics Coming Soon Given a quintic polynomial trying to find the correct window for graphing by Prof Ben Jones I hope to write this soon but until I do you can find it at https sage uwstout edu 8000 home pub 21 and https sage uwstout edu 8000 home pub 23 48 2 2 The Finance Example Mortgages This example assumes you know how to do mortgage calculations If you don t then you might want to read a different example Let us say that you want to sell some mortgage products and you want to compute the cost per t
3. Now the following command will plot the set of points where f x y z 0 and the variables range over the values 1 2 lt x lt 1 2 and 1 lt y lt 1 as well as 1 lt z lt 1 implicit_plot3d f x 0 5 0 5 y 1 1 z 1 1 color red That produces this 28 0 00 70 0 0 50 1 0 Before we wrap up our discussion of 3D plots it is important to point out that if you were to use plot_points 100 in the above command then the z interval would be divided into 100 parts as well as the y and z intervals for a total of 100 x 100 x 100 1 000 000 plotting points This is extremely unwise and can crash your web browser Backwards Compatibility The following syntax makes the plot command look like the plot3d com mand s syntax plot x2 x 2 2 where as we would normally have typed ploti z 2 2 2 as in the last section This useful for when you have to plot things in terms of t and not in terms of x For example var t plot 16xt72 400 t 0 3 is the height of a bowling ball dropped off of a 400 foot building t seconds after release Here is the plot 400 380 360 340 320 300 280 260 ala ee ee 2424222100222 022 aa 0 0 5 l La 2 2 5 3 29 Two Dimensional Implicit Plotting Once in a while you have to plot a curve implicitly even in two dimensions The syntax is basically identical to three dimensional plots For example 8 x y x 4
4. e Manual Restart and Re Evaluate 0 ee ees nario your Work Ba ap axa ht ee ess i ee oe we Sees oe e enamine ys 1OMING silla aa e OE EK a Oe ee A a B 10 Saving Loading your Work Locally B 11 Semicolons B 12 Displaying Source Code C Other Resources for SAGE D Installing SAGE on your Personal Computer 55 55 60 60 60 61 62 62 65 67 74 714 14 19 19 76 7 7 7 7 7 7 18 79 81 31 Sl 31 32 32 82 82 83 83 83 83 83 83 83 83 85 86 Chapter 1 Welcome to SAGE As the open source and FREE competitor to expensive software like Maple Mathematica Magma and Matlab SAGE offers anyone with access to a web browser the ability to use cutting edge mathematical software and display one s results for others This document is going to share with you several common mathematical tasks that are extremely easy and which may serve as your starting point into SAGE Pm sure that you will find SAGE far easier to use than a graphing calculator and vastly more powerful How to Use This Guide There is no need to read this entire document just as you would never read the dictionary cover to cover If you are handed this guide as part of a class then your professor will tell you which section numbers correspond with what you need to know Personally I recommend just reading Chapter One and then start playing around on your own Using the find feature of your pdf viewer you can alw
5. 0002080 0 0 0000000 18 7000041 5 0 030002000 49005000 3 which has some non zero entries but mostly contains zeros to mark where the blanks would go in an ordinary Sudoku game Now you can solve the game by typing sudoku A and obtain 14 51368724 9 8495215637 267349581 15846397 2 97421836 5 3267954 1 8 7829341 5 6 635172894 4918567 2 3 which is a valid Sudoku solution There might however be other solutions 3 9 Measuring the Speed of SAGE Factoring large numbers for example is supposed to be hard and you might want to know how long it takes SAGE to execute the command factor 2250000063000000041 In which case you should type timeit factor 2250000063000000041 and for me on my machine it says 625 loops best of 3 91 8 us per loop which means it tried 625 times and the best estimate for the running time is 91 8 microseconds or about 1 10 893 seconds Of course the response time when you operate with SAGE is much longer than that What s going on Most of the time using SAGE is wasted with the commands traveling over the internet to the server and traveling back Also some time is wasted displaying the data In this case you probably lose about 2 seconds on transit time 1 4th of a second on display and the interface and nearly 0 seconds on computation But for certain very hard problems you can spend 10 minutes on computation and 2 seconds on transit time and a 1
6. B 1 Copy and Paste You can use cut and paste normally in SAGE Just highlight anything you ve typed and go to the Edit menu of your web browser and select copy Next go somewhere else where you d want to type that thing and go to the Edit menu of your web browser and select paste This can save you a lot of typing Sometimes you get a red line in the box where you copied from even though you made no changes Just click evaluate under the red lined box and it will go away B 2 The Moving Slash Earlier you learned that the vertical green rectangle under a SAGE box is intended to indicate that SAGE is thinking about what you entered There s another feature that s meant to indicate to you that SAGE is thinking this usually is only important if you ask SAGE to do a very long computation or something very complicated At the very top of the web browser window during such a long computation you ll see four things a slash the filename of what you re working on a dash and the logo SAGE While SAGE is computing that slash will switch back and forth between the forward slash and the back slash about once per 2 3 seconds When you see this slash swinging back and forth you know that the computation is proceeding This way you can distinguish between the case of SAGE working hard on your problem versus an interrupted network connection or an outage at the SAGE server B 3 The Online Help Syst
7. solve x 8 1 x which produces x 1 2 I 1 2 1 7 1 8 sqrt 2 x I 1 7 1 8 x 1 2x 1 1 2 1 1 8 sqrt 2 x 1 7 1 8 x 1 2x 1 1 2 1 1 8 sqrt 2 x I 1 1 8 x 1 2x1 1 2 1 7 1 8 sqrt 2 x 1 1 8 giving us eight distinct complex numbers all of which have 1 as their eighth power This can be calculated by hand more slowly with DeMoivre s formula but for SAGE this is an easy problem Advanced Cases To see why we don t ask undergraduates to memorize Cardano s cubic formula try typing in solve x73 b x c 0 x That gives you Cardano s formula for a monic depressed cubic A cubic polynomial is said to be depressed if the quadratic coefficient is zero and monic means that the leading coefficient here the cubic coefficient is zero The full version of the cubic formula would be given by var d solve a x73 b x72 c x d 0 x which is just insanely complicated Try it and you ll see In fact it is so complicated that one would have to confess that it is not useable An even uglier formula would be the general formula for a quartic or degree four polynomial Surely any quartic can be written asx aga aX a x ag 0 and so we can ask SAGE to solve it with var a0 al a2 a3 a4 solve a4 x74 a3 x73 a2xx 2 al x a0 0 x and we get an answer that is a formula of horrific proportions 1 10
8. The Underscore One of the most useful commands in SAGE is the underscore and on most keyboards that can be found by pressing shift and the hyphen or dash It looks like this _ The numerical value of the special symbol _ is the value of the previous box So it is kind of like Ans on a graphing calculator For example you can add 100 to the previous box by typing _ 100 One of the ways that I use this is with the n command Imagine you type sqrt 75 and get back 5 sqrt 3 But suppose you wanted instead a decimal approximation Then you can type n _ and SAGE replies 8 66025403784439 A Financial Example Suppose you deposit 5000 in an account that earns 4 5 compounded monthly You are curious when your total will reach 7000 Using the formula A P 1 1 where P is the principal A is the amount at the end 7 is the interest rate per month and n is the number of compounding periods number of months we have 7000 5000 1 0 045 12 14 1 0 045 12 log1 4 log 1 0 045 12 log1 4 nlog 1 0 045 12 log 1 4 log 1 0 045 12 And so we type into SAGE log 1 4 log 1 0 045 12 and get the response 89 8940609330801 so that we know 90 months will be required Therefore we write the answer 7 years and 6 months This is an example of a computer assisted solution to a problem In addition to that approach SAGE is also willing to solve the problem from start to finish with no human intervention We ll see that on Pag
9. To find out what 1939 is mod 37 you can type 1939 Y 37 but the operator takes the modular reduction only of the nearest number This means that to evalulate 3 4 2 mod 5 you must not type 372x x4 2 5 which evaluates to 38 but rather 372x4 2 5 99 which gives 3 the correct residue The opposite of the mod operator is the integer quotient To find the integer quotient of a number use 25 4 to get 6 This is the quotient rounded down to the nearest integer You can define def is_even n return n 2 which will return True for even integers and False for odd ones This can then be used to wrap up into other if statements We ll talk much more about if statements in the chapter on programming 3 2 Minor Commands of SAGE Here we list a few commands that come up from time to time They can be quite useful in the situations that require them min x y Returns either x or y whichever is smaller max x y Returns either x or y whichever is greater floor x Just round x down or x ceil x Just round x up or x factorial n This is n also called n factorial binomial x m This is usually written 2 or Em Oir binomial x m m This is usually written Ay OF sky 3 2 1 Rounding Floors and Ceilings As you can see the floor command rounds a number down and the ceil command rounds a number up In older books the floor function is sometimes called The Greatest Integer Fun
10. just like we did in the previous section on 2 dimensional plots This results in 4 0 24 0 Note that we do not type var y again because you only have to declare each variable once Once you have declared it you do not have to declare it a second time And for a surface that is just plain weird try plot3d sin x y y cos x x 3 3 y 3 3 which creates This is an example of a surface that you really do have to see from many angles to understand Try dragging it around with the mouse to see what it looks like from many angles Another neat one is plot3d 4 x exp x72 y72 x 2 2 y 2 2 which creates 26 2 0 2 0 Clearly this too is a surface that can only be understood by rotating the object and examining it from several angles Questions of Resolution Density If instead you were to type plot3d sin x y y cos x x 10 10 y 10 10 you get a much rougher irregular surface that does not seem to properly resemble the previous After all we didn t change the function and we just went from the 6 x 6 region about the origin to the 20 x 20 region about the origin Have a look 10 0 10 0 The issue comes down to resolution The plot3d command by default chooses a certain number of points by some method and samples the function at those points Whatever method it uses which is not known to me produced enough points for a good 6 x 6 picture but not enough for a good 20 x 20 pi
11. 4th of a second on display and the interface So this feature of SAGE is there to help you measure that 3 10 Huge Numbers and SAGE Sometimes when studying combinatorics one has to deal with numbers that are huge For example asking SAGE for factorial 52 will result in 80658175170943878571660636856403766975289505440883277824000000000000 but it is hard for me to wrap my head around what that means So I can type ceil log _ 10 which will give me 68 That means that the number is a 68 digit number Why does this work Because log 10 7 and log 108 8 Accordingly any number x such that 107 lt x lt 10 has therefore 7 lt log x lt 8 But what numbers have 10 lt x lt 10 Precisely the eight digit numbers of course By taking the ceiling with ceil we round up and so every eight digit number has the ceiling of its common logarithm being precisely 8 Typing factorial factorial 7 to compute 7 is also amusing You get a very large number if you do this In fact using the above technique I can learn that 7 has 16 474 digits You ll also see that this very large number is displayed as dozens of lines each ending in a backslash The backslashes are SAGE s 19 way of saying that it really sees several lines as one large line but that the very large line of digits wouldn t fit on the screen Therefore SAGE has to break it to fit it on your screen It is also fun to type factorial factorial 7 and then fa
12. an abbreviation for yar a var b var c Also note that the following for forms are equivalent varias D C var a b c var Ca b c varl as D0 and you can use which ever one you prefer 31 Linear Systems of Equations You can solve several equations simultaneously whether they are linear or non linear An easy case might be to solve b 6 ct b 4 which would be done by typing var b solve x b 6 x b 4 x b Again note the useof square brackets and commas to form a list in SAGE This is our second example of a list in SAGE You can enclose any data with and and separate the entries with commas to make a list Our first example was in graphing multiple functions at once on Page 68 we ll see another example in Number Theory on Page 57 and in scatter plotting live data on Page 62 Also it is important to point out that there is no harm in declaring b twice The two var commands do no harm to each other On the other hand there is also no need to declare b twice as once it is declared SAGE will remember that it is a variable A more typical linear system might be 9a 3b 1e 32 4Ja 2b 1e 15 la 1b 1le 6 and to solve that we d type var ay by solve 9 a 3 b c 32 4xa 2b c 15 a b c 6 a b c and SAGE gives the answer La 4 b 3 5 Cc 5 which is correct Of course this is exactly the linear system
13. below the graph This is the plot 1000 oo a 400 j 6 T 8 9 10 Your hint that the location of the x axis in the display is not where it would be normally is that the axes do not intersect This is to tell you that the origin is far away When the axes do intersect on the screen then the origin is both in truth and on the screen where they intersect Also if you want to you can hide the axes with plot x73 5 10 axes false and that produces which is considerably less informative To Force the Y Range of a Graph If for some reason you want to force the y range of a graph to be constrained between two values you can do that For example to keep 6 lt y lt 6 we can do plot x73 x 3 3 ymin 6 ymax 6 in order to obtain 13 This is important because normally SAGE wants to show you the entire graph Therefore it will make sure that the y axis is tall enough to include every point from the maximum to the minimum For some functions either the maximum or the minimum or both could be huge Plots of Functions with Asymptotes The way that SAGE and all computer algebra tools computes the plot of a function is by generating a very large number of points in the interval of the xs and evaluating the function at each of those points To be precise if you want to graph f x 1 x between x 4 and x 4 the computer might pick 10 000 random values of x between 4 and 4 find the y values
14. break down of pag xr 5x 6 then you must do f x x 3 x x 2 5 x 6 followed by f partial_fraction and get the answer x 6 2 247 3 5 which is correct In fact it means 6 A 24 E 2 x 3 However you can also do the shortcut and ask instead integral x 3 x x 2 5 x 6 x which will result in 1 2 x 2 5 x 6 log x 2 24 xlog x 3 Improper Integrals While the following integral is mathematically valid 1 1 1 5 s 2 C and the right hand side can be evaluated at x 1 and x 1 there is no finite answer to because this improper integral is divergent The graph will reveal why that is the case Here is the plot generated by 45 plot 1 1 x72 10 10 ymin 2 ymax 2 Yet SAGE knows this and will say k A in response to integral 1 1 x72 x but instead will respond ValueError Integral is divergent to the inquiry integral 1 1 x72 x 1 1 Another type of improper integral is and the way to write that in SAGE is to type integral 1 x 2 x 2 00 to which SAGE responds 1 2 or another example is e da which in SAGE is integral exp x72 x 00 00 and it gives the correct response of 7 The idea is simply that oo is a special code for infinity It is two letter o s and the idea is that if you squint it looks like an infinity symbol The Function erf and Integrals Among mathematicians and especially among statisti
