MTH 141, Section 7 

Maple Assignment 2 

Calculus 1:  Theorems about Differentiation 

 

For further information about any command in the Calculus1 package, see the corresponding help page.  For a general overview, see Calculus1 . 

 

Read the folloiwng illustrations of Rolle's Theorem and The Mean Value theorem.  Then follow the directions under "Exercises" at the end of the worksheet. 

Getting Started 

While any command in the package can be referred to using the long form, for example, Student[Calculus1][MeanValueTheorem],  it is easier, and often clearer, to load the package, and then use the short form command names. 

> restart;
 

> with(Student[Calculus1]):
 

The following sections show how the various routines work. 

Tangents 

The Tangent routine returns the tangent to a curve at a given point. 

> Tangent( sin(x), x=1, output = line );
 

`+`(`*`(x, `*`(cos(1))), sin(1), `-`(cos(1))) (1.1)
 

We can evaluate the sines and cosines in the formula above. 

> evalf(%);
 

`+`(`*`(.5403023059, `*`(x)), .3011686789) (1.2)
 

We can see a plot of the tangent line on the same axes as the original function. 

> Tangent( sin(x), x=1, output = plot );
 

Plot_2d
 

Where the tangent is vertical, maple returns an Error. 

> Tangent( x^(1/3), x=0, output = line );
 

Error, (in Student:-Calculus1:-Tangent) the slope is not defined at the point `x` = 0
 

You can also learn about tangents using the TangentTutor command. 

> Typesetting:-mrow(Typesetting:-mi(
 

Plot_2d
 

Rolle's Theorem 

Rolle's theorem states that if Typesetting:-mrow(Typesetting:-mi( is a function that satisfies: 

1.  f is continuous on the closed interval Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(, 

2.  f is differentiable on the open interval (Typesetting:-mrow(Typesetting:-mi(), and 

3.  Typesetting:-mrow(Typesetting:-mi( 

then there exists a point Typesetting:-mrow(Typesetting:-mi( in the open interval (Typesetting:-mrow(Typesetting:-mi() such that f'(Typesetting:-mrow(Typesetting:-mi() = 0. 

The routine RollesTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative is zero.  The points output tells you the Typesetting:-mrow(Typesetting:-mi( coordinates of the points where the derivative is zero. 

> RollesTheorem(x*(x - 4), x=1..3, output=points);
 

[2] (2.1)
 

> solve(x*(x-4)=0);
 

0, 4 (2.2)
 

> RollesTheorem(x*(x - 4), x=1..3);
 

Plot_2d
 

> RollesTheorem(sin(x), 1..2*Pi + 1);
 

Plot_2d
 

The Mean Value Theorem 

The mean value theorem is a generalization of Rolle's theorem which states that if Typesetting:-mrow(Typesetting:-mi( is a function that satisfies: 

1.  f is continuous on the closed interval Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(, and 

2.  f is differentiable on the open interval (Typesetting:-mrow(Typesetting:-mi(), 

then there exists a point Typesetting:-mrow(Typesetting:-mi( in the open interval (Typesetting:-mrow(Typesetting:-mi() such that f'(Typesetting:-mrow(Typesetting:-mi() = Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi( where the right-hand side is the slope of the line connecting the points (Typesetting:-mrow(Typesetting:-mi() and (Typesetting:-mrow(Typesetting:-mi().  The Mean Value Theorem can be derived from Rolle's Theorem by considering the function Typesetting:-mrow(Typesetting:-mi(. 

The routine MeanValueTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative equals the slope of the secant line connecting the end points of the graph of Typesetting:-mrow(Typesetting:-mi( on Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(. 

> MeanValueTheorem(x^3 - 5*x^2 + 8*x - 1, x=1..3, output=points);
 

[`/`(7, 3)] (3.1)
 

> MeanValueTheorem(x^3 - 5*x^2 + 8*x - 1, x=1..3);
 

Plot_2d
 

> MeanValueTheorem(sin(x), x=-4..2*Pi);
 

Plot_2d
 

You can also learn about the Mean Value Theorem using the MeanValueTheoremTutor command. 

> MeanValueTheoremTutor();
 

Plot_2d
 

*******************  EXERCISES ************************** 

CREATE A NEW WORKSHEET.  Complete each exercise below.  Include the questions and be sure to show the commands you use for maple output. 

 

Try the 2 tutors mentioned above by entering "Typesetting:-mrow(Typesetting:-mi( 

You can get help by selecting "Maple Help" under the "Help" drop-down menu. 

 

1.)  As shown above, illustrate the tangent line to Typesetting:-mrow(Typesetting:-msqrt(Typesetting:-mrow(Typesetting:-mi(at Typesetting:-mrow(Typesetting:-mi(  What is the slope of the tangent at  Typesetting:-mrow(Typesetting:-mi(? 

 

2.)  Find the zeros of Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(between -10 and 10, see command below.  As shown above, illustrate Rolle's Theorem for Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(over an appropriate interval.  

3.)  As shown above, illustrate the Mean Value Theorem for Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( from Typesetting:-mrow(Typesetting:-mi( 

 

PRINT OUT THE WORK SHEET AND HAND IN THURSDAY. 

> Roots(cos(x),-5..5);
 

 

[`+`(`-`(`*`(`/`(3, 2), `*`(Pi)))), `+`(`-`(`*`(`/`(1, 2), `*`(Pi)))), `+`(`*`(`/`(1, 2), `*`(Pi))), `+`(`*`(`/`(3, 2), `*`(Pi)))] (3.2)