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Q18E

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Found in: Page 209

### Essential Calculus: Early Transcendentals

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

# 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 $$t = \pm \pi$$.

See the step by step solution

## 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 $$f( \pm \pi ) = - 1$$.