15. digits you can do 332 bits 200 digits is 664 bits and so on You can also abbreviate with n sqrt 2 prec 200 as we did earlier That s what the NO or nO function they are the same is an abbreviation of namely numerical approx Is SAGE Case Sensitive You ve probably had a situation in life where you entered your password correctly but discovered that it was rejected because you had the wrong capitalization For example perhaps you ve left the CAPS LOCK key on In any event SAGE does think of Pi as different from pi and Sin as different from sin The only two exceptions are i and nO First you can represent y 1 as either i or I and either way SAGE will know what you meant namely 1 Second you can write n sqrt 2 or N sqrt 2 and SAGE treats those as identical These easements only apply to 1 and nO as it turns out in all other cases capitalization matters Tab Completion Now would be a good time to explain a neat feature of SAGE If you are typing a long command like numerical approx and part way through you forget the exact ending is it approximation approximations appr then you can hit the tab button If only one command in the entire library has that prefix then SAGE will fill in the rest for you If it does not then you have to enter a few more letters On a slow internet connection the pause can be very noticeable but it is still a useful feature when you cannot remember exact command
16. first line of our analysis was 7000 5000 1 0 045 12 and translating that into the language of SAGE results in 7000 5000 1 0 045 12 n and so we should type var n find_root 7000 5000 1 0 045 12 n O 10000 Note that the second line means that I m searching for a value of n that makes the equation satisfied and that the value of n should be between 0 and 10 000 This seems reasonable because 10 000 months is a little over 833 years The first line is just declaring n as a variable which we saw on Page 31 In any case we get the answer 89 894060933080027 as before and learn that 90 months will be required 1 11 Using SAGE to take derivatives 3 The command for the derivative is just diff For example to take the derivative of f a x x you would type diff x 3 x xX to learn the answer 3 x 2 1 Or for f x sin x you would type diff sin x 2 x to receive the answer 2xx xcos x 2 Sometimes you might have earlier defined your own function by g x e 10 x earlier and so you can then type diff g x x to get the derivative which is naturally 10 e 10 x but you can also type gprime x diff g x 38 which is silent but then you can type gprime 2 or gprime 3 and it does what you expect It tells you the values of g 2 and g 3 Another 100 equivalent form is g x derivative SAGE can also do very difficult derivatives diff x x x
17. have zeros everywhere except a very noticeable diagonal of ones Here in C2 and C1 as well we see that column three differs from this pattern That column is called defective and its variable is going to be called the free variable Every other variable will be a formula in terms of the free variable In our case zz 0 ae yt2z 0 ES the final answer 0 0 pine 21 though I like to add the words z is free to the final answer under the equations for x and y even though it is obvious Now what does all this mean It means that you can choose any z that you want whatsoever Once you have chosen such a z then you can use those formulas for x and y to make a solution There is one solution for every z Some examples include iz 3 then 1 x gt 3 aud y 22 If z 2 then z z 2 and y 2z 3 6 The solution is z 3 y 6 z 3 2 4 The solution is z 2 y 4 z 2 If z 1 then z z 1 and y 2z 2 1 2 The solution is z 1 y 2 z 1 If z 0 then x z 0 and y 2z 2 0 0 The solution is z 0 y 0 z 0 If z 1 then z z 1 and y 2z 2 1 2 The solution is z 1 y 2 z 1 If z 4 21 then z z 4 21 and y 2 4 21 8 42 The solution is x 4 21 y 8 42 z 4 21 And so forth for infinitely many solutions DO NW WwW bw oe a 1 5 5 The Rare Case Coming soon 1 5 6 Geo
18. in particular a green rectangle that is very thin under the box where you just typed and next to the red line This is to indicate that your web browser is connecting to the SAGE server and has made its request and waiting for the answer Do not hit evaluate again On fast connections the green rectangle will appear and disappear very rapidly and you probably will not see it Note if you have a very slow connection then see The Moving Slash on Page 81 Saving or Printing your Work We will go into more detail about this later but for now notice the Save and Save amp Quit buttons as well as the Discard amp Quit button These do exactly what you think they do You can also use the Print button to print your work Like the undo button if you ve done a lot of work you ll have to scroll the window all the way to the top of the document to find it as this button sits at the top of the sheet Shortcut While I mentioned it before if you are tired of clicking evaluate all the time you can just press Shift and Enter at the same time Grouping Symbols When there are multiple sets of parentheses in a formula sometimes mathematicians use brackets as a type of super parentheses As it turns out SAGE needs the brackets for other things like lists and so you have to always use parentheses for grouping inside of formulas For example let s say you need to evaluate 1 1 0 05 9
19. integrals both numerically and symbolically 1 14 Matrices in SAGE Part TWO aladdin e A AAA A A Fun Exploration Projects using SAGE 2 1 The Pure Math Example Quintics a a a a 2 2 The Finance Example Mortgages e 2 3 The Physics Example Gravitation and Satellites 2 2 4 The Microeconomic Example Selling Price ee eee ees 2 5 The Engineering Example Two Stage Rockets ee 2 6 The Ecology Example Lake Dumping 0 0 002 2 eee eee ne 2 7 The Astronomy Example Point of Closest Approach 0 2 000000 2 8 The Earth Science Example Sunrise and Sunset e 2 9 The Medical Example Blood in an Artery 2 2 0 0 eee ee 2 10 The Business Example a Vehicle Fleet 2 11 The Biology Example Predator Prey Models e e e oR amp No 3 Advanced Features of SAGE 3 1 3 2 3 3 3 4 3 9 3 6 3 7 3 8 3 9 3 10 3 11 3 12 3 13 3 14 3 15 3 16 3 17 3 18 Using SAGE to work with integers gcd lcm factorization sigma tau etc Minor Commands of SAGE w2 4 4 6 se Ee eRe RES 3 2 1 Rounding Floors and Ceilings y x lt 4 4 bh oe E A 3 2 2 Combinations and Permutations 0048 3 2 3 Calculating Limits Expressly iaa sos oe ee HARES 3 2 4 The Hyperbolic Trigonometric functio
20. mostly in the study of the catenary curve which is the shape of a cable permitted to hang limp under its own weight but attached at each end Also there are some integrals which are much more compact in their solution if you know about the hyperbolic functions So don t feel bad if you ve never seen them These are the commands for all 12 of those functions SAGE func Mathematician s Words SAGE Long form SAGE Short form Mathematician s Words sinh x Hyperbolic Sine arcsinh x asinh x Inverse Hyperbolic Sine cosh x Hyperbolic Cosine arccosh x acosh x Inverse Hyperbolic Cosine tanh x Hyperbolic Tangent arctanh x atanh x Inverse Hyperbolic Tangent coth x Hyperbolic Cotangent arccoth x acoth x Inverse Hyperbolic Cotangent sech x Hyperbolic Secant arcsech x asech x Inverse Hyperbolic Secant csch x Hyperbolic Cosecant arccsch x acsch x Inverse Hyperbolic Cosecant 3 3 Scatter Plots in SAGE Let s suppose your examining some data given by the points First you should tell SAGE about the data using the following command which is another example of how lists are written in SAGE datapoints 0 7 1 Clio 20 20 2 2 9 3 1 05 4 0 9 Notice how the data namely those five points are separated by commas and enclosed by brackets This is an example of a list in SAGE You can enclose any data with and and separate the entries with commas to make a list This is our fourth example the previous thr
21. of this or alternatively consider the following example If we want to know what denominators result in a two decimal place or fewer exact terminating decimal we would just need to see what positive integers divide 100 In SAGE we d do divisors 100 and then get 1 2 4 5 10 20 25 50 100 but in binary the analogous rule would be 2 It is obvious that the only positive integers which divide 2 are the numbers 1 2 4 8 2 Therefore we can use this criterion for seeing which number bases are convenient for fractions and which are not Now we ll consider the Babylonian numbering system which is base 60 Two symbols would be sufficient to describe any fraction with the following denominators EL 2 3 de Oy Oe 8 9 10 12 15 16 18 20 24 25 30 36 40 45 48 50 60 72 75 80 90 100 120 144 150 180 200 225 240 300 360 400 450 600 720 900 1200 1800 3600 Did you notice how in decimal there were 9 denominators but in base 60 there were 45 denominators Or if you do not count one then 8 versus 44 What we really wanted to do was count the divisors The function for counting the number of positive integer divisors of an integer is 7 and in SAGE is given by sigma 3600 0 which just tabulates 1 for each divisor In other words each divisor is raised to the Oth power thus becomes 1 and then if you add those up you learn how many divisors there are Modular Arithmetic
22. other hand consider 12 Then among 1 2 3 4 5 6 7 8 9 10 11 we see that both 2 and 12 are divisible by 2 both 3 and 12 are divisible by 3 both 4 and 12 are divisible by 2 both 6 and 12 are divisible by either 2 or 3 both 8 and 12 are divisible by 2 both 9 and 12 are divisible by 3 both 10 and 12 are divisible by 5 Finally this means that only 1 5 7 and 11 are coprime to 12 and so 12 4 It would be very annoying to calculate of a 100 digit number such as when working with cryptography the science of codes All you need to do in SAGE is type euler_phi 12 and then you learn that the answer is 4 It is called euler phi because it was discovered by Leonhard Euler ov The Divisors of a Number Meanwhile consider the set of divisors of a number If you want to know the set of divisors of 12 which means the set of positive integers dividing 12 you would type divisors 12 and get then Es Ds Ds 205 253 50 125 2901 and likewise divisors 312500 gives you Liy 2 4 5 10 20 25 50 100 125 250 500 625 1250 2500 3125 6250 12500 15625 31250 62500 78125 156250 312500 but remember it is very important to not have any commas separating the thousands when inputting large numbers into SAGE Sometimes in number theory we want to calculate the sums of the squares or cubes of the divisors I can t think of an example why at the moment but I m sure there s a good one Consider that 45 has a
23. sqrt 4 all True to obtain 2xTI 2 I where the capital letter I represents y 1 the imaginary constant You might be interested to know that you can use the lowercase i instead of the capital one if you prefer While SAGE is quite good at complex analysis and can produce some rather lovely plots of functions of a complex variable we will not go into those details in this document 2To be mathematically correct I should say an integer or a fraction with denominator writable as a product of a power of 5 and a power of 2 For example 25ths 16ths and 80ths can be written exactly with decimals where as 3rds 15ths and 14th cannot Observe that 25 5 and 16 24 as well as 80 24 x 5 Those denominators have only 2s and 5s in their prime factorization Meanwhile 3 3 and 3 5 x 3 while 14 7 x 2 As you can see those denominators have primes other than 2 and 5 in their prime factorization If you find this interesting you should read Using SAGE to work with Integers on Page 55 Special Constants Pi and e We ve learned that the imaginary constant is capital I It turns out that m is built in as pi both letters lower case and e is built in as e again lower case You can do things like numerical_approx pi prec 200 to get many digits of pi or likewise numerical_approx sqrt 2 prec 200 to get a high accuracy expansion of v2 These expansions are to 200 bits and basically if you want 100
24. statistics package but has not created a scatter plot You of course are mindful of your scientific obligations and so you type scatter_plot otherdata to obtain the plot below on the left Then you can add in the line of best fit and you get the plot below on the right O 10 O O 10 O 4 4 0 2 o O 2 o 3 2 1 0 1 2 3 3 2 1 0 I 2 3 Without Best fit Line With Best fit Line 63 As you can see this is absurd The data is clearly an ordinary parabola Therefore a line is a terrible approximation for this data However of all the lines that could ever be drawn even though all of them have unacceptably high error there is one that has the least error It turns out that this line for this data set is y 0x 5 And it is manifestly clear that this is horrible fit as shown in the plot on the right By the way that plot was produced by the command scatter_plot otherdata plot 0 x 5 3 3 Now is a good time to explain that the object which scientists call the best fit line is actually better known among mathematicians as a linear regression By linear regression we mean analyzing a set of data to find the line that best fits the data Here we should have done a quadratic regression which would find the parabola which best fits the data A Chemistry Example Now you might think that the parabola example is quite cooked You re right I did think it up just to prove a point However the follow
25. the universal constant of gravitation the distance to the satellite might change quite often and the mass of the satellite could change also particularly if it has fuel aboard which will be burned during maneuvers So for those we should make a function Force Ms rs G Me Ms rs 2 and you could do things like Force 1000 10 1076 to get answers This is great if you have a large number of points that you re interested in and want to save a lot of typing Suppose now that you really wanted the function to work in terms of altitude as measured from the earth s surface and not from the center of the earth Well it is easy to realize that you just have to add the radius of the earth First you put in the radius of the earth which is 6378 1 km but of course we should switch it to meters to avoid unit conflicts re 6378100 Then you can define Forcei Ms alt Force Ms alt re and then you can just go ahead and use Force1 For example Forcei 1000 3621900 is roughly equivalent to what we had before I suppose it depends upon if the problem you are given is in terms of altitude or distance from the center of the earth This idea of building functions on top of functions is very powerful I have often written large suites of functions in this way one function on top of the other function on top of the next to model complex phenomena in SAGE 50 2 4 The Microeconomic Example Selling Price Onc
26. w are missing in the second equation We treat this as if the coefficients were zero With these two repairs the second equation is now 0w 0x y z 5 The third equation has parentheses That s not allowed We have to remember that 3 x y 3z 3y Furthermore the z is on the wrong side A third sin in the third equation is that there is no constant We treat this as a constant of zero Correcting these three issues we have w Pott ay 2 0 Now the real delinquent is the fourth equation We have w on the wrong side and what is worse is that there are two occasions of x When we move the 3x from the right of the equal sign to the left it becomes 3x and combines with the 2x that was already there to form 5x As if that were not enough the constant is on the wrong side Last but not least z is entirely absent Correcting all this we have W k30 Y al a With these equations in mind we have the matrix l 2 1 5 6 0 0 1 1 8d a 1 3 3 1 0 sak a o eel That would be very tedious to solve by hand using matrices it would be far worse to solve it by hand without matrices using medieval algebra However using SAGE we will solve it in the next subsection For now it turns out that the answer is w 107 7 x 12 7 y 54 7 a 19 7 which you can easily verify now if you like 1 5 3 Doing the RREF in SAGE This subsection is going to tell you how to compute the RREF of a matrix presumably w
27. y 4 16 implicit_plot g x 3 3 y 4 4 is a quartic curve favored by furniture designers for making conference room tables As you can see its mathematical formula is y 16 Here is the plot 4 3 2 1 9 Using SAGE to solve problems symbolically When we say that a computer has solved a problem for us that can come out to be in one of two flavors symbolically or numerically When we solve numerically we get a decimal expansion for a very good approximation of the answer When we solve symbolically we get an exact answer often in terms of radicals or other complicated functions It is a matter of saying if the solutions to r er 5 are 1 v5 and 1 V5 or saying that they are 1 23606797749979 and 3 23606797749979 The former is an example of a symbolic solution and the latter of a numerical solution If the answer is a formula then symbolic solution is the only way to go This section is about symbolic solution and then the next one is about numerical solution Univariate Formulae First let s try solving some single variable problems We can type solve x72 3 x 2 x and then we obtain x 2 x 1 But don t forget the asterisk To be precise to type solve x72 3x 2 x 30 would be wrong because of the absence of the asterisk between the 3 and the x Another more illustrative of a symbolic computation is solve x 2 9 x 15 0 x which gives x 1 2 sqrt 2
