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Q18E

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Essential Calculus: Early Transcendentals
Found in: Page 209
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\). (Use the graphs and transformations of Sections 1.2.)

18. \(f(t) = \cos t,\;\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\)

The absolute and local maximum occurs at \(t = 0\). The absolute and local minimum occurs at \(t = \pm \pi \).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

The given function is \(f(t) = \cos t,\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\).

Step 2: Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Step 3: Sketch the graph of the function

First draw the graph of the function \(f(t) = \cos t\) as shown below in Figure:

Since the function is restricted to \(\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\), draw the graph of the function in the closed interval of \(\left( {\frac{{ - 3\pi }}{2},\frac{{3\pi }}{2}} \right)\) as shown below in Figure 2 :

Generally the cosine curve attains the maximum at even multiples of \(\pi \) and attains the minimum at odd multiples of \(\pi \) as shown in Figure 1. According to the given interval, from Figure 2, it is observed that the cosine curve attains an absolute maximum at \(t = 0\) and attains an absolute minimum at \(t = \pm \pi \). Therefore, the absolute maximum occurs at \(t = 0\). Since any absolute maximum can be a local maximum, the local maximum is \(f(0) = 1\).

Similarly, the absolute minimum occurs at \(t = \pm \pi \). Since any absolute minimum can be a local minimum, the local minimum is \(f( \pm \pi ) = - 1\).

Most popular questions for Math Textbooks

(a) Use a graph of \[f\] to estimate the maximum and minimum values. Then find the exact values.

(b) Estimate the value of \[x\] at which \[f\] increases most rapidly. Then find the exact value.

\[f\left( x \right) = {x^2}{e^{ - x}}\]

A cone-shaped drinking cup is made from a circular piece of paper of radius \(R\) by cutting out a sector and joining the edges \(CA\) and \(CB\). Find the maximum capacity of such a cup.

Find the absolute maximum and absolute minimum values of on the given interval.

46. \(f(t) = t + \cot \left( {\frac{t}{2}} \right)\), \(\left( {\frac{\pi }{4},\frac{{7\pi }}{4}} \right)\)

The family of bell-shaped curves

\(y = \frac{1}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{(x - \mu )}^2}}}{{2{\sigma ^2}}}}}\)

occurs in probability and statistics, where it is called the normal density function. The constant \(\mu \) is called the mean and the positive constant \(\sigma \) is called the standard deviation. For simplicity, let’s scale the function so as to remove the factor \(1/\left( {\sigma \sqrt {2\pi } } \right)\) and let’s analyze the special case where \(\mu = 0\). So we study the function

\(f\left( x \right) = {e^{ - {x^2}/\left( {2{\sigma ^2}} \right)}}\)

(a) Find the asymptote, maximum value, and inflection points of \(f\).

(b) What role does \(\sigma \) play in the shape of the curve?

(c) Illustrate by graphing four members of this family on the same screen.

A number a is called fixed point of a function f if f(a)=a. Prove that If \({f^\prime }(x) \ne 1\) for all real numbers x, then f has at most one fixed point.
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