   Chapter 4.2, Problem 16E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function. the secant line through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel?f(x) = e−x, [0, 2]

To determine

To find: The number c that satisfies the conclusion of Mean value Theorem on the given interval and to sketch the function that has secant line through endpoints, tangent line at (c,f(c)) and prove the lines are parallel to each other.

Explanation

Given:

The function f(x)=ex on the given interval [0,2] .

Calculation:

Obtain the derivative of f(x) .

f(x)=ddx(ex)=ex

Replace x by c in f(x) .

f(c)=ec

If a=0 , then the value of f(0) is obtained as follows.

f(a)=f(0)=e0=1

If b=2 , then the value of f(2) is obtained as follows.

f(b)=f(2)=e2

Compute the values of c such that f(c)=f(b)f(a)ba . Substitute the corresponding values.

ec=e2120ec=e212ec=(1e2)2ec=1e22

Take logarithm on both the sides,

lnec=ln(1e22)c=ln(1e22)c=ln(1e22)c=ln(1e22)1

Thus, the value of c=ln(21e2) .

The equation of the secant line through the end points is, yf(a)xa=f(b)f(a)ba .

y1x0=e2e020y1x=e212y1=e212xy=e212x+1

Thus, the required secant line is y=e212x+1

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 