28. 1 9 2 x 1 2 sqrt 21 9 2 One way to really see what symbolic versus exact is about is to compare 3 1 7 1 15 1 1 1 292 1 1 1 1 1 6 which returns 1 354 394 431 117 with n 3 1 7 1 154 1 1 1 292 1 1 1 1 1 6 which returns 3 14159265350241 That could easily be mistaken for m In fact it is really close to m with relative error 2 78 x 1071 There is no restriction to polynomials You can also var theta solve sin theta 1 2 theta to get theta 1 6x pi You might be confused to see that var command The idea is that you need to tell SAGE that this variable is not yet known We ve used variables before but that s because we had assigned values to them When you want a variable to represent some unknown quantity you must use var The variable x is always pre declared you do not need to declare it SAGE assumes that x is an unknown Multivariate Formulae Now we re going to use SAGE to re derive the quadratic formula First we must declare our variables with vara b c and then we type solve a x 2 b x c 0 x and get instead back x 1 2 b sqrt 4xaxc b72 a x 1 2 b sqrt 4 ax xc b 2 a which has some terms in an unusual order but is undoubtedly correct Note that the square brackets and comma indicate a list in SAGE something which SAGE inherited from the computer language Python Declaring Variables in Bulk By the way var a b c is merely
29. 3 in the variable c You can use it anytime such as 23 2 c which evaluates to 561 113 If you go back and change the value of c however you have to make sure you use the evaluate button to re evaluate all other boxes that make use of c In general it is easy to avoid this by not changing intermediate variables once you set them You can also just tell SAGE to re evaluate every single box in order This is described on Page 83 If we wanted to find the relative error of c as an approximation of 7 we would remember the formula approximation truth relative error truth and type N c pi pi and learn that the relative error is around 84 parts per billion Not bad This approximation was found by the Chinese Astronomer Zu Chongzhi also known as Tsu Ch ung Chih We can compare 335 113 to 22 7 with N 22 7 pi pi and learn that this much more common approximation of 7 as 22 7 has a relative error around 402 parts per million Much worse The intermediate variables technique is great for things like the strength of gravity in physics or the speed of light which so far as we know do not change For things that do change try substitution which Pl tell you about in just a moment In finance you could use this for interest rates or the coefficients in a cost function 1 7 Using SAGE to Manipulate Polynomials Coming Soon Meanwhile here is a command list if you need to look up the commands e factor
30. AGE to go and find If you have no idea where the root is you can type find_root x75 x74 x 3 x72 x 1 10712 10712 which is asking SAGE to find a root that is between plus minus one trillion Extremely sophisticated problems can be asked about Some favorites of mine include if someone asks you about x 5 then you could try to find where x 5 0 You would type find_root x x 5 1 10 or equivalently find_root x x 5 1 10 36 Another one is to find where e 1 x and you d do that by finding a root of e 1 x 0 You could graph that with plot ex 1 x 3 3 ymin 10 ymax 10 obtaining the plot and then you can see the root is between 1 and 1 This means that we should type find_root e x 1 x 1 1 will result in finding out that the root is at 0 56714329040979539 Last but not least another favorite is to find sing cosx It happens to be between 0 and 7 2 which you can see by graphing The graph is 0 5 0 5 Now here you could ask for a root of sin x cosa Alternatively you can also type find_root cos x sin x 0 pi 2 which is just easier notation 37 Returning to an Old Problem Earlier on Page 8 we saw a problem involving compound interest and finding how many months will be required to turn 5000 into 7000 Now let s see how to solve that with SAGE in only one line rather than the previous approach which required the human to do some algebra The
31. GE s way of telling us that the model is y 62 7551x7 666 864a 1925 57 and then we can draw that model with scatter_plot cobalt_data plot 62 7551 x 2 666 864 x 1925 57 0 6 which produces the image 20000 1500 1000 500 While that s clearly not a very bad fit it is unwise to model radioactive decay with a quadratic function This is not only because many of the data points were missed The issue can be most easily seen by allowing time to have the range 0 to 15 instead of O to 6 In that case we type scatter_plot cobalt_data plot 62 7551 x 2 666 864 x 1925 57 0 15 where as you can see the only change to the command was changing the 6 into a 15 at the very end We get the following image which is clearly not an example of radioactive decay 3 5 Additional Topics in Plotting and Graphing Here are some additional notes and features about making plots and graphs in SAGE come out exactly as you want them to Graphing with Colors An example of graphing with colors would be plot sin x cos x 0 10 color purple producing 0 5 1 which looks like DNA to me As you can see by that you can plot multiple functions at the same time on the same graph To do that the list of functions should be separated by commas and enclosed in brackets 67 This is an example of a list in SAGE You can enclose any data with and and separate the entries with comm
32. R Multiple Precision Floating point Reals designed in part by Prof Paul Zimmerman you can go up to 10 000 bits of precision which is roughly 3010 digits of precision And MPFR comes with SAGE though it is a bit hard to use so we will leave it to the MPFR manual to explain how to use it http mpfr loria fr mpfr current mpfr html 3 17 Using SAGE to work with Cramer s Rule Coming Soon This will cover taking the determinant of matrices where one parameter is missing and is instead a variable and looking at the rational functions which come out of that T 3 18 Using SAGE to work with Resultants Coming Soon 18 Appendix A Obtaining a SAGE Account If you want to use the SAGE web browser tool you ll need an account on a server There are several free servers which you can use and some universities have their own server However an account on one is entirely separate from an account on another Unlike ultra large systems like GMAIL or FaceBook your files your username and your password do not transfer from one server to another Just pick one server and use it all the time This way you can always reach all your files anytime from anywhere in the world There s an advantage to choosing a server near you geographically because that way the connection will be faster Step One Some popular servers at this time September of 2011 include http sagenb org Seattle Washington USA http www sagenb org
33. SAGE for Undergraduates How to Get Started Gregory V Bard January 17 2013 Contents 1 2 Welcome to SAGE LL Usine SAGE as aca lCUla tor 0 tee e we a EB RS A a es a a a A 1 2 Using SAGE with Common Functions 44 44 246 e28les tah hid deme be ood B s 13 Using SAGE for trigonometry 44 amp 4 44 be ee aye e eee Be OS 1 4 Using SAGE to graph 2 dimensionally ee es Lo Matrices a d SAGE Part One eee peaa res Se 2 PRASA SS EGS Bee ew PEPE EL ae ForsG Taste or MACOS dy e 2 4 4 a E amp ee ae Bae bo EE Grek BA ig a EZ COmplcalons reis osa bee ees Bs ee OS S L53 Dome the RREP USAGE 23 30 00 Ee A E A a l low Theben RIG Cases adas try ane AAA eee ee at ee BS Peotone are Mase di Se a ae Bok Thos he Gre eae A Pewee ie eek ds Ge EY de da e 1 5 6 Geometrical Interpretations 2 a e e a a a a LD Mnr NOTES 2 bdo a E a a A TA o o raid e ed 1 6 Making your Own Functions Iin SAGE s s ss amp t indeed a ere ee Bw 1 7 Using SAGE to Manipulate Polynomials 0 0 00 eee ee ee 1 8 Using SAGE to graph a 3 dimensional function or an implicit function 1 9 Using SAGE to solve problems symbolically 2 2 2 0 0 000 ee ee 1 10 Using SAGE asa numencalsolyer a d sa a oe abba Be RE RB acer ek ok ww A 1 11 Using SAGE to take derivatives cig woh Sh 1 a EE A ee 1 12 Derivatives and Gradients in Multivariate Calculus 0 0 00000 1 13 Using SAGE to calculate
34. Seattle Washington USA http alpha sagenb org Seattle Washington USA http demo sagenb org Seattle Washington USA http nt sagenb org Seattle Washington USA http uw sagenb org Seattle Washington USA http flask sagenb org Seattle Washington USA http nb femhub org Reno Nevada USA http sage luther edu Decorah lowa USA https sage uwstout edu 8000 Menomonie Wisconsin USA http sagenb mc edu Clinton Mississippi USA http sagenotebook mc edu Clinton Mississippi USA http sagenb kaist ac kr Daejeon South Korea http sage math canterbury ac nz Christchurch New Zealand https sagenb gursey gov tr 7000 Istanbul Turkey http sage mathmu com Beijing China but the first two are so popular that they are overburdened and extremely slow and at times even painful to use Therefore it is better to randomly choose one of the others near to you Important It should be noted that the South Korean site completely resets each day and does not keep files from day to day between reset events Therefore you cannot store your work there in a long term way This is much like the 1980s classic movie Groundhog Day Be sure to save your work see Page 5 The following URL explains the resetting how to circumvent it and why it exists http sagenb kaist ac kr about html 19 Step Two To obtain a SAGE account for yourself point your web browser to the address you chose in Step One When the web pag
35. Using SAGE as a numerical solver Let s suppose you need to know the value of a root of oat lt a be ae ge whose graph is 30 Furthermore you are interested in a root between 1 and 1 This could come about because you graph it with or without SAGE and see that there is a root in that region or it could come about because you see that f 1 4 but f 1 2 and so clearly f crosses O at some point between 1 and 1 though we don t know where yet Perhaps the problem in the textbook simply tells you that the root is between 1 and 1 Okay to solve the problem you need only type this find_root x75 x74 x73 x72 x 1 1 1 and SAGE tells you that 0 71043425578721398 is the root This is a numerical approximation because quintic equations have a very special property if you don t know what the special property is just ask any math teacher While it is an approximation the computer does several billion instructions per second and so it is an approximation that is trustworthy to around an accuracy of 10718 which is one tenth of a quadrillionth So it is a good approximation Meanwhile it turns out there s no root between 1 and 0 Perhaps you are told that or perhaps you graph it with or without SAGE If you type find_root x75 x74 x 3 x72 x 1 1 0 then SAGE tells you RuntimeError f appears to have no zero on the interval which is perfectly honest because there s no root in the interval for S
36. a Vehicle Fleet Coming soon 2 11 The Biology Example Predator Prey Models Coming soon 54 Chapter 3 Advanced Features of SAGE 3 1 Using SAGE to work with integers gcd lem factorization sigma tau etc The study of problems that deal with the integers in pure mathematics is called number theory and in certain types of applied mathematics called integer programming This subject can be started at a very young age e g what is a prime number but can be very advanced too Many Ph D dissertations are written every year with new results in number theory Mostly number theory is about writing proofs and so you d imagine that SAGE is not very useful for that Actually SAGE is very good at working with the integers to help you work out specific examples on what the instruments of number theory do to specific integers This can help you understand new concepts more intimately and I find it can form a bridge between the lecture on a topic and the time when you are really ready to write serious proofs about a topic For example you can type factor 2007 to find the factorization of that integer which is 1 372 223 and likewise factor 2008 is 273 251 To find the next prime number after one million you can do next_prime 1076 and discover the answer is 1 000 003 You can also test a number for primality Consider is_prime 101 or alternatively is_prime 102 Of course SAGE says True fo
37. agemath org quickref Feature Tour If you would like to show friends parents or non mathematicians what SAGE is about there are two single page feature tours One is math heavy http www sagemath org tour quickstart html and the other is more about the beautiful graphics that SAGE can create http www sagemath org tour graphics html 9 Appendix D Installing SAGE on your Personal Computer While installing SAGE on your machine takes a lot of time and effort the result will be that it runs much faster Also you can then use SAGE even when the machines dedicated to web users are busy or when your internet connection is down or non existant such as in the subway or on an airplane Sadly the installation process is quite challenging First go to the main SAGE website http www sagemath org then click Documentation and then Installation Guide Follow the instructions from then onward 86 Acknowledgements First I would like to thank Martin Albrecht who first introduced me to the SAGE community and who lead to the inclusion of my dissertation code into the SAGE libraries in what has become the M4RI project Second I would like to thank William Stein the creator of SAGE for his encouragement and access to supercomputing resources which have made my cryptanalytic research possible Third I would like to thank Andrew Novocin for teaching me how to include png files in LaTeX documents which saved me the t
38. as to make a list This is our first example but we ll see another one on Page 32 in solving equations in Number Theory on Page 57 and in scatter plotting data on Page 62 However it is important not to get carried away Rarely does it make sense for four functions to appear together in one graph For example plot sin x cos x sin 2 x cos 2 x 0 10 T Jl l To do multiple functions in multiple colors the command is actually to add the plots plot sin x 0 10 color purple plot cos x 0 10 color blue which produces something quite readable Or perhaps plot sin x 0 10 color purple plot cos x 0 10 color blue plot sin 2 x O 10 color green plot cos 2 x 0 10 color red which is not readable but certainly is very pretty as you can see 68 A a Labels and Legends Sometimes when you have several functions in the same graph it is nice to label them with a legend An example plotting f x x x and the tangent line at x 1 is done via plot x 3 x 3 3 color blue legend_label f plot 2 x 2 3 3 color green legend_label tangent which produces You can also make the tangent line really short if you like plot x 3 x 3 3 color blue legend_label f plot 2 x 2 0 25 1 75 color green legend_label tangent 69 10 10 70 That last bit of code requires an explanation The
39. at the point x b 2a e For the revenue function we see that maximum revenue occurs at 180 cones a cost of 2 25 per cone and a revenue of 405 The costs would be 190 for a profit of 215 e For the profit function we see that maximum profit occurs at 160 cones a cost of 2 50 per cone and a revenue of 400 The costs would be 180 for a profit of 220 e The point of minimum costs would be 0 cones of ice cream for a cost of 100 However the revenue is O and the profit is negative a loss of 100 Definitely this is not a good business plan Even if the costs are lower in this bullet than in the previous two bullets e Did you notice how the point of maximum profit had lower revenue 400 vs 405 but higher profit 220 vs 215 than the point of maximum revenue Did you notice how the point of maximum revenue had lower profit 215 vs 220 but higher revenue 405 vs 400 than the point of maximum profit 29 66 In conclusion we see that maximizing revenue minimizing costs and maximizing profit are three 9 9 9 9 different objectives 2 5 The Engineering Example Two Stage Rockets Coming soon 2 6 The Ecology Example Lake Dumping Coming soon 2 7 The Astronomy Example Point of Closest Approach Coming soon 59 2 8 The Earth Science Example Sunrise and Sunset Coming soon 2 9 The Medical Example Blood in an Artery Coming soon 2 10 The Business Example
