Maple and Calculus Operations Although our initial Intro to Maple included discussion of a number of Maple functions related to calculus, you may not have noticed that fact if you didn't look carefully at what Maple can do. This handout is, in the first instance, just a summary of Maple's calculus-related functionality. As always, I'm only scratching the surface. The Maple help command (say ?diff; to get help on diff, for instance) offers a lot more information on any of the commands used in this worksheet. Basic Commands Limits Maple's command for limits is, logically enough, limit. limit(sin(x)/x, x=0);

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limit(1/x, x=0);

SSp1bmRlZmluZWRHJSpwcm90ZWN0ZWRH

limit(1/x^2, x=0);

SSlpbmZpbml0eUclKnByb3RlY3RlZEc=

limit(sin(1/x), x=0);

OyEiIiIiIg==

It is useful to plot all the functions above, and to make sure you understand what Maple is telling you with all these answers. Our investigation of derivatives could have been speeded up substantially had we been willing to use Maple uncritically as a black box for taking limits. Here, for instance, are calculations of the derivatives of the sine, exponential, and cube root functions. It would be useful review to try to remember how we obtained these results in class. Are you sure Maple doesn't have bugs that affect these calculations? limit((sin(x+h)-sin(x))/h, h=0);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRic=

limit((exp(x+h)-exp(x))/h, h=0);

LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRic=

g := x -> x^(1/3);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQpRiQjIiIiIiIkRixGJUYlRiU=

(g(x)-g(a))/(x-a);

KiYsJiokKUkieEc2IiMiIiIiIiRGKUYpKiQpSSJhR0YnRihGKSEiIkYpLCZGJkYpRi1GLkYu

limit(%, x=a);

LCQqJiMiIiIiIiRGJSlJImFHNiIjIiIjRiYhIiJGJQ==

For limits, as for most calculus functions, Maple has an "inert" form of the limit command that writes the limit without evaluating it. This command is called Limit. It's often useful in writing down formulas in Maple worksheets. Limit(x^x, x=0);

LUkmTGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JClJInhHRidGKi9GKiIiIQ==

Limit(x^x, x=0) = limit(x^x, x=0);

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQpSSJ4R0YoRisvRisiIiEiIiI=

Thinks about why the value of this limit wasn't obvious. (What's LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMEYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE=? What's LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy? What should LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjhGK0Y+Rjg= be, if anything?) Derivatives Maple has two functions for computing derivatives. If f is an expression (like 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), then diff(f, x) is the derivative of f with respect to the variable x. If f is a function (like sin, or like the user-defined function g above), then the function D(f) is the derivative of f as a function. Using D can save typing, and gets the derivative as a function. diff(sqrt(x+sqrt(x+sqrt(x))), x);

LCQqKCMiIiIiIiNGJSksJkkieEc2IkYlKiQpLCZGKUYlKiQpRilGJEYlRiVGJEYlRiVGJCEiIiwmRiVGJSooRiRGJUYsRjAsJkYlRiUqJkYkRiVGL0YwRiVGJUYlRiVGJQ==

f := x -> x^x;

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKUYkRiRGJUYlRiU=

fprime := D(f);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYpRiRGJCIiIiwmLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGI0YrRitGK0YrRiVGJUYl

plot({f(x), fprime(x)}, x=0..2);

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

D(sin)(x);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRic=

D(D(sin))(x);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCEiIg==

D(D(D(sin)))(x);

LCQtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCEiIg==

(D@@4)(sin)(x) = D(D(D(D(sin))))(x);

Ly1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YoRiM=

These operators can also be used with undefined functions to get general differentiation rules. Right now, f and g have value, but we can remove those g(x);

KiQpSSJ4RzYiIyIiIiIiJEYn

g := 'g';

SSJnRzYi

g(x);

LUkiZ0c2IjYjSSJ4R0Yk

f := 'f'; f(x);

SSJmRzYi

LUkiZkc2IjYjSSJ4R0Yk

Here are Maple versions of the product rule, the chain rule, and an extended chain rule for 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. Play with things like this, and make sure the notation makes sense. If you need help, try ?D or ?diff. D(f*g);

LCYqJi1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiZkdGKSIiIkkiZ0dGKUYsRiwqJkYrRiwtRiU2I0YtRixGLA==

D(f@g);

KiYtSSJARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSJER0YlNiNJImZHRihJImdHRigiIiItRis2I0YuRi8=

diff(f(g(h(x))), x);

KigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJmR0YpNiMtSSJnR0YpNiMtSSJoR0YpNiNJInhHRikiIiItLUYlNiNGLkYvRjQtSSVkaWZmR0YnNiRGMEYzRjQ=

