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Chapter 3: Q. 60 (page 249)

Determine whether or not each function fsatisfies the hypotheses of the Mean Value Theorem on the given interval [a,b]. For those that do, use derivatives and algebra to find the exact values of all c鈭�(a,b)that satisfy the conclusion of the Mean Value Theorem.

f(x)=tanx,[a,b]=[-蟺,蟺].

Short Answer

Expert verified

The function f(x)=tanxsatisfies the Mean Value Theorem and the value isc=sec-1(0).

Step by step solution

01

Step 1. Given Information.

The function is,

f(x)=tanx,[a,b]=[-蟺,蟺].

02

Step 2. Mean Value Theorem.

The function f(x)=tanxis continuous and differentiable on [-蟺,蟺]. The Mean Value Theorem applies to this function on the interval [-蟺,蟺].

The slope of the line from (-蟺,f(-蟺))to (蟺,f(蟺))is:

f(蟺)-f(-蟺)蟺-(-蟺)=tan蟺-tan-蟺蟺+蟺=0-02蟺=02蟺=0

By the Mean Value Theorem, there must exist at least one point c鈭�(-蟺,蟺)with f'(c)=0.

We have to find the value of cwith f'(c)=0we solve it:

sec2(x)=0鈬�sec(x)=0鈬�x=sec-1(0)

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Most popular questions from this chapter

Find the possibility graph of its derivative f'.

Find the critical points of the function

f'(x)=(x-1)(x-2)x-3

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

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Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=e3x-e2x
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