Calculus on Maple

Purdue Engineering Computer Network

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Introduction

If you are entering the fields of the physical science, engineering, business, or quantitative psychology, you probably have to take a calculus course. Calculus can be frustrating to many. Maple may be able to help some of your frustrations. This document addresses the different commands that can be used for calculus problems and shows how to use these commands through examples.

The topics that will be discussed are:

• Derivatives
• Partial Derivatives
• Indefinite Integration
• Definite Integration

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Derivatives

If you want to find the rate of change or find the slope of a line, you take the derivative of that function. The derivative takes the general form:

f^'(x)=nx^n-1

For example, the derivative of x³ would be 3x². In Maple, the diff command can be used to take the derivative of a function. The diffcommand take the general form:

diff( function, variable)

Example 1.1

Consider the function:

f(x)=56x²³+sin(2x³)*x^e2x+(x+1)²

Find the derivative of this function.

-Solution-

1. First, type the diff command for this function.
   diff((56*x^23)+(sin(2*x^3)*x^exp(2*x))+((x+1)^2),x);

2. Press return and the answer should yield:
   1288x²²+6cos(2x³)*x²*x^e2x+sin(2x³)*x^e2x*(2e^2x*ln(x)+e^2x/x)+2x+2

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Partial Derivatives

In multivariale calculus, the student is introduced to the partial derivative. The partial derivative takes the general form:

f^'(x,y)=nx^n-1yⁿ+xⁿny^n-1

As above when we introduced the diff command for derivatives, the diff command is also used for partial derivatives. The genereal form of the diff command for partail derivatives is as follows:

diff ( function, variable1, variable2)

Example 1.2

Consider the function:

f(x,y)=x³y+e^2y2(x²+2x+5)³+cos(y)sin(x)

find f^'(x,y).

-Solution-

1. First, enter the diff command as follows:
   diff((x^3)*y+((e^(2*y^2))*((x^2)+2*x+5)^3)+cos(x)*sin(x),x,y);

2. Press return and the answer should be as follows:
   3x²+12e^2y2yln(e)(x²+sx+5)²(2x+2)

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Indefinite Integration

Integration is when you take a function and preform what is known as an antiderivative. The general form of an antiderivative is as follows:

f(x)=(1/(n+1))xⁿ⁺¹+constant

An Indefinite Integral is simply taking the antiderivative of a function without boundary conditions. In Maple, you can use th int to perform an Indefinite Integral. The int command takes the following general form:

int (itegrand, variable of integration)

Example 1.3

Dr. John Doe calculated the velocity of an airplane to be the following:

V(t)=t³+3t²+2t+5

Find the distance function.

-Solution-

1. In order to find the distance function, you must take the integral of the velocity function. In Maple, type the int command as follows:
   int((t^3)+(3*t^2)+2*t+5,t);

2. Press return and the answer should yield:
   1/4 t⁴ + t³ + t² + 5t

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Definite Integral

A Definite Integral is preforming an integration with boundary conditions. To preform a definite integral in Maple you use the int which takes the general form as follows:

int ( integrand, variable of intergration = low .. high)

Example 1.4

A satellite in orbit around Jupiter in not conserved. The torque of the satellite is as follows:

T(t)=Cos(2t)+t

Find the Angular Momentum and evaulate the Angular Momentum from the times from 0 to 3.

-Solution-

1. In order to find the angular momentum and evaulate from 0 to 3, you must take the integral of torque. Enter the int as follows:
   int(cos(t)+t,t=0..3);

2. Press return and the answer should yield as follows:
   sin(3)+9/2