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:**
Find the limit algebraically.
\[
\lim_{{h \to 0}} \frac{{f(1 + h) - f(1)}}{h}
\]
where \( f(x) = -9x^2 + 4x - 4 \).
---
**Solution Explanation:**
To solve this limit problem, we need to find the derivative of \( f(x) \) at \( x = 1 \) using the definition of the derivative:
1. **Substitute \( f(x) \):**
\( f(x) = -9x^2 + 4x - 4 \)
2. **Calculate \( f(1 + h) \):**
- \( f(1 + h) = -9(1 + h)^2 + 4(1 + h) - 4 \)
3. **Simplify \( f(1 + h) \):**
- Expand: \( (1 + h)^2 = 1 + 2h + h^2 \)
- Substitute: \( -9(1 + 2h + h^2) + 4 + 4h - 4 \)
- Simplify: \( -9 - 18h - 9h^2 + 4 + 4h - 4 \)
- Result: \( -9h^2 - 14h - 9 \)
4. **Calculate \( f(1) \):**
- \( f(1) = -9(1)^2 + 4(1) - 4 \)
- Simplify: \( -9 + 4 - 4 = -9 \)
5. **Substitute into limit:**
\[
\lim_{{h \to 0}} \frac{{(-9 - 9h^2 - 14h) - (-9)}}{h}
\]
6. **Simplify the expression:**
- Numerator becomes: \( -9h^2 - 14h \)
- Limit: \(\lim_{{h \to 0}} \frac{{-9h^2 - 14h}}{h} = \lim_{{h \to 0}} (-9h - 14) \)
7. **Evaluate the limit:**
- As \( h \to 0 \), the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fa35088-fa5f-472d-877a-bca1a109875f%2F55302aa6-3c64-4b93-a92e-bac16be85f9d%2Fcacwld_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the limit algebraically.
\[
\lim_{{h \to 0}} \frac{{f(1 + h) - f(1)}}{h}
\]
where \( f(x) = -9x^2 + 4x - 4 \).
---
**Solution Explanation:**
To solve this limit problem, we need to find the derivative of \( f(x) \) at \( x = 1 \) using the definition of the derivative:
1. **Substitute \( f(x) \):**
\( f(x) = -9x^2 + 4x - 4 \)
2. **Calculate \( f(1 + h) \):**
- \( f(1 + h) = -9(1 + h)^2 + 4(1 + h) - 4 \)
3. **Simplify \( f(1 + h) \):**
- Expand: \( (1 + h)^2 = 1 + 2h + h^2 \)
- Substitute: \( -9(1 + 2h + h^2) + 4 + 4h - 4 \)
- Simplify: \( -9 - 18h - 9h^2 + 4 + 4h - 4 \)
- Result: \( -9h^2 - 14h - 9 \)
4. **Calculate \( f(1) \):**
- \( f(1) = -9(1)^2 + 4(1) - 4 \)
- Simplify: \( -9 + 4 - 4 = -9 \)
5. **Substitute into limit:**
\[
\lim_{{h \to 0}} \frac{{(-9 - 9h^2 - 14h) - (-9)}}{h}
\]
6. **Simplify the expression:**
- Numerator becomes: \( -9h^2 - 14h \)
- Limit: \(\lim_{{h \to 0}} \frac{{-9h^2 - 14h}}{h} = \lim_{{h \to 0}} (-9h - 14) \)
7. **Evaluate the limit:**
- As \( h \to 0 \), the
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