40. ate i r m r 12 One student once described this to me by saying that the n and i are math friendly whereas the t and r are human friendly This can be fixed by doing CostPerThousandi r t CostPerThousand i r 12 n 12 t and now we have made a new formula which uses the rate in terms of r as we liked and the number of years t not the number of compounding periods And now for a 30 year mortgage at 6 5 compounded monthly we can do CostPerThousand1 0 065 30 and get a more user friendly input and output The Grouping Symbols Just a reminder while a mathematician might write PV PMT1 1 3 0 for the present value formula using the and as a sort of auxiliary and in SAGE the only grouping operators are parentheses So that formula is PV PMT 1 1 1 7 n i with additional explanation found on Page 5 49 2 3 The Physics Example Gravitation and Satellites Let s say you re analyzing a satellite in its orbit around the earth Newton s formula for gravity is _ GMM F 2 S where G 6 77 x 107 Nm kg is the universal constant for gravitation Me 5 9742 x 10 kg is the mass of the earth next r is the distance from the satellite to the center of the earth in meters and finally M kg is the mass of the satellite Surely G and M will not change so we can set those with G 6 77 107 11 Me 5 9742 107 24 just like we set c 355 113 on Page 23 Unlike the mass of the earth or
41. ays search this guide for whatever command you would like to use I recommend that you never try to read more than 3 entire sections in one day Otherwise there is too much for your brain to absorb while still keeping the experience fun and new Chapter 1 contains the basics what you really need to know Chapter 2 has various topics of advanced mathematics and its sections are meant to be read independently of each other as needed Chapter 3 will cover writing your own programs in SAGE and Python but it isn t written yet The appendices contain lots of useful information particularly Appendix C convenience features of SAGE and Appendix D which lists other resources for SAGE Last but not least the Table of Contents can be found at the back of the guide and serves as an easy substitute for an index I assume you do not already have an account on the SAGE notebook server If you have one then log in now but if not then see Appendix A which will tell you how to get an account In Appendix E are the directions for installing it on your personal computer but this a very long and difficult process and it is not recommended to beginners 1 1 Using SAGE as a calculator First off you can always use SAGE as a simple calculator For example if you type 2 3 and click evaluate then you will learn the answer is 5 The expressions can become as complicated as you like To calculate the amount on a simple interest loan at 6 per year fo
42. benefits While this is great the situation is actually quite a bit better than that All of SAGE is open source software This means that not only is it free but you can look inside of it and see how it does everything that it does There are no trade secrets and you get to know precisely and exactly what it is doing The open source distinction is actually even better than that There are licenses that govern the use of free software and SAGE uses a particular GNU Public License so that you can take pieces of it and use them in your own non commericial software for free Think how much time that can save you Now for the pragmatics Let us say that you were eager to learn how SAGE calculates determinants If this were the case then you use the command and it will display the source code written in Python Just type A matrix 2 2 1 2 3 4 A determinant and you ll get all the source code as well as a ton of extra information Note that sometimes you have to look deeper and run the command on more and more pieces to get a complete picture of the entire puzzle For example typing 83 A det yields the rather unsatisfactory information that A det is just an abbreviation for A determinant 34 Appendix C Other Resources for SAGE There are quite a few resources for SAGE Here are some major ones Book The book SAGE for Newbies by Ted Kosan would be a great thing for you to read after reading this pamp
43. both calculate 1 Late Z Ls _ 1 Sea a 4 2 4 2 A 2 4 1 1 1 r r dz qa 3 0 A 2 but if you wanted to ask SAGE to calculate this numerically then you should type numerical_integral x73 x O 1 which would output 0 24999999999999997 2 775557561562891e 15 where the first number is the best guess SAGE has for the answer while the second number is the uncertainty In this case the uncertainty is 2 77 quadrillionths which is very impressive Of course SAGE can do this integral exactly as well using the commands we learned moments ago What is interesting about numerical integration are the cases when you cannot do the indefinite integral because like the Fresnel above it cannot be written but you can find good numerical estimates Consider again 3 ene sin x dx 1 but now with bounds so that it can be done numerically We would type numerical_integral exp x73 sin x72 1 3 and that returns back 0 077997011262087815 8 6840496851995037e 16 with as before the first number being the answer and the second the accuracy In this case the uncertainty is 868 quintillionths which is very impressive Another way to say it is that the uncertainty is 1 1152 trillionths 44 Partial Fractions One thing that is very important in problems that arise from Differential Equations and even Calculus II is the question of integration by partial fractions If you want to know the partial fraction
44. by plugging them into f x and then finally drawing the dots in the appropriate spots on the graph So if the graph has a vertical asymptote then near that asymptote the value will be huge Because of this when you graph a rational function be sure to restrict the y values Compare Plot 3 Plot 4 14 Plot 1 plot 1 x73 x 2 2 Plot 2 plot 1 x 3 x 2 2 ymin 5 ymax 5 Plot 3 plot 1 x73 x x 2 2 detect_poles true ymin 5 ymax 5 Plot 4 plot 1 x73 x x 2 2 detect_poles show ymin 5 ymax 5 As you can see the first is a disaster The second one cuts off the very high and very low y values but it keeps trying to connect the various limbs of the graph When you set detect poles to true then it will figure out that the pieces are not connected One minor point Did you notice in the four plots above that show is in quotes but true is not in quotes That s because true and false occur so often in math that they are built in keywords in SAGE However the show occurs less often and therefore is not built in and we must put it in quotes just as we do with colors while plotting Multiple Graphs at Once Now we re going to see how to superimpose plots on each other using a plus sign Suppose you were investigating polynomials and you wanted to know the effect of varying the last numeral in y x 1 1 2 a 5 on its graph Then you could c
45. certain bounds such as to calculate 3 e dt 2 then we would type integral t720 e t t 2 3 and get instead 329257482363600896 e 2 12112 059051462881 e 3 which is getting toward an answer On the other hand if you really want a numerical answer then you should use our old friend n and type 43 n integral t720 e t t 2 3 which would return the number 8 79797452800000e9 and that means 8 79797452800000 x 10 This highlights the difference and similarities of the various types of answer A Theoretical Side Note That number which we just calculated looks like an integer because it ends with 528 when you carry out the 10 part but when we look at the exact answer we can see that no it is not an integer because e and e are involved and e is an irrational number It is quite possible that you haven t yet learned what irrational numbers are but if not then ask any math teacher They will be happy to discuss such an advanced topic with you Whenever dealing with computations it is critical to remember what is an approximation and what is exact Irrational number can be expressed exactly on occasion like we just did here using e and e but that is the exception and not the rule In any case this paragraph has nothing to do with SAGE and so we return to the topic at hand Numerical Integrals Speaking of approximations you can also jump to the numerical integral very rapidly I m sure we can
46. cians the integral Fee de ett e x erf y o vT which is just a slight variant of the one that I told you earlier cannot be found using the elementary functions of mathematics is so important that they gave it a name and that name is erf While being friendly and pronounceable erf stands for the Error Function Thus if you type into SAGE integral 2 sqrt pi exp x72 x 00 2 it will respond sqrt pi erf 2 sqrt pi sqrt pi 46 which is correct because 2 0 2 Bee dx eee dx f E dx 1 erf 2 VE 7 erf 2 va yrerf 2 VT oo VT 0 yT YT yvT Perhaps you might find that 1 erf 2 is a more compact and simpler answer I would be inclined to agree I m not quite sure why SAGE does not simply its final answer further In any case the point is that there are some functions out there which do not have integrals writable using elementary functions but which SAGE calculates anyway using the erf function Not all calculus textbooks are anticipating that and there can be some unexpected surprises when you compute an answer to a textbook problem and look in the back of the book only to find that your textbook tells you that the problem is unsolvable It really is a matter of whether or not you accept erf as a legitimate or illegitimate member of the family of functions 1 14 Matrices in SAGE Part Two Coming Soon This will include using SAGE to multiply and invert matrice
47. ction This is because it can be defined by the greatest integer that is less than or equal to x Some older mathematics books will write x instead of x I suppose if you wanted you could define the ceiling function by the least integer that is greater than or equal to x 3 2 2 Combinations and Permutations In a lot of textbooks particularly for Finite Mathematics or Quantitative Literacy as well as a basic Probability ES Statistics a lot of time is spend talking about combinations and permutations For example if I have 52 distinct cards in a deck of cards I might ask how many 5 card hands I can build with them If order doesn t matter such as in Poker then I would have 52 52 52 x 51 x 50 x 49 x 48 5205 ani eg lot 5 547 5x4x3x2x1 possibilities and SAGE calls this binomial 52 5 This is an example of the combinations formula However if order does matter which is not the case for poker then I would have 52 52 52P 5 8 Fey 52 x 51 x 50 x 49 x 48 even more possibilities In SAGE you can type factorial 52 factorial 52 5 60 for that or you can type binomial 52 5 factorial 5 both of which return the same answer This is an example of the permutations formula You might be wondering why SAGE does not have a command for permutations but that it does have a command for combinations Well there is a command for permutations but it does something a bit m
48. ctor _ I am impressed at how rapidly SAGE can factor that 16 474 digit number And observe that every single prime number less than 5040 is present in the factorization though raised to various powers Fractions can also be huge For example if you type 7200000 21 20 2011 1867 you get back 22585044527 136650669987 277 16366996290277307841436194047 14135419872297991 159890730122325475848713199097965808127266523705668417267736363165100450 844089107959562425378283156323856617090505699529 27875931498163278926919 647840810451882475520000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000 Notice that there s a forward slash roughly 2 3rds of the way through the middle line So that five line number is really a 2 67 line integer divided by a 2 33 line integer In any case typing n _ gives me the decimal approximation 8 10198738242156e9 and that s about 8 1 billion something that I can wrap my head around even if but partially In case you were wondering what that came from look at the dates and you ll quickly realize exactly what it is In the year 1867 the USA purchased Alaska from Russia for 7 2 million dollars If that sum had instead been invested at 5 compounded annually for the same period of time it would be worth 8 1 billion dollars today 3 11 Advanced Linear Algebra Coming Soon It is hoped that this will cover the follow
49. cture The situation is much improved by typing plot3d sin x y y cos x x 10 10 y 10 10 plot_points 90 which forces SAGE to use 90 points on the x axis and 90 points on the y axis for a total of 8100 points To exaggerate the effect for the purposes of explanation consider the following images 21 10 0 10 0 10 0 10 0 with plot_points 90 with plot_points 30 that s 8100 points that s 900 points Perhaps you might be wondering why we do not always use a large number of points Well even with 90 you might find that your internet connection slows down and your computer starts acting slowly unless you have a large amount of memory These 3D plots require an enormous amount of computation power especially if you rotate them Implicit Functions Sometimes in calculus texts a function is defined implicitly instead of explicitly What I mean is that instead of z f x y one could have some polynomial for instance in x y and z equal to zero An example might be Ag oe ba eg doz ay be 1 0 This should not be too surprising After all while most functions in the univariate calculus are defined as y f x on the other hand we write a circle like x y 4 or alternatively 2 y 4 0 as a polynomial in x and y set equal to zero In any case here s what you would do in SAGE First we define the function that we wish to draw f x y z 4 x72 x 2 y 2 z 2 z y 2 y 2 z 2 1
50. e 38 1 3 Using SAGE for trigonometry For trigonometry SAGE works in radians So if you want to know the sine of 7 3 you should type sin pi 3 and you will get the answer 1 2 sqrt 3 This is an exact answer rather than a mere decimal approximation You will find that SAGE is very oriented toward exact rather than approximate answers Sometimes this is irritating because if you ask for the cosine of 7 12 then you would type cos pi 12 and obtain 1 12 sqrt 3 3 sqrt 6 which is especially irritating if you want a decimal Instead if you type N cos pi 12 then you will obtain 0 965925826289 a rather good decimal approximation You will discover that SAGE is fairly savvy when it comes to knowing when functions will go wrong In particular just try evaluating tangent at one of its asymptotes For example tan pi 2 will produce the helpful answer Infinity The rare or reciprocal trigonometric functions of cotangent secant and cosecant which are important in calculus but annoying on hand held calculators are built into SAGE They are identified as cot sec and csc The inverse trigonometric functions are also available They are used just like the trigonometric functions For example if you type arcsin 1 2 you will obtain 1 6 pi as expected Likewise arccos 1 2 produces 1 3 pi The usual abbreviations are all known by and used by SAGE Here is a complete list Math Notation Long form SAGE c
51. e I was told that the goal of the first microeconomics course should be twofold First to teach the concept of supply and demand Second to teach that the three actions of maximizing profit versus maximizing revenue or minimizing costs are three extremely different things This example will look at that second objective Here is an extended study of the micro economics of the sale of ice cream We imagine a vendor with an ice cream cart at a busy intersection in Central Park An ice cream salesman observes that when he charges 3 per ice cream he only sells 120 cones per day On the other hand when he charges 2 per ice cream he sells 200 cones per day If he makes his price to high he will sell too few and won t make much profit If he makes his price too low he will sell more but he still won t make much profit The costs of doing business are basically just 50 cents per cone because he has long since paid off his start up expenses plus a 100 fee per day for the rental of the ice cream cart The ice cream salesman is going to assume that the demand varies linearly with price This often is not quite true but it is often a reasonable approximation There are several different ways to realize that the demand n and the price p will have the relationship n 360 80p or equivalently p 4 50 n 80 and you can check that by plugging in the given data points The easy way to find that equa
