Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please explain how to solve, thank you!
![## Problem Statement
Use integration to find \( f(4) \) if \(\frac{df}{dx} = 2x^2\) and \( f(0) = 5 \).
### Options:
- ⃝ 42.667
- ⃝ 43.667
- ⃝ 47.667
- ⃝ 48.667
## Explanation:
We will solve the problem step-by-step using the given differential equation and initial condition:
\[ \frac{df}{dx} = 2x^2 \]
and
\[ f(0) = 5. \]
First, integrate \(\frac{df}{dx}\) with respect to \(x\) to find \( f(x) \):
\[
f(x) = \int 2x^2 \, dx.
\]
\[
f(x) = \frac{2x^3}{3} + C,
\]
where \( C \) is the constant of integration.
To find \( C \), use the initial condition \( f(0) = 5 \):
\[
f(0) = \frac{2(0)^3}{3} + C = 5,
\]
\[
C = 5.
\]
Thus, the function \( f(x) \) is:
\[
f(x) = \frac{2x^3}{3} + 5.
\]
Now, to find \( f(4) \):
\[
f(4) = \frac{2(4)^3}{3} + 5,
\]
\[
f(4) = \frac{2 \times 64}{3} + 5,
\]
\[
f(4) = \frac{128}{3} + 5,
\]
\[
f(4) = 42.67 + 5 = 47.67.
\]
Thus, \( f(4) \) is approximately 47.67.
The correct answer is ⃝ 47.667.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F883933e7-6864-477d-8bb2-712659e12b95%2F101f3188-9021-4451-9245-c33f4625301d%2Flc2afzi_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Use integration to find \( f(4) \) if \(\frac{df}{dx} = 2x^2\) and \( f(0) = 5 \).
### Options:
- ⃝ 42.667
- ⃝ 43.667
- ⃝ 47.667
- ⃝ 48.667
## Explanation:
We will solve the problem step-by-step using the given differential equation and initial condition:
\[ \frac{df}{dx} = 2x^2 \]
and
\[ f(0) = 5. \]
First, integrate \(\frac{df}{dx}\) with respect to \(x\) to find \( f(x) \):
\[
f(x) = \int 2x^2 \, dx.
\]
\[
f(x) = \frac{2x^3}{3} + C,
\]
where \( C \) is the constant of integration.
To find \( C \), use the initial condition \( f(0) = 5 \):
\[
f(0) = \frac{2(0)^3}{3} + C = 5,
\]
\[
C = 5.
\]
Thus, the function \( f(x) \) is:
\[
f(x) = \frac{2x^3}{3} + 5.
\]
Now, to find \( f(4) \):
\[
f(4) = \frac{2(4)^3}{3} + 5,
\]
\[
f(4) = \frac{2 \times 64}{3} + 5,
\]
\[
f(4) = \frac{128}{3} + 5,
\]
\[
f(4) = 42.67 + 5 = 47.67.
\]
Thus, \( f(4) \) is approximately 47.67.
The correct answer is ⃝ 47.667.
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