Q. 69 For each function f and interval... [FREE SOLUTION]

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Chapter 4: Q. 69 (page 386)

For each function f and interval [a, b] given in Exercises 68鈥�73, find a real number c 鈭� (a, b) such that f(c) is the average value of f on [a, b]. Then use a graph of f to verify that your answer is reasonable.
$f (x) = 2 x^{2} + 1, [a, b] = [鈭� 1, 2]$

Short Answer

Expert verified

The real number is $1$ and the average value is localid="1648721084595" $3$ and it is verified from the graph.

Step by step solution

Step 1. Given Information.

The given function and the interval is $f (x) = 2 x^{2} + 1, [a, b] = [鈭� 1, 2] .$

Step 2. Finding a real number c ∈ (a, b).

To find a real number c 鈭� (a, b) such that f(c) is the average value of f on [a, b]. We will use the average value formula: $\frac{1}{b - a} {鈭�}_{a}^{b} f (x) d x .$

$\frac{1}{b - a} {鈭�}_{a}^{b} f (x) d x = \frac{1}{2 - (- 1)} {鈭�}_{- 1}^{2} (2 x^{2} + 1) d x = \frac{1}{3} {[\frac{2 x^{3}}{3} + x]}_{- 1}^{2} = \frac{1}{3} [\frac{2 {(2)}^{3}}{3} + 2 - (\frac{2 {(- 1)}^{3}}{3} + (- 1))] = \frac{1}{3} [\frac{16}{3} + 2 + \frac{5}{3}] = \frac{1}{3} [\frac{16 + 6 + 5}{3}] = \frac{1}{3} [\frac{27}{3}] = 3$

Thus, the average value is $3 .$

Now,

$f (c) = 3 2 {(c)}^{2} + 1 = 3 2 c^{2} = 2 c^{2} = 1 c = 卤 1$

The real number will be $1, 1 鈭� [- 1, 2] .$

Step 3. Verification.

By using the graphing utility, the graph off is

From the graph, we can depict that $f (c) = 3 .$ Thus, the real number is $1, 1 鈭� [- 1, 2] .$ Hence, the answer is right.

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91影视

For each function f and interval [a, b] given in Exercises 68鈥�73, find a real number c 鈭� (a, b) such that f(c) is the average value of f on [a, b]. Then use a graph of f to verify that your answer is reasonable.
$f (x) = 2 x^{2} + 1, [a, b] = [鈭� 1, 2]$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding a real number c ∈ (a, b).

Step 3. Verification.

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91影视

For each function f and interval [a, b] given in Exercises 68鈥�73, find a real number c 鈭� (a, b) such that f(c) is the average value of f on [a, b]. Then use a graph of f to verify that your answer is reasonable.f(x)=2x2+1,[a,b]=[鈭�1,2]

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding a real number c ∈ (a, b).

Step 3. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Geometry

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

For each function f and interval [a, b] given in Exercises 68鈥�73, find a real number c 鈭� (a, b) such that f(c) is the average value of f on [a, b]. Then use a graph of f to verify that your answer is reasonable.
$f (x) = 2 x^{2} + 1, [a, b] = [鈭� 1, 2]$