Integrals Maple does both indefinite and definite integrals with int. The inert form of int is, logically enough, Int. As with limits, the inert forms are useful in writing formulas. int(sin(x), x=0..Pi);

IiIj

int(x/(1+x^2), x);

LCQqJiMiIiIiIiNGJS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkYlRiUqJClJInhHRixGJkYlRiVGJUYl

int(x/(1+x^4), x);

LCQqJiMiIiIiIiNGJS1JJ2FyY3Rhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiQpSSJ4R0YsRiZGJUYlRiU=

int(x/(1+x^3), x);

LCgqJiMiIiIiIidGJS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsKCokKUkieEdGLCIiI0YlRiVGMSEiIkYlRiVGJUYlKigjRiUiIiRGJSlGNiNGJUYyRiUtSSdhcmN0YW5HRik2IywkKihGNUYlLCYqJkYyRiVGMUYlRiVGJUYzRiVGN0YlRiVGJUYlKiZGNUYlLUYoNiMsJkYlRiVGMUYlRiVGMw==

Int(x^3*exp(x),x);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJilJInhHRiciIiQiIiItSSRleHBHRiQ2I0YrRi1GKw==

Int(x^3*exp(x),x) = int(x^3*exp(x),x);

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYpSSJ4R0YoIiIkIiIiLUkkZXhwR0YlNiNGLEYuRiwqJiwqIiInISIiKiZGNEYuRixGLkYuKiZGLUYuKUYsIiIjRi5GNSokRitGLkYuRi5GL0Yu

Int(x^n,x) = int(x^n,x);

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkieEdGKEkibkdGKEYrKiYpRissJkYsIiIiRjBGMEYwRi8hIiI=

The values of inert forms can also be taken using the command value. Int(ln(x)^n/x, x) = value(Int(ln(x)^n/x, x));

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYpLUkjbG5HRiU2I0kieEdGKEkibkdGKCIiIkYvISIiRi8qJilGLCwmRjBGMUYxRjFGMUY1RjI=

Sums Sums are done with sum and Sum, just as integrals are done with int and Int. sum(k, k=1..10);

IiNi

sum(k, k=1..n);

LCgqJiMiIiIiIiNGJSksJkkibkc2IkYlRiVGJUYmRiVGJSomRiRGJUYpRiUhIiJGJEYs

factor(%);

LCQqKCMiIiIiIiNGJUkibkc2IkYlLCZGJ0YlRiVGJUYlRiU=

sum(k^2, k=1..n);

LCoqJiMiIiIiIiRGJSksJkkibkc2IkYlRiVGJUYmRiVGJSomI0YlIiIjRiUpRihGLUYlISIiKiYjRiUiIidGJUYpRiVGJUYxRiU=

factor(%);

LCQqKiMiIiIiIidGJUkibkc2IkYlLCZGJ0YlRiVGJUYlLCYqJiIiI0YlRidGJUYlRiVGJUYlRiU=

Sum(k^3, k=1..n);

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJClJImtHRiciIiQiIiIvRis7Ri1JIm5HRic=

Sum(k^3, k=1..n) = factor(sum(k^3, k=1..n));

Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQpSSJrR0YoIiIkIiIiL0YsO0YuSSJuR0YoLCQqKCNGLiIiJUYuKUYxIiIjRi4pLCZGMUYuRi5GLkY3Ri5GLg==

The range of sums Maple understands is pretty impressive. Here's an identity that should look beautiful and surprising: Sum(1/n^2, n=1..infinity) = sum(1/n^2, n=1..infinity);

Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQpSSJuR0YoIiIjISIiL0YsOyIiIkkpaW5maW5pdHlHRiYsJComI0YxIiInRjEpSSNQaUdGJkYtRjFGMQ==

If Maple can do all this, then Maple ought to be able to handle the nasty work we encountered with Riemann sums. Here, for instance, is Maple's treatment of a computation of 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 using Riemann sums. f := x -> x^3;

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQpRiQiIiQiIiJGJUYlRiU=

RSum := Sum(f(k*x/n)*(1/n), k=1..n);

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKClJImtHRiciIiQiIiIpSSJ4R0YnRixGLSlJIm5HRiciIiUhIiIvRis7Ri1GMQ==

RSum = value(RSum);

Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKigpSSJrR0YoIiIkIiIiKUkieEdGKEYtRi4pSSJuR0YoIiIlISIiL0YsO0YuRjIsKCoqI0YuRjNGLkYvRi5GMUY0KSwmRjJGLkYuRi5GM0YuRi4qKiNGLiIiI0YuKUY7Ri1GLkYvRi5GMUY0RjQqKkY5Ri5GL0YuRjFGNClGO0Y+Ri5GLg==