52. e expand e combine and collect e gcd e lcm e adding and subtracting polynomials e multiplication of polynomials e how to do polynomial long division e composition of polynomials 1 8 Using SAGE to graph a 3 dimensional function or an implicit function Three dimensional images are not only visually stimulating but they can help demonstrate a lot of important effects in the multivariate calculus Instead of f x we will have functions of the form f x y Before we had only one variable x and now we re adding in the variable y This means we need the variable y so we first type 24 var y and now y is declared The declaration of variables will be explained on Page 31 in case you are curious but for now just type it In general you need to declare every variable that you use except that x is so commonly used so it is built in If we wanted to view the bottom part of the paraboloid z y with x and y both ranging from 2 to 2 we would type var y plot3d x2 y 2 2 2 Y 2 2 2 to obtain 2 0 22 0 Try that and try dragging the plot around with the mouse You can view the plot from all sorts of angles and directions Now we can view things from 4 to 4 instead with plot3d x 2 y72 x 4 4 y 4 4 to obtain 4 0 74 0 For a hyperbola of one sheet try plot3d 2 372 x 2 2 y 2 2 color green 20 where you can see that we changed the color
53. e loads on the right you ll see a login area with a place for a username a password and a remember me box Below that is the sign in button and below that is Sign Up for a new Sage Notebook account Just click Sign Up for a new Sage Notebook account now Step Three You will now be prompted to enter a username I suggest that you use your email address This guarantees that your username is not taken and you will be less likely to forget it Then you will have to choose a good password and enter it twice When you have entered all of that click on the button create account Step Four You re now back to the screen that you were at in Step One Now enter your username and password in the appropriate spots and click Sign In You re Done Once you are inside click create a new worksheet You will be prompted to enter the filename for your very first worksheet You can call it what you like for example physics homework or taylor series You now have a SAGE account for life and you need never again repeat this process of acquiring an account Unless you want to use a different server 80 Appendix B Convenience Features of SAGE Earlier in this document we learned about some basic convenience features of SAGE such as the Undo button the Print button the Save button and pressing Shift Enter instead of clicking Evaluate Now here are some other major timesavers
54. ee being while plotting several functions at once on Page 68 while solving equations on Page 32 and in Number Theory on Page 57 Now we can easily make a scatter plot by just saying scatter_plot datapoints 62 which will produce for us the scatter plot shown below on the left Without Best fit Line With Best fit Line Now perhaps you have the custom of computing best fit lines in a spreadsheet tool or in a statistical package If so that s great If not then SAGE will be very able to do this for you We ll explain how to do that in the next section Meanwhile the actual best fit line for this example is y 7 2x and the scatter plot on the right includes this line As you can see it is a rather good fit but an imperfect fit That plot with the line included was created by typing scatter_plot datapoints plot 7 2 x 0 4 Now let s consider another data set This example will explain to you why we are bothering to make a plot at all when there are so many tools out there including SAGE to give us the line of best fit whenever we want one First enter the data otherdata 3 10 2 5 Ls 2 O 1 1 2 2 5 3 10 and note that cut amp paste usually works quite well if you don t want to retype the above list Imagine that this is some data from a scientific experiment and that you lab partner has copied the data and has found the line of best fit y Ox 5 using some sort of
55. em The best bet for any help that you may require which this document might not contain is the Google Group called SAGE Support and that can be found at the address http groups google com group sage support You are also free to check the SAGE user s manual However that manual is written for faculty not for students and so it assumes a bachelor s degree or realistically much higher level of mathematics The manual can be found by going to http www sagemath org doc reference Another webpage that collects useful guides about SAGE can be found at 81 http www sagemath org help html There is also a built in help command If you want some help for the command plot or diff for example then you would type plot or diff and you will get a document about that command in response Fetching this document is a bit slow on the order of 3 5 seconds at times The various help articles have been written by very different people and so they assume various levels of ability Some are perfectly clear and straight forward Others are advanced Of course the flaw with the question mark system is that you need to know the full name of the command before you can pull up the help document Realistically you only need to know the start of the command by using tab completion We ll tell you what that is momentarily B 3 1 Forgetting Long Commands If you ve forgotten a command especially a long one it can be quite frustra
56. er 5 The last one is checked similarly Note that it is crucial to check all three equations Otherwise you d fail to detect the imposter solution x 30 y 29 2S which satisfies the first two equations but not the third one This is how I remember Columns such as on a bank Congress or a Greek temple are vertical Rows by process of elimination are horizontal But do we enter the rows first or the columns first I always think RC Cola a minor brand of soda which reminds me rows first columns second Thankfully there does not exist a CR Cola 19 Now let s consider the complicated example from the previous subsection It was k 2 1 5 6 0 0 1 1 3 Re 1 3 3 1 0 mil Joe Ol and we can enter that into SAGE with B matrix 4 5 1 2 1 5 6 0 O 1 1 5 1 3 3 1 0 1 5 1 O 1 By the way be certain to enter that above all on its own line You cannot have any linebreaks Then we check with B alone in its own box and check to see that we have entered that which we had intended to enter That being done we type B rref into a box and we then learn 1 0 0 0 107 7 0 1 0 0 12 7 0 0 1 0 54 7 0 0 0 1 19 7 Giving the final answer w 107 7 x 12 7 y 54 7 z 19 7 Now we must check our work For example we check the first equation with 2 12 7 5 19 7 54 7 and 6 107 7 both of which come out to 65 7 The point is not
57. finite integral and an indefinite integral The first two types result in numbers but which will be found approximately or exactly respectively The third one is the antiderivative If this is confusing then let us consider a simple example Background What integral might you want to consider in order to find the area between the x axis and f x x 1 on the interval 3 lt x lt 6 Well that integral would be 1 1de Zata 3 3 and that is a number Here we ve done it analytically using calculus This is an exact definite integral It is definite because it ends with a number It is exact because we did no approximations On the other hand 6 1 3 l 3 6 6 433 3 72 6 9 3 866 s 3 3 1 rro C is an indefinite integral It produces a function and not a number Of course we calculate an indefinite integral on the way to calculating a definite one but they are two distinct categories If you want a function you should use the indefinite and if you want a number then you should use the definite Some integrals are famously impossible The two most famous are the Fresnel E i sin dt 0 2 which is important in optics and the Gaussian dez Ya which is pivotally important in probability There is no function built up of addition subtraction multipli cation division roots exponents logarithms trigonometric functions and inverse trigonometric functions as well as their hy
58. functions are also built in For example type sqrt 144 then click evaluate and you ll learn that the answer is 12 From now on I m not going to say click evaluate because repeating that might get tiresome Pll assume that you know you have to do that Since SAGE likes exact answers try sqrt 8 and get 2 sqrt 2 as your answer If you really need a decimal try instead NC sqrt 8 and obtain 2 82842712475 which is a decimal approximation The NO function which can also be written n will convert what is inside the parentheses to a real number if possible Usually that is a decimal expansion and so unless what is inside the parentheses is an integer then it will be necessarily an approximation SAGE will assume that all decimals are mere approximations so for example sqrt 3 4 will evaluate to 1 84390889145858 without need of using N Higher Order Roots Higher order roots can be calculated like exponents are For example to find the sixth root of 64 do 64 1 6 and obtain that the answer is 2 Getting Both Square Roots In the spirit of absolute pomposity you may know that 2 4 as well as 22 4 So if you want all the square roots of a number you can do sqrt 4 all True to obtain 2 2 where the square brackets indicate a list We ll see other examples of lists on Page 68 Page 32 Page 57 and Page 62 A Taste of the Complex Numbers If you know about complex numbers you can do
59. guages Now consider this problem suggested by Dr Grout To solve p q 9 qy pr 6 qy px 24 De we would type var p q y eql ptq eq2 q y p x 6 eq3 qty 2 pxx 2 24 solve eq1 eq2 eq3 eq4 P q x y which produces lp 1 q 8 x 4 3 sqrt 10 2 3 y 1 6 sqrt 2 sqrt 5 2 31 lp 1 q 8 x 4 3 sqrt 10 2 3 y 1 6 sqrt 2 sqrt 5 2 3 As you can see that is a list of lists again using square brackets and commas Clearly that is a very hard to read mess Using the technique that we learned when analyzing the degree six polynomial above we replace the last line with answer solve eqi eq2 eq3 eq4 l p q x y Now we can type print answer 0 print answer 1 and we get p 1 q 8 x 4 3x sqrt 10 2 3 y 1 6 sqrt 2 sqrt 5 2 3 p 1 q 8 x 4 3 sqrt 10 2 3 y 1 6 sqrt 2 sqrt 5 2 3 This is SAGE s way of telling you Solution 1 p 1 q 8 x v10 3 y v10 6 Solution 2 p 1 q 8 x v10 2 ee Since there are only two answers we cannot ask for a third In fact if we type print answer 2 then we get Traceback click to the left of this block for traceback IndexError list index out of range which is SAGE s way of telling you that it has already given you all the answers for that problem the list does not contain a 3rd element f Now let s try to intersect the hy
60. han automatic The reason I say that is because the human must be very careful to first convert the word problem into a system of linear 17 equations We will not cover that here The second step is to convert that system into a matrix and that isn t quite trivial In fact it is often the case that student errors occur in this step and so we will invest a bit of time with a detailed example Consider the system of equations 2I 02 Yy 6 w otz y O wt3 a y z l 2r y w 32 As it turns out this system of equations has the following traps In the first equation the variables are out of order which is extremely common Very often both students and professional mathematicians will fail to notice this and put the wrong coefficients in the wrong places You can use any ordering that you want so long as you use the same ordering for every equation However to avoid making an error mathematicians usually put the variables in alphabetical order Also in the first equation the w is on the wrong side and we must move it to the left of the equal sign It is necessary that all the variables end up on one side of the equal sign and all the numbers on the other side of the equal sign With these two repairs the first equation is now w 2x y 9z 6 The second equation has the constant on the wrong side and so we must move it across the equal sign remembering to negate it As if that were not enough both x and
61. he product of all the numbers in the list divided by their ecd raised to the power of one less than the length of the list equals the lem Since we have four numbers in our list we would divide by the cube of the gcd This is done by 120 55 25 35 573 More about Prime Numbers One neat trick is that if you wanted to find the 54 321th prime number you could type nth_prime 54321 and discover that it is 670177 You can double check this with factor 670177 which returns itself confirming that the number is prime Phi Sometimes it is useful to know how many integers are coprime to x from the range 1 to x For example how many numbers from 1 to 100 share a factor with 10 and how many are coprime with 10 Well the prime factors of 100 are 2 and 5 and so any number composed only of twos and fives will work This would be 2 4 5 6 8 10 and so the other numbers 1 3 7 9 are coprime with 10 In particular if you know what modular arithmetic is then these are the numbers that have multiplicative inverses or reciprocals mod 10 Since 4 numbers are coprime to 10 we will write 10 4 Even if you don t know what modular arithmetic is consider 7 Each of 1 2 3 4 5 6 is coprime to 7 because the only prime that divides 7 is 7 and 7 does not divide any of those So 6 out of the 6 numbers are coprime to 7 and we would then write 7 6 It will always be the case for a prime number that p p 1 On the
62. hlet It is fairly large at 150 pages but will describe things that we don t have room to describe here You can find the book at http sage math washington edu home tkosan newbies_book sage_for_newbies_v1 23 pdf Manual The official SAGE Reference Manual can be found at http www sagemath org doc reference but this is not for beginners Be warned it is literally thousands of pages This is like an encyclopedia you don t read it cover to cover instead you look up what you need Tutorial The following web tutorial by Mike O Sullivan Ryan Rosenbaum and David Monarres has an emphasis on writing larger programming projects in SAGE which we will only touch on during this document http www rohan sdsu edu mosulliv sagetutorial index html Tutorial The following web tutorial by Paul Lutus is also about SAGE and does a good job of explaining the open source culture of SAGE development http arachnoid com sage index html Tutorial If you re doing a PhD or research in numerical computing the following tutorial is for you http www sagemath org doc numerical_sage Web Tour Aimed at mathematics faculty the following web tour can be quite informative about some of SAGE s advanced features as well as its basic ones too http www sagemath org doc a_tour_of_sage Quick Reference Cards Several SAGE users have made quick reference cards to help them learn the commands and have them handy while using SAGE http wiki s
63. hould not just assume that a linear regression is what you want Dividing the world of functions into linear and non linear functions is like dividing the set of living things on the earth into armadillos and non armadillos 3 4 Making Your Own Regressions in SAGE This lesson assumes that you ve read the previous section on scatter plots In particular we ll be using the cobalt data from the previous lesson For your convenience we repeat that data now cobalt_data 0 2000 0 75 1414 1 5 1000 2 25 707 3 00 500 3 75 353 4 50 250 5 25 177 6 00 125 First we re going to try to fit a linear regression to that data This means that we re going to ask SAGE to give us a function y ax b that models as best as possible these 9 points The commands for that are var a b model x a x b find_fit cobalt_data model where the var command is one we ve seen before see Page 31 but the next two commands are new to us The reply we get is la 290 33333269040816 b 1596 1111098318909 which is SAGE s way of telling us that the linear regression is y 290 333x 1596 11 Now we can plot that with scatter_plot cobalt_data plot 290 333 x 1596 11 0 6 Okay that s not a very good fit at all The image can be found in the previous section so we aren t going to waste space with it here Of course basic chemistry tells us that radio active decay follows an exponential cur
64. housand for several loans at interest rate r and compounded monthly If the length is for t years then the number of compoundings is n mt 12t The interest rate per compounding period would be i r m r 12 and value would be V 1000 So we would start with the basic formula E Vi o 1 14 i and plug in all that data Since V 1000 is not going to change we can assign that one with PMT V 1000 to make it a long term constant Note SAGE accepts your constant quietly it does not give a response Then we could define our own function as described earlier CostPerThousand i n Vx i Y 1 1 i1 7 n and so for a 30 year mortgage at 6 5 compounded monthly we can do CostPerThousand 0 065 12 30 12 and then if the rate rises to 7 we can do CostPerThousand 0 07 12 30 12 and so on and so forth Notice that the function name has to be only one word but by varying the capitalization that way which programmers call Camel Case we could express an idea that was more than one word Surely it is much easier to read CostPerThousand than costperthousand or COSTPERTHOUSAND While this above function is nice for example it will save us from agonizing each time over the placement of the parentheses it also isn t quite perfect Our data will be given in terms of the number of years of the loan t not the number of payments n mt 12t Likewise the rate will be the nominal published rate r not the per period r
