Q 80. Consider the function f(x)=2+tan... [FREE SOLUTION]

91影视

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 $c 鈭� [- 4, 4]$ 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.

$\begin{array}{rcl} 鈭� (2 + \tan^{- 1} x) d x & = & 鈭� 2 d x + 鈭� \tan^{- 1} x d x \\ = & 2 x + [\tan^{- 1} x 鈭� d x - 鈭� \{\frac{d}{d x} (\tan^{- 1} x) 鈭� d x\} d x] \\ = & 2 x + [\tan^{- 1} x 路 x - 鈭� \frac{1}{x^{2} + 1} 路 x d x] \\ = & 2 x + x 路 \tan^{- 1} x - \frac{1}{2} 鈭� \frac{1}{t} d t 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� \{x^{2} + 1 = t, 2 x d x = d t, x d x = \frac{1}{2} d t\} \\ = & 2 x + x 路 \tan^{- 1} x - \frac{1}{2} \ln |t| \\ = & 2 x + x 路 \tan^{- 1} x - \frac{1}{2} \ln |x^{2} + 1| 鈰� 鈰� 鈰� 鈰� \{t = x^{2} + 1\} \end{array}$

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

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

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

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

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 $0 鈭� [- 4, 4]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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 $c 鈭� [- 4, 4]$ 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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

91影视

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鈭�-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

Mechanics Maths

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Calculus

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 $c 鈭� [- 4, 4]$ art which $f (x)$ achieves its average value.