RSum = factor(value(RSum));

Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKigpSSJrR0YoIiIkIiIiKUkieEdGKEYtRi4pSSJuR0YoIiIlISIiL0YsO0YuRjIsJCoqI0YuRjNGLkYvRi4pLCZGMkYuRi5GLiIiI0YuKUYyRjxGNEYu

This works out the Riemann sum for n slices. Now we want to take the limit of this sum as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGNy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPkYrRldGPg==. Int(t^3, t=0..x) = Limit(RSum, n=infinity); Int(t^3, t=0..x) = Limit(factor(value(RSum)), n=infinity); Int(t^3, t=0..x) = limit(factor(value(RSum)), n=infinity);

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQpSSJ0R0YoIiIkIiIiL0YsOyIiIUkieEdGKC1JJkxpbWl0R0YlNiQtSSRTdW1HRiU2JCooKUkia0dGKEYtRi4pRjJGLUYuKUkibkdGKCIiJSEiIi9GOztGLkY+L0Y+SSlpbmZpbml0eUdGJg==

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQpSSJ0R0YoIiIkIiIiL0YsOyIiIUkieEdGKC1JJkxpbWl0R0YlNiQsJCoqI0YuIiIlRi4pRjJGLUYuKSwmSSJuR0YoRi5GLkYuIiIjRi4pRj1GPiEiIkYuL0Y9SSlpbmZpbml0eUdGJg==

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQpSSJ0R0YoIiIkIiIiL0YsOyIiIUkieEdGKCwkKiYjRi4iIiVGLilGMkYtRi5GLg==

The student calculus package. Maple also has a whole library of functions useful for the calculus student and contained in the package Student. We're interested in a sub-package called Student[Calculus1]. You need to load the package using with(Student[Calculus1]) before you can use the functions. with(Student[Calculus1]);

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

It's worth asking for help about most of these functions, just to see what they do. Here's a quick teaser: The midpoint approximation to the integral of 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 between 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 and 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 is shown below RiemannSum(Zeta(x), x=-20..-19, partition=10, method=midpoint, output=plot);

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If we don't want the picture, we can write down the sum of the areas of the reactangles RiemannSum(Zeta(x), x=-20..-19, partition=10, method=midpoint);

LDYqJiMiIiIiIzVGJS1JJVpldGFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMhJCpSIiM/RiVGJSomRiRGJS1GKDYjIyEkKFJGMEYlRiUqJkYkRiUtRig2IyMhI3oiIiVGJUYlKiZGJEYlLUYoNiMjISQkUkYwRiVGJSomRiRGJS1GKDYjIyEkIlJGMEYlRiUqJkYkRiUtRig2IyMhJCpRRjBGJUYlKiZGJEYlLUYoNiMjISQoUUYwRiVGJSomRiRGJS1GKDYjIyEjeEY7RiVGJSomRiRGJS1GKDYjIyEkJFFGMEYlRiUqJkYkRiUtRig2IyMhJCJRRjBGJUYl

and then approximate the sum numerically. % = evalf(value(%));

Lyw2KiYjIiIiIiM1RiYtSSVaZXRhRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjISQqUiIjP0YmRiYqJkYlRiYtRik2IyMhJChSRjFGJkYmKiZGJUYmLUYpNiMjISN6IiIlRiZGJiomRiVGJi1GKTYjIyEkJFJGMUYmRiYqJkYlRiYtRik2IyMhJCJSRjFGJkYmKiZGJUYmLUYpNiMjISQqUUYxRiZGJiomRiVGJi1GKTYjIyEkKFFGMUYmRiYqJkYlRiYtRik2IyMhI3hGPEYmRiYqJkYlRiYtRik2IyMhJCRRRjFGJkYmKiZGJUYmLUYpNiMjISQiUUYxRiZGJiQiK1x3J1FtIyEiKQ==

By comparison, a more exact value of the integral is Int(Zeta(x), x=-20..-19) = evalf(int(Zeta(x), x=-20..-19));

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUklWmV0YUdGJTYjSSJ4R0YoL0YtOyEjPyEjPiQiK0UjM3JsIyEiKQ==

The error in the midpoint sum is therefore only RiemannSum(Zeta(x), x=-20..-19, partition=10, method=midpoint) - Int(Zeta(x), x=-20..-19) = evalf(value(RiemannSum(Zeta(x), x=-20..-19, partition=10, method=midpoint)))- evalf(int(Zeta(x), x=-20..-19));

