Chapter 5: Q 80. (page 429)

Consider the function $f (x) = 2 + \tan^{- 1} x$ shown in the next figure.
(a) Find the average value of $f (x)$ on $[- 4, 4]$ .
(b) Find a value $c \in [- 4, 4]$ art which $f (x)$ achieves its average value.

Short Answer

Expert verified

Part (a). The average value is $2$ .

Part (b). The value of constant is $c = 0$ .

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = 2 + \tan^{- 1} x$ .

Part (a) Step 2. First, we solve the indefinite integral.

$\begin{array}{rcl} \int (2 + \tan^{- 1} x) d x & = & \int 2 d x + \int \tan^{- 1} x d x \\ = & 2 x + [\tan^{- 1} x \int d x - \int \{\frac{d}{d x} (\tan^{- 1} x) \int d x\} d x] \\ = & 2 x + [\tan^{- 1} x \cdot x - \int \frac{1}{x^{2} + 1} \cdot x d x] \\ = & 2 x + x \cdot \tan^{- 1} x - \frac{1}{2} \int \frac{1}{t} d t \dots \dots \dots \dots \dots \dots \dots \{x^{2} + 1 = t, 2 x d x = d t, x d x = \frac{1}{2} d t\} \\ = & 2 x + x \cdot \tan^{- 1} x - \frac{1}{2} \ln |t| \\ = & 2 x + x \cdot \tan^{- 1} x - \frac{1}{2} \ln |x^{2} + 1| \dots \dots \dots \dots \{t = x^{2} + 1\} \end{array}$

Part (a) Step 3. Now find the average value by using the formula.

The average value of a function is $\frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

In this problem $a = - 4, b = 4$ and $f (x) = 2 + \tan^{- 1} x$ .

$\begin{array}{rcl} \frac{1}{4 - (- 4)} \int_{- 4}^{4} (2 + \tan^{- 1} x) d x & = & \frac{1}{8} {[2 x + x \cdot \tan^{- 1} x - \frac{1}{2} \ln |x^{2} + 1|]}_{- 4}^{4} \\ = & \frac{1}{8} [2 (4) + 4 \cdot \tan^{- 1} 4 - \frac{1}{2} \ln |4^{2} + 1|] - \frac{1}{8} [2 (- 4) + (- 4) \cdot \tan^{- 1} (- 4) - \frac{1}{2} \ln |{(- 4)}^{2} + 1|] \\ = & \frac{1}{8} [8 + 4 \cdot \tan^{- 1} 4 - \frac{1}{2} \ln |17|] - \frac{1}{8} [- 8 + (- 4) \cdot \tan^{- 1} (- 4) - \frac{1}{2} \ln |17|] \\ = & \frac{1}{8} \{8 + 8 + 4 \tan^{- 1} 4 - 4 \tan^{- 1} 4 - \frac{1}{2} \ln |17| + \frac{1}{2} \ln |17|\} \\ = & \frac{1}{8} (16) \\ = & 2 \end{array}$

Part (b) Step 1. Given information.

The given function is $f (x) = 2 + \tan^{- 1} x$ .

Part (b) Step 2. Find a value c∈-4,4.

$\begin{array}{rcl} F_{a v g} & = & {[2 + \tan^{- 1} x]}_{x = c} \\ 2 & = & 2 + \tan^{- 1} c \\ \tan^{- 1} c & = & 0 \\ c & = & 0 \end{array}$

Hence, the required value of $c$ is $0 \in [- 4, 4]$ .

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Consider the function $f (x) = 2 + \tan^{- 1} x$ shown in the next figure.
(a) Find the average value of $f (x)$ on $[- 4, 4]$ .
(b) Find a value $c \in [- 4, 4]$ art which $f (x)$ achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. First, we solve the indefinite integral.

Part (a) Step 3. Now find the average value by using the formula.

Part (b) Step 1. Given information.

Part (b) Step 2. Find a value c∈-4,4.

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Consider the function f(x)=2+tan-1xshown in the next figure.(a) Find the average value of f(x)on -4,4.(b) Find a valuec∈-4,4art whichf(x)achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. First, we solve the indefinite integral.

Part (a) Step 3. Now find the average value by using the formula.

Part (b) Step 1. Given information.

Part (b) Step 2. Find a value c∈-4,4.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Geometry

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider the function $f (x) = 2 + \tan^{- 1} x$ shown in the next figure.
(a) Find the average value of $f (x)$ on $[- 4, 4]$ .
(b) Find a value $c \in [- 4, 4]$ art which $f (x)$ achieves its average value.