65. idea is that we told SAGE to draw the curve x x using the domain z 3 to x 3 while the line 2x 2 is to be drawn from x 0 25 to x 1 75 Since we have given the line a smaller domain it appears shorter in the graph Grids and Graphing Calculator Style Graphs Sometimes it is nice to show the grid lines as if the graph were drawn on a piece of graph paper This is done with plot x73 x 3 3 gridlines true which produces The previous graph can be shown in a window that looks remarkably similar to the screen of a graphing calculator plot x73 x 3 3 ymin 6 ymax 6 show frame True resulting in TO Labeling the Axes of Graphs Sometimes it is very useful to label the axes of a graph directly on the graph This is particularly handy when there is a risk of misunderstanding the units involved or in a scientific application where many functions are discussed In any case this code provided by Martin Albrecht and Keith Wojciechowski accomplishes that objective p plot sin pitx pi pi color red legend_label f x sin pi x p axes_labels x y f x p show Graphs of Hyperactive Functions The function sin 1 x wiggles like crazy near x 0 but SAGE knows this and plots a lot of extra points There are many other hyperactive functions in mathematics Here are a few examples and their SAGE images As you can see SAGE is imperfect but accomplishes the objec
66. ime of manually converting the file formats of 50 some odd screen captures in this guide I would also like thank Robert Beezer and Robert Miller for giving me encouragement in using SAGE in my teaching and research respectively I have received a great deal of encouragement in this project including detailed suggestions from Jason Grout Benjamin Jones and Susan Schmoyer The funding for SAGE comes in large part from the National Science Foundation and in particular the Division of Mathematical Sciences The SAGE community is grateful for the support of the National Science Foundation Grants DMS 0821725 DMS 0713225 and DMS 1015114 as well as funding from Microsoft and Google 87
67. ing commands kernel A A eigenvalues A eigenvectors_left A eigenvectors_right A echelon_form rows columns charpoly rank nullity transpose factor A charpoly A hessenberg_form A gram_schmidt A hermite_form Dope 76 3 12 Taylor Series or MacLaurin Polynomials Coming soon It is hoped this will cover taylor sin x x 0 20 taylor cos x x pi 2 20 and perhaps even var m v taylor 0 5 m v72 sqrt 1 v c 72 v 0 7 3 13 Super Advanced Plotting Coming soon It is hoped that this will cover e parametric_plot e polar plot e parametric_plot3d 3 14 Minimizations and Lagrange Multipliers Coming Soon Here s a sample example anyhow var x y z f x y z 120 y x72 1 72 2 x 72 110 z y72 72 150 x 3 y 72 minimize f 1 3 4 3 15 Infinite Series Sums and Products Coming soon In the meantime you can type sum for the online help with sums and that is an outstanding resource 3 16 Ultra High Precision Arithmetic Really SAGE is about exact computations which means keeping fractions as genuine ratios of integers and not their decimal approximations thus rounding error is avoided When floating point numbers are needed e g log q where both p and q are distinct primes then you get 53 bits of precision or 15 decimal places That s really enough for almost anything However if you need more precision SAGE will deliver Using MPF
68. ing data shows how a 2 kilogram sample of Cobalt 62 might decay Here the x coordinate is time in minutes and the y coordinate is weight in grams cobalt_data 0 2000 0 75 1414 1 5 1000 2 25 707 3 00 500 3 75 353 4 50 250 5 25 177 6 00 125 Then we can type scatter_plot cobalt_data and get the nice scatter plot 20000 1500 1000 O 500 O 0 1 2 3 4 5 6 A weak student might be tempted to do a linear regression especially if the last few lab reports required linear regressions The linear regression of this data happens to be y 290 333x 1596 11 In that case one gets the plot shown below on the left However a stronger student should remember that radioactive decay follows an exponential curve Instead of asking for y az b which is a line the strong student would ask for y ae which is an exponential This is part of the beauty of SAGE s regression tools namely that a linear regression is just as easy hard as a quadratic regression an exponential regression a cubic regression or a logarithmic regression We ll see how to do those regressions in the next section Meanwhile the exponential regression happens to be y 1999 96e0 6216 7 and you get the curve shown below on the right 64 20000 1500 1000 O 500 O Linear Regression Exponential Regression The moral of the story is that while it is great to calculate regressions of data you s
69. ing the 1970s and 1980s What if you make a mistake Let s say that I really meant to say 13 000 and not 11 000 Click on the mistake and correct it using the keyboard and change the 11 into 13 Now click evaluate again and all is well There s also an Undo button and it does exactly what you think it does However if you ve done a lot of work you have to scroll the window all the way to the top of the document to find it as this button sits at the top of the sheet The Red Line You may have noticed a vertical red line next to the box after you had begun to change the formula but before you had clicked evaluate After you have clicked evaluate the red line vanished The purpose of this red line is to tell you that the information in the box had changed since the last time evaluate had been clicked and so you need to hit evaluate again to get an accurate answer l An irrelevant historical aside Because rewriting software is a painful and time consuming process many important pieces of scientific software remained written in FORTRAN until a few years ago and likewise the same is true of business software in COBOL Each new programming language incorporates features of the previous generation of languages to make it easier to learn and accordingly one sees some truly ancient seeds of dead languages once in a while The Green Rectangle Also you may have noticed if you have a slow internet connection
70. ith the goal of solving some linear system of equations Doing this in SAGE requires only four steps the last of which is optional First we re going to define the matrix 3 4 5 14 A 1 1 8 5 2 Al 1 fi which came from the previous subsection As you can see it has 3 rows and 4 columns We ll do that with the SAGE command A matrix 3 4 3 4 5 14 1 1 8 5 2 1 1 7 2 The key to understanding that SAGE command is to see that the first number is the number of rows followed by the number of columns Then follows a list of all the entries separated by commas and enclosed in brackets Notice that SAGE does not respond when this command is given The only reaction is the green rectangle which appears and disappears The second step is to verify that the matrix you typed was the matrix that you had hoped to enter This step is not avoidable as typos are extremely common To do this just type A in an empty box by itself Then the matrix displays and you can verify that you have typed what you had wanted to type The third step is that you want to compute the RREF or Reduced Row Echelon Form This is done with the command A rref and SAGE tells you the Reduced Row Echelon Form or OS ee a E E O Ww The fourth and optional step is to check your work but I highly recommend it First we type 3 3 4 0 5x1 and learn that the answer is 14 Next we type 1x3 1 0 8 1 and learn that the answ
71. ives and Gradients in Multivariate Calculus In the previous section we learned about derivatives of single variable functions Now we ll learn about multivariate derivatives Partial Derivatives You re probably not surprised to learn that SAGE can do partial derivatives very easily For example if you have g a y xy sin a e you can type g x y x y sin x72 e x diff g x y x and get 2 x cos x72 y e x which is 0g Ox and you can type derivative g x y y to learn that 0g 0y is just x Note here this really shows how diff and derivative are synonyms You can even find 8 Ta by typing derivative g x y x y Gradients If you want Vf then you type g derivative and SAGE will correctly calculate the gradient In fact SAGE displays x y gt 2 x xcos x 2 y e x x which is to remind you that this is a map from a point x y to a pair of real numbers and not an ordinary function Interestingly the gradient of a univariate function done this way is just the ordinary first derivative which is quite correct according to the laws of calculus An Extended Example Marginal Cost in a Factory Coming Soon 40 1 13 Using SAGE to calculate integrals both numerically and symbolically The only thing you need to understand in order to do integrals in SAGE is that there are really three types of integrals that can be done a numerical approximate integral an exact de
72. lities amp Linear Programming This section is almost complete but there is a bug in how SAGE plots inequalities There is some hope that this bug might be fixed very early in 2012 In which case I will restore this section Actually the bug is now fixed and much of this section is written but I haven t finished it yet 3 7 What about Octal Binary and Hexadecimal To find out the decimal values of hexadecimal 71 type Ox71 and get 113 Note that this is because 7 x 16 1 113 By the way that s a zero before the x not the letter o Alternatively for hexadecimal 0x1F you would type Ox1F and for octal 11 type 011 Binary is similar to hexadecimal except with a b in place of the x Thus to find out what binary 1101 is you would type 0b1101 and learn that the answer is 13 To go in reverse you can type hex 31 and you ll get 1F Likewise for binary you can type bin 63 and you ll get the binary expansion for 63 which is 11111 or five ones consecutively The octal in reverse is a bit messy Type oct int 31 to learn that the answer is 37 That means that 3Toctal 3 X8 7 24 7 31 octa 3 8 Can SAGE do Sudoku To solve the 9 x 9 Sudoku puzzle defined by a 9 x 9 matrix first enter the matrix A matrix ZZ 9 053 0 0 6 7 lt 3 0 0 0 0 0 0 O 0 0 0 4 9 0 O 2 bj v 2 2 v Then if you type A on a line by itself you get 5000800 4 9 000500030 06730000 1 150000000
73. m edu was far more efficient and useful than using the search tool on Fordham s own website This can be a great way to find the webpages of faculty Sometimes I will add site edu when I want to restrict a web search only to universities for example when I wanted to learn something about stock options what they are and how they worked I had to do this Otherwise I was flooded with hits about particular stock options that is to say options for particular stocks which was not what I was looking for 9 B 4 Inserting a Line Box Two insert a line or a box between two boxes just move the mouse to the white space just slightly above the lower of the two boxes Then you ll see a thick and wide dull blue line appear Clicking there will insert a new box 82 B 5 Deleting a Line Box To remove an unneeded line box just click inside it and backspace out all of the contents Once the box is empty an extra press of the backspace key will delete the box B 6 Combining or Splitting Boxes Coming Soon control semicolon and control backspace B 7 Manual Restart and Re Evaluate Coming Soon B 8 Sharing your Work Coming soon B 9 Renaming vs Cloning Coming soon B 10 Saving Loading your Work Locally Coming Soon B 11 Semicolons This is how you put two commands on one line Not interesting but coming soon B 12 Displaying Source Code As you know SAGE is free software and that s one of its great
74. metrical Interpretations Coming soon 1 5 7 Minor Notes Coming soon 1 6 Making your Own Functions in SAGE It is extremely easy to define your own functions in SAGE To me this is one of the most brilliant design features of SAGE that the creators made this very easy for the user You can make your own function using exactly the symbols that you would normally use in writing the function down with your pencil Defining your own Functions If you want to define f x x x then you type f x x 3 x and likewise if you wanted to define g x v1 x2 you would type g x sqrt 1 x72 which can be very handy Take a moment and see that f 0 and f 2 as well as g 0 do exactly what you expect them to do Plotting Functions Now that we ve defined f x and g x we can plot them with plot f 2 2 you get the expected plot 22 Likewise if you type plot g 0 1 which is equivalent to plot g x 0 1 then you get the graph of g x on 0 lt a2 lt 1 Last but not least if you type plot g which is rather abbreviated omitting the interval of the x axis that you want then SAGE will assume that you want 1 lt x lt 1 by default This produces the plot Using Intermediate Variables Sometimes you ll be using the same value a lot and you want to store it somewhere For example if you type 355 113 a lot then you can type c 335 113 which silently stores the value 335 11
75. nctions as well as their hyperbolic cousins which you might or might not have been taught about which will have as its derivative In other words f is integrable but the integral cannot be written in terms of common and not so common functions Thus since por sin 12 dx does not have a nice expression then when you type integral exp x 3 sin x 2 x the SAGE replies not with an answer but instead with integratel e x7 3 sin x 2 x which represents an admission of defeat Have no fear however because we can calculate numerical answers approximately with very high accuracy and we ll learn about that momentarily A nefarious integral famous among calculus teachers is edi where the 20 could be any decently large number This comes up in explaining what Euler s gamma function is but we aren t too concerned with that right now In any case we could just type integral t720 e t t but then we get back t720 20xt719 380 t718 6840 t717 116280 t 16 1860480 t 15 27907200 t 14 390700800 t 13 5079110400 t 12 60949324800 t7 11 670442572800 t 10 6704425728000 xt 9 60339831552000 t 8 482718652416000 t 7 3379030566912000 t 6 20274183401472000 t 5 101370917007360000 t 4 405483668029440000 t 3 1216451004088320000 xt 2 2432902008176640000 t 2432902008176640000 e t which is large enough that it isn t clear how to get an idea of what that means If instead we had
76. ns Scatter Plots in SAGE o be eRe A Se AR Making Your Own Regressions in SAGE 0000048 Additional Topics in Plotting and Graphing Graphing Inequalities amp Linear Programming What about Octal Binary and Hexadecimal 0 Catv SAGE do SUCK Lira A ep ie AAA ee Oe Sets Measuring the Speed of SAGE 1 aa a a aa Huge Numbers and SAGE 0 o Advanced Linear Algebra e Taylor Series or MacLaurin Polynomials Super Advanced Plotting 0 00 eee es Minimizations and Lagrange Multipliers Infinite Series Sums and Products 0 0 000080008 8 Ultra High Precision Arithmetic 4 244 6644054404008 82 was Using SAGE to work with Cramer s Rule 2 Using SAGE to work with Resultants 00 A Obtaining a SAGE Account B Convenience Features of SAGE B 1 Cop and astes reas dora E A as ario a B2 he Mo neol hi a edito ARA e Ge aa B 3 B 4 B 5 B 6 B 7 B 8 B 9 The Online Help System a cris Kaha wR OR A A B 3 1 Forgetting Lone Commands 20 2 a rr A ees B 3 2 When You Don t Know which Command B 3 3 A Superb Google Trick 20 00 0000 a Inserting a Line Box dda dr a Se ee Deleting a Line BOX marena ee ie ee A Se Se ds Bode a aed Combining or Splitting Boxes
77. of equations that you would use if someone asked you to find the parabola connecting the points 3 32 and 2 15 as well as 1 6 That would be f x 4a 3x 5 Naturally linear systems of equations can also be solved with matrices In fact SAGE is quite useful when working with matrices and can remove much of the tedium normally associated with matrix algebra That will be discussed in the matrix section on Page 17 Non Linear Systems of Equations While linear equations are very easy to solve via matrices the non linear case is usually much harder First we will warm up with just one equation but a highly non linear one Let us try to solve z 21x 1752 735z 16242 1764x 720 0 which can be done by solve x76 21 x75 175xx 4 735 x73 1624 x 2 1764 x 720 0 x which gives x 5 x 6 x 4 x 2 x 3 x 1 32 That is a list of six answers because that degree six polynomial has six roots It was easy to read this time but you can also access each answer one at a time Instead we type answer solve x76 21 x75 175 x74 735 x73 1624 x 2 1764 x 720 0 x and then we can type print answer 0 or print answer 1 to get the first or second entries To get the fifth entry of that list we d type print answer 4 This is because SAGE is built out of Python and Python numbers its lists from 0 and not from 1 There are reasons for this based on some very old computer lan