Lyw4KiYjIiIiIiM1RiYtSSVaZXRhRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjISQqUiIjP0YmRiYqJkYlRiYtRik2IyMhJChSRjFGJkYmKiZGJUYmLUYpNiMjISN6IiIlRiZGJiomRiVGJi1GKTYjIyEkJFJGMUYmRiYqJkYlRiYtRik2IyMhJCJSRjFGJkYmKiZGJUYmLUYpNiMjISQqUUYxRiZGJiomRiVGJi1GKTYjIyEkKFFGMUYmRiYqJkYlRiYtRik2IyMhI3hGPEYmRiYqJkYlRiYtRik2IyMhJCRRRjFGJkYmKiZGJUYmLUYpNiMjISQiUUYxRiZGJi1JJEludEdGKjYkLUYpNiNJInhHRi0vRl9vOyEjPyEjPiEiIiQiKEIlZm4hIik=

Integration Tutor One of the neatest uses of the Student[Calculus1] package is that it lets one explore possible choices of u when doing substitution and choices of u and v' when doing integration by parts. This is easier to see on your own than it is to explain here, but here are the commands you would use to open the integration tutor for the integrals 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 and 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. IntTutor(x^2/(1+x^6),x); IntTutor(x^3*exp(x^2),x); You might enjoy exploring, for instance, what happens with the first integral if you let 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 and then if you let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiUtSSNtbkdGJDYkUSIzRidGOS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZW5GOQ==. The only tricky bit might be that Maple calls substitution "change of variable," so that in the IntTutor you do it by hitting the Change button. You might also enjoy exploring Parts as a technique for the second integral, letting u be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQEY+, then trying 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, then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjk=, then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiQtSSNtbkdGJDYkUSIyRidGOUY5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y5. The Undo button will be useful in this play. There is lots more functionality in Maple that is relevant to calculus, and there are lots more fun and useful functions in the Student[Calculus1] package. A bit of time spent playing with Maple's tutorials and messing about with the online help can be very rewarding. Sample Problems to play with. If you want to mess about and to get comfortable, here are some problems you migt have fun with. 1. Use Maple to explore the sums 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 for various values of s. What do you observe? 2. As efficiently as possible, compute the 10th derivative of 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. 3. What happens when you use the substitution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEnYXJjc2luRicvRjVGQkY+LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zobi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0Y0RjcvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GPkZib0Y+RitGYm9GPkYrRmJvRj4= on the integral 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? What if you use 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 on the integral 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? Can you understand what's going on by doing these integrals by hand? Can you find other integral where these substitutions work? Be creative! 4. Use Maple to explore sums like 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 for various values of s, and see if you observe anything. Also, use Maple to look at sums like 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KS1JK211bmRlcm92ZXJHRiQ2Jy1JI21vR0YkNi9RJiZTdW07RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Jy1GLDYlUSJrRicvJSdpdGFsaWNHRkgvRjxRJ2l0YWxpY0YnLUY1Ni1RIj1GJ0Y7RkFGRC9GR0ZDRkkvRkxGQy9GTkZDRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGYW8tSSNtbkdGJDYkUSIxRidGOy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOy1GLDYlUSJuRidGZm5GaG4vJSdhY2NlbnRHRkMvJSxhY2NlbnR1bmRlckdGQ0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSQ1LjBGJy8lJmRlcHRoR0ZmcC8lKmxpbmVicmVha0dRJWF1dG9GJy1GIzYpRlktRjU2LVExJkludmlzaWJsZVRpbWVzO0YnRjtGQUZERl1vRklGXm9GX29GT0ZRL0ZVRlMtSShtZmVuY2VkR0YkNiQtRiM2J0ZZLUY1Ni1RIitGJ0Y7RkFGREZdb0ZJRl5vRl9vRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGXnJGY29GZ29GO0Y7RmFxLUZmcTYkLUYjNidGWUZqcS1GZG82JFEiMkYnRjtGZ29GO0Y7RmdvRjtGK0Znb0Y7RitGZ29GOw==, 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, . . . See if you notice anything here. It might help if you factor the expressions you compute. 5. Let f(x) be a parabola f := x -> a*x^2 + b*x + c;

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJkkiYUdGJSIiIilGJCIiI0YsRiwqJkkiYkdGJUYsRiRGLEYsSSJjR0YlRixGJUYlRiU=

What must a, b, and c be in order for f to have the same value, the same first derivative, and the same second derivative as 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=? (COmpute the derivatives of both functions at 0, set them equal to one another, and solve the resulting system of equations.) Plot sin(x) together with this parabola. Now take a generic cubic, 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, and try to make its value and its first, second, and third derivatives equal to the value and the first 3 derivatives of sin(x) at 0. Again, plot this cubic together with sin(x). Keep doing this with polynomials of increasing degree. You should be getting curves that approximate the sine function better and better. 6. Ask Maple about the sum 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. What has this got to do with Problem 5?