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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![**Problem Statement:**
We are given a derivative function \( f'(t) = t^5 + \frac{1}{t^7} \), with the condition \( t > 0 \). Additionally, we know that \( f(1) = 4 \).
**Objective:**
Find the function \( f(t) \).
**Solution Approach:**
To find \( f(t) \), we need to integrate the given derivative \( f'(t) \):
1. **Integrate** \( f'(t) \):
\[
f(t) = \int \left( t^5 + \frac{1}{t^7} \right) \, dt
\]
2. **Integrate Each Term**:
- \(\int t^5 \, dt = \frac{t^6}{6} + C_1\)
- \(\int \frac{1}{t^7} \, dt = \int t^{-7} \, dt = \frac{t^{-6}}{-6} + C_2\)
3. **Combine Integrals**:
\[
f(t) = \frac{t^6}{6} - \frac{1}{6t^6} + C
\]
4. **Apply Initial Condition** \( f(1) = 4 \) to find \( C \):
\[
f(1) = \frac{1^6}{6} - \frac{1}{6 \cdot 1^6} + C = 4 \\
\frac{1}{6} - \frac{1}{6} + C = 4 \\
C = 4
\]
Hence, the function is:
\[
f(t) = \frac{t^6}{6} - \frac{1}{6t^6} + 4
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c35614e-f53f-486b-9e87-afb8487cbd26%2F623dd3df-c5c6-434d-8114-e32ba88ff825%2Flizugf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
We are given a derivative function \( f'(t) = t^5 + \frac{1}{t^7} \), with the condition \( t > 0 \). Additionally, we know that \( f(1) = 4 \).
**Objective:**
Find the function \( f(t) \).
**Solution Approach:**
To find \( f(t) \), we need to integrate the given derivative \( f'(t) \):
1. **Integrate** \( f'(t) \):
\[
f(t) = \int \left( t^5 + \frac{1}{t^7} \right) \, dt
\]
2. **Integrate Each Term**:
- \(\int t^5 \, dt = \frac{t^6}{6} + C_1\)
- \(\int \frac{1}{t^7} \, dt = \int t^{-7} \, dt = \frac{t^{-6}}{-6} + C_2\)
3. **Combine Integrals**:
\[
f(t) = \frac{t^6}{6} - \frac{1}{6t^6} + C
\]
4. **Apply Initial Condition** \( f(1) = 4 \) to find \( C \):
\[
f(1) = \frac{1^6}{6} - \frac{1}{6 \cdot 1^6} + C = 4 \\
\frac{1}{6} - \frac{1}{6} + C = 4 \\
C = 4
\]
Hence, the function is:
\[
f(t) = \frac{t^6}{6} - \frac{1}{6t^6} + 4
\]
Expert Solution
Step 1: Determine the given function
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