78. ommand Short form SAGE command A a Cl eee PN em ER S a sin g arcsin x asin x cos az arcsin x acos x tan x arcsin x atan x cot x arccot x acot x sect x arcsec x asec x esc g arccsc x acsc x You can also use SAGE to graph the trigonometric functions We ll do that in the section entitled Using SAGE to graph 2 dimensionally on Page 12 Converting to Degrees and Back Remember to covert radians to degrees just multiply by 180 7 and to multiply from degrees to radians just multiply by 7 180 Here s what you type to convert 7 3 to degrees pi 3 180 pi produces 60 while 60 pi 180 produces 1 3 pi The way I like to remember this is that one protractor is 180 degrees and protractor begins with p while 7 is the Greek p For example 90 degrees is half a protractor and 60 degrees is a quarter protractor That s why I can remember that they are 7 2 and 7 2 Likewise if I see pi 6 then I know that this is one sixth of a protractor or 30 degrees There is actually an elaborate package in SAGE for converting among different units and it can be quite useful I hope to add it to this document in the near future 1 4 Using SAGE to graph 2 dimensionally My favorite shape is the parabola so let s start there Type plot x72 and you get a lovely plot of a parabola going in the range la 1 which is the default range It will bound the y values to whatever is needed to sh
79. onsider looking at x 3 and x 4 as replacements for the x 5 as well as x 6 and a 7 This would be done by plot x 1 x 2 x 3 2 10 plot x 1 x 2 x 4 2 10 plot 1 2 Ob 2 10 where I chose the domain x 2 until x 10 arbitrarily Now we obtain 500 400 300 200 100 Hii i 10 100 It seems that I should zoom in a bit to see more detail So I type plot x 1 x 2 x 3 1 6 plot x 1 x 2 x 4 1 6 plot x 1 x 2 x 5 1 6 and obtain as a result 15 Now I can see better what that last numeral does It controls the location or x coordinate of the last time that the curve crosses the x axis As you can see the plus sign among the plot commands means to superimpose them There is a technicality here You must not break a newline among the plots and plus signs The various plot commands and their plus signs must be strung together like a paragraph wrapping naturally from one line to the next You do not use the enter key to put each plot on its own line Thus you cannot do plot x 1 x 2 x 3 1 6 plot x 1 x 2 x 4 1 6 plot x 1 x 2 x 5 1 6 plot x 1 x 2 x 6 1 6 plot x 1 x 2 x 7 1 6 which SAGE thinks is five separate commands instead of one big command Therefore if you have problems getting those last two commands to work then check to see that you have no acciden
80. ore than the above In the above work we calculated how many permutations but sometimes you might want to know what those permutations are For this you can type myset ace king queen jack permutations myset and it will obediently display all 24 possibilities Lbvace kinig queen Jack Mace king jack queen ace queen king jack ace queen jack king ace jack king queen ace jack queen king king ace queen Jack kine ace jack queen king queen ace jack king queen jack ace king jack ace queen king jack queen ace quee ac e king jack I queen ace jack king queen king ace jack queen king jack ace queen jack ace king L queen jack king ace gt jack ace king queen jack ace queen king L jack king ace queen jack king queen ace jack queen ace king L jack queen king ace Try in
81. ould simply type integral x sin x72 x and then you d learn the answer is 1 2 cos x72 Similarly if you wanted to know dx J zr 1 you would simply type integral x x72 1 x and then you d learn the answer is 1 2xlog x 2 1 The Definite Integral Alternatively we could put a lower bound of 0 and an upper bound of 1 on that integral and then get 1 x d ee 1 becomes integral x x72 1 x 0 1 and so we discover 1 2 108 2 is the answer but remember that log means the natural logarithm not the common logarithm More Difficult Cases Consider the following function f f a et sin x which is a nice continuous function We can find its plot with plot exp x73 sin x 2 5 5 ymin O ymax 1 which yields the plot 0 8 0 6 42 The function is interesting but the integral is quite frustrating I cannot solve that integral with my pencil and I am willing to wager that you cannot either Try it if you like While it is the case that there is some function somewhere which has f as its derivative which is a logical consequence of some advanced real analysis and the fact that f is continuous the practicalities of the matter are that there is no way to write the integral To be really precise there is no function built up of addition subtraction multiplication division roots exponents logarithms trigonometric functions and inverse trigonometric fu
82. ow those x values Here s what you will get 0 8 0 6 0 4 0 2 Likewise you can do plot x 3 x which is nice and visually appealing as you can see 0 4 0 3 0 4 For x 1 2 you can do 10 plot abs x 1 2 which can come up from time to time This produces 1 4 1 2 0 4 0 2 l 0 5 0 5 1 But what if you wanted a different x range For example to graph in 2 lt x lt 2 you would type plot x 3 x 2 2 and you get the desired graph namely A very cool graph is plot x 4 3 x 2 2 2 2 but notice the asterisk between 3 and x in 3x x You will get an error if you leave that out The plot is 11 Another minor sticking point is that for y sin t you have to say plot sin x or better yet plot sin x 10 10 because plot is not expecting t it expects x The images that you get are 0 5 0 5 10 5 10 0 5 1 plot sin x plot sin x 10 10 In fact if you were to type plot sin t you would see Traceback click to the left of this block for traceback NameError name t is not defined because in this case SAGE does not know what t means An alternative way of handling this is given on Page 29 Going Off the Scale Consider the plot of x from x 5 to x 10 which is given by plot x 3 5 10 12 and as you can imagine the function goes from 5 125 up to 10 1000 and so the origin should appear very far
83. perbola x y 1 with the ellipse gt 1 We type 4 var y solve x 2 y 2 1 x72 4 y72 3 1 1 x y and obtain x 4 7 sqrt 7 y 3 7 sqrt 7 x 4 7 sqrt 7 y 3 7 sqrt 7 Lx 4 7 sqrt 7 y 3 7 sqrt 7 Lx 4 7 sqrt 7 y 3 7 sqrt 7 which is unreadable Once again we change the command to be answer solve x 2 y 2 1 x72 4 y72 3 1 x y and then type print answer 0 print answer l1 print answer 2 print answer 3 print answer 4 and that produces four answers and an error message x 4 7 sqrt 7 y 3 7 sqrt 7 x 4 7 sqrt 7 y 3 7 sqrt 7 x 4 7 sqrt 7 y 3 7 sqrt 7 x 4 7 sqrt 7 y 3 7 sqrt 7 Traceback click to the left of this block for traceback IndexError list index out of range which is SAGE s way of telling you Solution 1 x V TY 2317 Solution 2 x EVT Y mE T 3 VT y 2Vv7 FV Gy V1 but that there is no Solution 5 Last but not least note that we aren t limited to polynomials We can do Solution 3 zx Solution 4 x solve sin xty 0 5 x and obtain x 1 6 pi y 34 Complex Numbers Normally we expect an 8th degree polynomial to have 8 roots over the complex plane unless some are repeated roots Therefore if we ask what are the roots of z8 1 surely we expect 8 roots These are the 8 eigthth roots of 1 in the complex numbers We can find them with
84. perbolic cousins which you might or might not have been taught about which will have as lts derivative either the Fresnel or the Gaussian Yet in applied mathematics we need to calculate these integrals In particular the Gaussian is superbly important in statistics So what is done is that the definite integral can be computed by dividing the interval into many many tiny regions Lots of infinitesimal rectangles and trapezoids can be used to approximate the curve Each of these has a small amount of error but when you add up all the areas the hope is that the area cancels out a bit You probably did this during your calculus class at some point or will at some future point Even more sophisticated methods include using polynomials instead of trapezoids and SAGE uses extraordinarily intricate techniques to make the best approximation possible These approximations are an old and deep subject going back to Simpson s Rule where Simpson used tiny parabolas instead of trapezoids The above might have been a bit confusing but the whole point of SAGE is that it will worry about these details for you so long as you know what you want from SAGE I hope this background was interesting and at least partially clear If not you should probably discuss it with your calculus instructor before proceeding further 41 The Indefinite Integral The indefinite integral is the simplest and for example if you wanted to know J x sin x dz you w
85. r 101 and False for 102 Last but not least you can find all the primes in a certain interval For example prime_range 1000 1100 gives you all the prime numbers between 1000 and 1100 59 The gcd and the lcm To find the greatest common denominator or gcd of two numbers perhaps 120 and 64 just type gcd 120 64 and you will learn that the answer is 8 This makes sense because 120 8 15 while 64 8 8 and as you can see 15 and 8 share no common factors More explicitly we can look at the prime factorizations 15 5 x 3 and 8 2 x 2 x 2 Another way of looking at that is 120 2 x 3 x 5 while 64 2 and so all that 120 and 64 have in common is just 2 which is 8 Likewise you can find the least common multiple or lcm This is done by typing lcm 120 64 from which you learn that the lcm is 960 This makes sense because 960 120 8 and also 960 64 15 and again 8 shares no factors with 15 A mathematician would say 8 is coprime to 15 Some authors say mutually prime instead of coprime Another way to look at it is that the lcm should be 26 x 3 x 5 960 because the 2 appears 6 times in 64 and 3 times in 120 thus a maximum of 6 while the 3 and 5 both appear 0 times in 64 and 1 time in 120 thus a maximum of 1 each Raising each of those primes 2 3 and 5 to those powers 6 1 and 1 gives the lcm Before we continue one more example is helpful Let s work
86. r 90 days and principal 900 we all know the formula to be A P 1 rt so we would just type in 900 1 0 06 90 365 and click evaluate You will learn that the answer is 913 315068493151 or 913 32 after rounding to the nearest penny Notice that the symbol for addition is the plus sign and the symbol for subtraction is the hyphen The symbol for multiplication is the asterisk and the symbol for division is the forward slash that s the one found with the question mark not the backslash For compound interest we d need to take exponents The symbol for exponents is the caret found above the number six on most keyboards It looks like this Let s consider 12 compounded annually on a signature loan for 3 years and a principal of 11 000 We all know the formula to be A P 1 1 and so we would just type in 11000 1 0 12 73 and click evaluate You will learn that the answer is 15454 2080000000 or 15 454 21 after rounding to the nearest penny By the way instead of clicking evaluate you can just press shift enter on the keyboard and it will do the same thing Warning It is very important not to enter a comma in large numbers when using SAGE You have to type 11000 and not 11 000 For those who don t like the caret you can use two asterisks in a row 11000 1 0 12 x 3 which is actually a throw back to the historical programming language FORTRAN which was most popular dur
87. s Exponentials Just as ZO gives you 2 and likewise 3 gives you 3 if you want to say e then just say e 3 and that s fine Or for a decimal approximation you can do N e 3 Also it is worth mentioning sometimes books will write exp 5 11 8 instead of e t and SAGE thinks that s just fine For example exp 5 11 8 e 5x11 8 evaluates to 0 Don t forget the asterisk between the 5 and the 11 3This turns out to be because 10200 3 2664 if you are curious Logarithms Of course SAGE knows about logarithms But there s a problem There are several types of logarithm including the common logarithm the natural logarithm the binary logarithm and the logarithm to any other base you feel like using In high school and middle school log refers to the common logarithm and In to the natural logarithm However during and after calculus log refers to the natural logarithm Since SAGE was mainly meant for university and higher level work then it is only natural they chose to use the natural logarithm for log So to find the natural logarithm of 100 type NC log 100 for the common logarithm of 100 type NC log 100 10 and for the binary logarithm of 100 type NC log 100 2 which of course generalizes To find the logarithm of 100 base 42 type NC log 100 42 Note that SAGE is quite good at getting exact answers Try log sqrt 10073 10 and you will obtain 3 the exact answer
88. s and take determinants as well as adding and subtracting matrices Also the backslash operator Meanwhile here is a command list if you need to look up the commands e The two ways to declare a matrix e vector e augment e A solve_right v e A solve_left v e A echelon_form e A rref e A inverse e The backslash operator e Matrix Multiplication e The identity matrix e det A 4T Chapter 2 Fun Exploration Projects using SAGE Here are some extended examples of using SAGE to explore a non trivial problem in an area outside of computer algebra We present one example from each of several disciplines All of these projects can be explored using information taught in Chapter One and do not require any additional familiarity with SAGE Pure Mathematics Analyzing a quintic polynomial s features for graphing Finance Analyzing a mortgage Physics Measuring the effect of gravity on a space satellite Microeconomics Modeling the effect of the selling price on the sales of ice cream Engineering Exploring the effects of a two stage rocket where the first stage fails to detach Ecology Approximating the impact of toxic waste on the oxygen content of a large lake Astronomy Finding the point of closest approach of a comet to the orbit of the earth Earth Sciences Discovering the relationship between the time of sunrise and sunset with latitude and time of year Medicine Modeling the flow of blood in an artery
89. s its divisors 1 3 5 9 15 45 Then we have sigma coming to 78 because 1 3 5 9 15 45 78 and then the sum of squares would be 1 3 5 9 15 457 2366 and with cubes A EEA 95 382 but in SAGE the commands for that are sigma 45 2 for squares and sigma 45 3 for cubes Likewise sigma 45 1 is the same as sigma Now there s a fun game that the Arabs used to play in the Medieval Period Consider two particular numbers 220 and 284 The divisors are divisors 220 1 2 4 5 10 11 20 22 44 55 110 220 and then they would calculate 1 2 4 5 10 11 20 22 44 55 110 284 but on the other hand divisors 284 1 2 4 71 142 284 which yields 1 2 4 471 4142 284 220 and so we see 220 and 284 are intimately related The Arabs found this so moving that they inscribed these numbers on jewelry that they would give to their spouses and so forth These are called Amicable Pairs 220 representing the young lady and 284 representing the young gentleman You can find this in SAGE via 58 sigma 220 220 and sigma 284 284 Tau As you know some fractions like 1 4 and 1 2 as well as 1 25 can be expressed as decimal fractions which terminate Others like 1 3 or 1 9 will repeat It is pretty easy to see why a fraction can be expressed as an exact terminating decimal fraction of n decimal places or fewer if and only if the denominator divides 10 Take a moment to convince yourself
90. stead myset ace king queen jack ten permutations myset if you are curious 3 2 3 Calculating Limits Expressly If you wanted to calculate P r 0gr 3 you should type limit x 2 x 3 x 0 to learn that the answer is 2 3 which is perhaps obvious Perhaps less obvious would be the limit qa2 Adx 3 lim _ a1 z 3r 2 for which you should type limit x72 4 x 3 x72 3 x 2 x 1 61 to obtain the answer 2 If you do not know how to do that without a computer or calculator in other words only with pen and pencil then let me suggest that you try factoring those two polynomials as a first step Last but not least if you wanted to verify the strange and amazing fact 2 lim cos r a then you should type limit cos x 1 x72 x 0 to obtain the desired answer e 1 2 To show that the last limit there given in terms of cosine and z is true you could try values like x 0 001 on any scientific calculator or with SAGE But to prove the limit true is quite difficult It requires two applications of L H pital s Rule or a Taylor Series but not both so far as I am aware 3 2 4 The Hyperbolic Trigonometric functions If you ve learned the hyperbolic trigonometric functions then that s great Personally I have never had them in a course and I have a PhD but I do know that they come in handy at times They come up in physics
