Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Mathematics Problem Analysis**
**Problem Statement:**
Given the function \( f(x) = x - 4x^2 \) and \( h \neq 0 \), perform the following tasks and simplify the results.
**Tasks:**
(a) Find and simplify \( f(x + h) \)
(b) Compute and simplify \( \frac{f(x + h) - f(x)}{h} \)
**Solution Approach:**
1. **Task (a): Compute \( f(x + h) \)**
Given \( f(x) = x - 4x^2 \), to find \( f(x + h) \):
\[
f(x + h) = (x + h) - 4(x + h)^2
\]
Expand and simplify the expression:
\[
(x + h) - 4(x^2 + 2xh + h^2)
\]
2. **Task (b): Compute \( \frac{f(x + h) - f(x)}{h} \)**
Using the simplified form of \( f(x + h) \) from part (a):
\[
\frac{[(x + h) - 4(x^2 + 2xh + h^2)] - [x - 4x^2]}{h}
\]
Simplify the numerator before dividing by \( h \).
**Additional Support:**
For further guidance, use the provided resources:
- Read additional explanations: **Read It**
- Practice more examples: **Master It**
By following these steps, students can deepen their understanding of function manipulation and limit-related calculations in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f9e4cd1-015a-4a96-a677-2fe562598422%2Faf9716fb-258d-4e3c-8b54-530cfce3974f%2Flxs7pf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem Analysis**
**Problem Statement:**
Given the function \( f(x) = x - 4x^2 \) and \( h \neq 0 \), perform the following tasks and simplify the results.
**Tasks:**
(a) Find and simplify \( f(x + h) \)
(b) Compute and simplify \( \frac{f(x + h) - f(x)}{h} \)
**Solution Approach:**
1. **Task (a): Compute \( f(x + h) \)**
Given \( f(x) = x - 4x^2 \), to find \( f(x + h) \):
\[
f(x + h) = (x + h) - 4(x + h)^2
\]
Expand and simplify the expression:
\[
(x + h) - 4(x^2 + 2xh + h^2)
\]
2. **Task (b): Compute \( \frac{f(x + h) - f(x)}{h} \)**
Using the simplified form of \( f(x + h) \) from part (a):
\[
\frac{[(x + h) - 4(x^2 + 2xh + h^2)] - [x - 4x^2]}{h}
\]
Simplify the numerator before dividing by \( h \).
**Additional Support:**
For further guidance, use the provided resources:
- Read additional explanations: **Read It**
- Practice more examples: **Master It**
By following these steps, students can deepen their understanding of function manipulation and limit-related calculations in calculus.
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