Chapter 5: Q. 81 (page 418)

Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c 鈭� [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.

Short Answer

Expert verified

Part (a) The value is $0 .$

Part (b) The value is $c = 1 .$

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = \frac{\ln x}{x}$

Part (a) Step 2. Explanation.

The average value of the function will be,

$Avg. = \frac{1}{b - a} {鈭�}_{a}^{b} f (x) d x = \frac{1}{2 - \frac{1}{2}} {鈭�}_{\frac{1}{2}}^{2} \frac{\ln x}{x} d x = \frac{2}{3} {[\frac{1}{2} (\ln (x))^{2}]}_{\frac{1}{2}}^{2} + c = \frac{2}{3} [\frac{1}{2} [(\ln (2))^{2}] - {(\ln (\frac{1}{2}))}^{2}] + c = 0$

Part(b) Step 1. Explanation.

Now,

c = 1, f (x) = \frac{\ln 1}{1} = 0

Therefore,

The value of $c = 1 .$

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91影视

Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c 鈭� [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part(b) Step 1. Explanation.

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91影视

Consider the function f(x)=lnxxshown here: (a) Find the average value off(x) on role="math" localid="1649276701979" 12,2. (b) Find a value c鈭�12,2 at which f(x) achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part(b) Step 1. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Logic and Functions

Decision Maths

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c 鈭� [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.