91. tal line breaks in the command For SAGE these commands must be entered without a line break kind of like typing a long paragraph in an ordinary word processor you can wrap around the end of the line but you cannot insert a line break Further Graphing amp Plotting Topics The following additional topics on graphing and plotting can be found in the section Advanced Topics in Plotting and Graphing on Page 67 They include e Graphing with Colors e Labels and Legends e Grids and Graphing Calculator Style Graphs e Labeling the Axes of Graphs e Graphs of Hyperactive Functions e Odd Roots of Negative Numbers e Restricting the X Range of Plots 16 1 5 Matrices and SAGE Part One 1 5 1 A First Taste of Matrices This section assumes that you ve never worked with matrices before in any way or alternatively that you have forgotten them Experts in linear algebra can skip to the next subsection Doing the RREF in SAGE Let us suppose that you wish to solve the following linear system of equations on 4yHksz 14 IU O E 129 2 de E a First you would convert this into the following matrix 3 4 5 14 A 1 1 8 5 2 oh 1 7 Notice that the coefficients of the xs all appear in the first column the coefficients of the ys all appear in the second column the coefficients of the zs all appear in the third column The fourth column gets the constants Thus we have encapsulated and abbreviated all of the informa
92. that they come out to 65 7 but rather that they come out equal Thus the first equation is satisfied We check the other three equations similarly We really do have to plug each of the four values into each of the four equations by the way For example the imposter solution w 139 7 x 8 7 y 64 7 2220 7 satisfies the first three of those equations but fails to satisfy the last one If you happened to read the previous subsection you will recall that we did quite a lot of work to get B Those many steps might have contained an error if we were sloppy That s another reason that we should check our work 1 5 4 The Semi Rare Cases Now we re going to examine the two semi rare cases These occur if there is one row of all zeros at the bottom of the matrix ending in either a non zero number or ending in zero The system of equations is CEL oe Y 4x 5y 6z 16 fx 8y 9z2 24 That results in the matrix Ow a _ _ SBR 00 UTN No So _ a 20 and that we shall enter into SAGE with Gh matrix 35 45 i 2 3 Tr 4 5 63 16 7 84 9 24 3 To find the solution we type Ci rref and receive back 1 0 1 0 0 1 2 0 E O0 O 70 1 This translates into T YR 22 0 Now we ll examine shortly how to think about those top two equations However the bottom equation says 0 1 which is clearly not true There is no way to satisfy the requirement that 0 1 Therefore this equa
93. ting If you type the first few letters of the command then you can hit tab This is called tab completion and was introduced on Page 7 B 3 2 When You Don t Know which Command A major flaw in the design of all computer algebra systems help features is that the help system is based around the name of the command which you want help with However if you do not know the name of the command then you cannot use a system organized this way That is a problem and SAGE has come up with a partial solution If you type search_doc combinations then you get a list of links which contains every page in the SAGE documentation system that contains the word combinations One of these is very likely to contain the information that you want Suppose you wanted to know about how to use SAGE to compute Lagrange Multipliers You could type search _doc Lagrange and you get four things none of which have anything to do with Lagrange multipliers However I will now explain an alternative B 33 A Superb Google Trick My favorite way of learning about a broad category of math and how to do it in SAGE is to type something like Lagrange Multipliers site sagemath org into Google You get some great hits that way The site command restricts the search to all web pages on that website Here you can search any webpage on sagemath org for what you need This can be used for other purposes For example doing a search and adding site Fordha
94. tion has no solutions Instead of the answer being a solution a list of solutions or infinitely many solutions the answer is the words this system has no solutions That will be the outcome whenever the RREF has a row with all zeros except the last column which is a non zero number Whenever you see a row with all zeros except in the last column you must remember that the system has no solutions Another interesting case is to change the 24 in the last equation to 25 That results in the matrix gt a 6 16 9 25 o SBR 00 UTN and that we shall enter into SAGE with 62 matriz 3 4 ly 2 3 Te Li 56 16 T 879 251 To find the solution we type C2 rref and receive back 1 05 1 0 0 1 2 0 0 O 0 0 As you can see this RREF has a row of all zeros ending in a zero That indicates infinitely many solutions Some instructors allow you to write infinitely many solutions and in certain application problems you would not care to provide a way of saying what those solutions are However most of the time you d like to know what the solutions are This is especially true in industrial problems In any case you can t make a list because there infinitely many of them Instead mathematicians use a dummy variable usually t to get around the problem Did you notice how the RREFs of A and B all have a certain structure to the matrix In particular all the columns excepting the last column
95. tion is to compute the slope connecting the points 200 2 and 120 3 and then find the equation of the line between those two points Let s pause a moment and see if this makes sense First plugging in n 200 and n 120 we do get the prices of 2 and 3 that we expect If we set p 2 50 then we get 160 as we would anticipate that s the midpoint At a price of 4 50 then we sell no ice cream This could be believable if people are usually expecting to pay around 2 then you cannot charge them more than double what they are accustomed to pay If the price is 0 then we sell 360 cones per day that s called the saturation point and that s a lot of ice cream The model breaks down for prices over 4 50 because it predicts a sale of a negative number of cones which does not make sense Likewise if you set a demand above 360 cones per day the saturation point then a negative price is called for which also does not make sense So the model is valid for 0 lt p lt 4 50 or equivalently 0 lt n lt 360 cones Clearly if one sells n cones at price p then the revenue is np So then we have r n np 4 50n n 80 as the revenue function Meanwhile the cost function is 50 cents per cone plus 100 for a total of e n 0 50n 100 and that gives us a profit function of 2 N e do 100 p n g An which is a parabola It makes sense that we should get some sort of curve because if
96. tion of the problem Furthermore observe that additions are represented by a positive coefficient and subtractions by a negative coefficient Matrices have a special form called Reduced Row Echelon Form often abbreviated as RREF The RREF is from a certain perspective the simplest description of the system of equations that is still true The Reduced Row Echelon Form of this matrix A is 1 0 0 3 0 1 00 0 0 1 1 We can translate this literally as lr 0y 0z 3 Or ly 0z 0 Ox O0y i1iz 1 or in simpler notation TE y 0 ak which is the answer You should take a moment now plug in these three values and see that it comes out correct 1 5 2 Complications You might be looking at the previous discussion and imagine that what we re doing is a kind of silver bullet quickly capable of solving any of a large number of extremely tedious problems that take with a pencil a very long time Indeed it is now that we have computers Prior to the computer large linear algebra problems were considered extremely unpleasant Yet now that we have computers and because computers can carry out these operations essentially instantly this topic is a great way to rapidly solve enormous systems of linear equations That in turn means that skilled mathematicians can address important problems in science and industry because real world problems often have many variables This topic however is better described as semi automatic rather t
97. tive 71 Plot 1 Plot 2 Plot 1 plot sin 1 x 0 5 0 5 Plot 2 plot exp 1 2 x 1 4 ymin 1 ymax 5 Plot 3 plot sin exp 1 2 x 1 4 Plot 4 plot sin 10 x x 2 2 Odd Roots of Negative Numbers This is a freakish technicality To plot the negative real cube root you have to use something like the following plot lambda x RR x nth_root 3 x 1 1 and not plot x 1 3 D 5 which objects to negative reals being raised to fractional powers Hopefully we ll get that one sorted out shortly Certainly SAGE is a work in progress The two commands produce 12 0 5 0 8 0 6 0 4 l 0 l 2 3 4 5 Plot using Lambda Plot without Lambda To Restrict the z values of a Graph Once in a great while you need to restrict the x values of a graph This occurs when part of the domain is missing Consider the function f x Vda 8 log 3 9x V1 4z which clearly has a domain of 8 5 lt x lt 1 4 If you type plot sqrt 5 x 8 log 3 9 x sqrt 1 4 x 2 2 gridlines true even though you asked for 2 lt x lt 2 you only get 8 5 lt x lt 1 4 It is also accompanied by an error message Instead you must do plot sqrt 5 x 8 log 3 9 x sqrt 1 4 x 2 2 gridlines true show xmin 2 xmax 2 where xmin and xmax explicitly force the x coordinates to be what you want 15 l 0 5 0 2 l 0 1 2 Plot using Show Plot without Show 13 3 6 Graphing Inequa
98. ut that at roughly 292 664 cones and 27 3350 cones we break even Between these values is a profit and outside these values there is a loss That seems to match the data in the table e The other definition of the break even point is when profit is zero So we could have typed find_root x 2 80 4 x 100 10 50 and also 92 find_root x 2 80 4 x 100 250 350 to find those same values 292 664 and 27 3350 and you ll see that the green curve crosses the x axis there e The cost function is never zero That s because the salesman has to pay the 100 per day fee e The revenue function is zero at two points First when no ice cream is sold that makes sense We found earlier that is at 4 50 per cone and O cones per day Second revenue is zero when the ice cream is sold for 0 cents per cone Clearly if you don t charge anything then there is no revenue That occurs at the saturation point of 360 cones per day and 0 per cone In SAGE we would type find_root 4 50 x x72 80 O 50 and also find_root 4 50 x x72 80 300 400 to get the values 0 0 and 360 0 e Now we ve found when profit is zero and when revenue is zero and determined that cost is never zero The next step is to try to minimize or maximize these functions If you ve had calculus then you know how to minimize or maximize a polynomial If not then you might recall the fact that a parabola y ax bx c has its optimum
99. ve To do an exponential regression we type 65 var a b model x a exp b x find_fit cobalt_data model and as you can see only the middle line of those three lines has changed We have changed the model from y az b into y ae Now SAGE s response to these commands is la 1999 9607020042693 b 0 46216218069391302 which is SAGE s way of telling us that the model is y 1999 96e 0 462162 and you can immediately see that there s some rounding error That leading coefficient of 1999 96 really should be 2000 as we started with 2000 grams But that s part of the pros and cons of numerical methods You can do almost anything with numerical methods but you very rarely can be exact In any case we can see the quality of this regression by the command scatter_plot cobalt_data plot 1999 96 exp 0 462162 x 0 6 and again the image is found in the previous section so we will not repeat it here to save space Last but not least one might consider a quadratic regression That s a regression of the form y ax bz c and the command for that would be var a bc model x ax xx 2 bx x c find_fit cobalt_data model where you can see that the only real change is the middle of the three lines again but also we had to declare c as a variable which we did in the var command In any case the response from SAGE is la 62 755170737557854 b 666 86435772164759 c 1925 5757574133333 and that is SA
100. which provides the answer log x 1 x x Now if you re interested in giving your calculus skills a workout but possibly a very intense one you can try to see how you would calculate the derivative of x with your pencil In fact if you happen to know Newton s Method you can use that derivative and Newton s Method to calculate the self logarithm of a number Of course many calculus teachers omit Newton s Method so do not feel bad if you haven t been taught it yet Plotting f x and f x Together One of my favorite bits of code in SAGE is f x x 3 x fprime x f x derivative plot f x 2 2 color blue plot fprime x 2 2 color red and I use this quite frequently This plots f x and f x on the same graph with f x in blue and f x in red You can very easily change what function you are analyzing but just changing that first line Here we are analyzing f x x x in blue and its derivative f x 3x 1 in red The plot is as follows Higher Order Derivatives While you can find the second derivative by using diff twice you can jump directly to the second derivative with diff x 3 x x 2 to get the answer 6 X 39 Third or higher derivatives are also no problem For example the third derivative can be found with diff x 3 x x 3 For those who don t mind typing more you can do derivative x 3 x x 2 which is more readable than writing diff 1 12 Derivat
101. with slightly larger numbers perhaps 3600 and 1000 First we do the prime factorizations 3600 2 x 3 x 5 while 1000 2 x 5 and we can compute the lcm by taking the maximum exponent each time 2 x 37 x 5 18 000 and the gcd by taking the minimum exponent each time PR Xo 200 Take a moment to verify that 200 goes into both 1000 and 3600 as well as the fact that no higher number will Next take another moment to verify that both 1000 and 3600 go into 18 000 but there is no lower number for which that s true And of course the product of the numbers divided by the gcd is the lcm and the product of the numbers divided by the lcm is the gcd The key here is to realize that the prime factorization of a number communicates an enormous amount of information about that number Several gcds or lcms at Once If you want to take the gcd of many numbers at once for example 120 55 25 and 35 there s no need to do gcd 120 gcd 55 gcd 25 35 because you can do instead gcda 120 55 25 35 to get 5 and likewise 56 lent 1203 55 20 35 1 to get 46 200 This is another example of a list in SAGE You can enclose any data with and and separate the entries with commas to make a list We saw this in graphing multiple functions at once on Page 68 and in solving equations exactly on Page 32 We ll see this again in scatter plotting live data on Page 62 And you can even verify the theorem that says t
102. you make the price very low or very high then the profit is zero or worse you have a loss Somewhere in the middle is the sweet spot Before we graph this if you are inexperienced in this kind of analysis let s make a table and see that it makes sense and matches our prior data ol Price Cones Sold Revenue Costs Profit 0 00 360 0 00 280 00 280 00 0 50 320 160 00 260 00 100 00 1 00 280 280 00 240 00 40 00 1 50 240 360 00 220 00 140 00 2 00 200 400 00 200 00 200 00 2 50 160 400 00 180 00 220 00 3 00 120 360 00 160 00 200 00 3 50 80 280 00 140 00 140 00 4 00 40 160 00 120 00 40 00 450 0 0 00 100 00 100 00 I recommend that you pick 2 or 3 values and manually verify those 2 or 3 rows of the table before continuing onward Okay did you really verify 2 or 3 rows of the above It is really useful to do so before reading further In any case here s the graph 400 300 200 100 100 where you can see that the revenue function is in black the cost function is in red and the profit function is in green There are lots of facts that can be gleaned from this e When red cross black that means that cost and revenue cross each other that s the break even point Here we have two break even points one just under 30 and one just under 300 Let s find them We would type find_root x 2 80 4 5 x 0 5 x 100 250 350 and also find_root x 2 80 4 5 x 0 5 x 100 10 50 to find o

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