Purpose

The purpose of this week's lab is to investigate the Mean Value Theorems (MVTs) for Derivatives and Integrals. This investigation will emphasize the hypotheses and conclusions of the two theorems and the connections between the two results. This will be done by way of several examples. The main Maple tools that will be used are the MeanValueTheorem  and FunctionAverage  commands provided in the Student[Calculus1]  package.

This lab can be prepared in any format -- Word document, Maple worksheet, text file, handwritten (provided it is neat!). All reports are due by 5:00pm Thursday, November 21, 2002.

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Example:

>	restart; with( plots ): with( Student[Calculus1] ):

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Recall the statements of the two Mean Value Theorems:

To illustrate these two results, consider the function  on the interval [ 0, 10 ].

>	f := sin(x) + x/2;

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a := 0;

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b := 10;

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The hypotheses for the MVT for Derivatives are clearly satisfied.  A visualization of the MVT for Derivatives applied to this problem can be obtained with the MeanValueTheorem with optional argument output=plot :

>	MeanValueTheorem( f, x=a..b, output=plot );

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To obtain the (three) values of  guaranteed by the MVT for Derivatives, change the optional argument to output=points :

>	mvtDpts := MeanValueTheorem( f, x=a..b, output=points );

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The points on the curve can be found using a loop such as the following:

>	for c in mvtDpts do   pt := [ c, eval( f, x=c ) ];   print( pt = evalf( pt ) ); end do:

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Turning our attention to the MVT for Integrals, the hypotheses are again satisfied.  The FunctionAverage  command with optional argument output=plot  produces a picture showing the rectangle whose ``area'' agrees with the ``area'' under the curve.  (I like to include the additional argument averageoptions=[color=cyan,filled=true]  to emphasize the ``area'' of the rectangle. The view  optional argument is included to overcome an apparent bug in FunctionAverage .)

>	FunctionAverage( f, x=a..b, output=plot,                  averageoptions=[color=cyan,filled=true],                  view=[0..10, 0..6] );

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The height of the rectangle, i.e., the average value of the function over this interval, can be obtained with the FunctionAverage  command with optional argument output=value :

>	ht := FunctionAverage( f, x=a..b, output=value );

The (in this case unique) value of  in this interval where the function has this height can generally be found with the command

>	mvtIpts := solve( f=ht, x );

In this case, the point, c, is approximately

>	c := evalf( mvtIpts );

(In examples with more than one solution it might help to use the allvalues  command.)

The point on the graph corresponding to this value of  is found to be

>	pt := [ c, eval( f, x=c ) ];

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This result can be verified by noting that

>	evalf( ht ) = pt[2];

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Exercise 1:

Note that there is no connection between the  values for the two Mean Value Theorems when applied to the same function (as in the Example). This is to be expected. However, it is possible to find a related function, say , with the property that the MVT for Derivatives applied to  on [ , ] and the MVT for Integrals applied to  on [ , ] both produce the same  values.

a. With , find a function  that satisfies the condition in the previous paragraph. That is,

         f'(c) =    if and only if     =             

>	f := sin(x) + x/2;

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b. With , find a function  with the property that the MVT for Integrals applied to  on [ , ] and the MVT for Derivatives applied to  on [ , ] produce the same  values.  That is,

              =      if and only if   F'(c) =           

>	f := sin(x) + x/2;

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Your lab report should clearly indicate the functions  and  with appropriate Maple commands (and output) showing that your functions satisfy the stated conditions.

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Exercise 2:

What is the general relationship between the function  and the functions  and  found in Exercise 1?

Your answer to this questions must be one or more complete English sentences, using appropriate mathematical terminology, that answer this question.

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Exercise 3:

Consider the function  on the interval [ -1.5, 1.5 ].

>	f := abs( x^2-1 ); a := -1.5; b := 1.5;

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Which, if either, of the Mean Value Theorems can be applied to this function?

For each MVT that applies, list all values of c that satisfy the conclusion of the theorem. For each MVT that does not apply, explicitly indicate all hypotheses that are not satisfied.

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Your response to this question should consist of at least two complete English sentences, using appropriate mathematical terminology and plots, that explain your answer to the question.