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...
Related questions
Question
![**Problem:**
Find and simplify the expression:
\[
\frac{f(x+h) - f(x)}{h}
\]
for each function.
**Given Function:**
\[
f(x) = \frac{1}{x}
\]
**Solution:**
To solve this, you will substitute \( f(x) \) and \( f(x+h) \) into the difference quotient and simplify.
1. Find \( f(x+h) \):
\[
f(x+h) = \frac{1}{x+h}
\]
2. Substitute \( f(x+h) \) and \( f(x) \) into the expression:
\[
\frac{\frac{1}{x+h} - \frac{1}{x}}{h}
\]
3. Simplify the expression:
- Find a common denominator for the fractions in the numerator:
\[
\frac{x - (x+h)}{(x+h)x} = \frac{-h}{(x+h)x}
\]
- Divide by \( h \):
\[
\frac{-h}{h(x+h)x} = \frac{-1}{(x+h)x}
\]
Thus, the simplified form of the expression is:
\[
\frac{-1}{(x+h)x}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25c04d59-ed01-4203-b4a0-5c8469f37968%2Ffc5f083a-a819-4195-8681-9a366d13cb4e%2F9yrfvsv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find and simplify the expression:
\[
\frac{f(x+h) - f(x)}{h}
\]
for each function.
**Given Function:**
\[
f(x) = \frac{1}{x}
\]
**Solution:**
To solve this, you will substitute \( f(x) \) and \( f(x+h) \) into the difference quotient and simplify.
1. Find \( f(x+h) \):
\[
f(x+h) = \frac{1}{x+h}
\]
2. Substitute \( f(x+h) \) and \( f(x) \) into the expression:
\[
\frac{\frac{1}{x+h} - \frac{1}{x}}{h}
\]
3. Simplify the expression:
- Find a common denominator for the fractions in the numerator:
\[
\frac{x - (x+h)}{(x+h)x} = \frac{-h}{(x+h)x}
\]
- Divide by \( h \):
\[
\frac{-h}{h(x+h)x} = \frac{-1}{(x+h)x}
\]
Thus, the simplified form of the expression is:
\[
\frac{-1}{(x+h)x}
\]
![**Task:**
Find and simplify the expression
\[
\frac{f(x+h) - f(x)}{h}
\]
for each function.
**Given Function:**
\( f(x) = 3x^2 - 4x - 5 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25c04d59-ed01-4203-b4a0-5c8469f37968%2Ffc5f083a-a819-4195-8681-9a366d13cb4e%2F3oyyad_processed.png&w=3840&q=75)
Transcribed Image Text:**Task:**
Find and simplify the expression
\[
\frac{f(x+h) - f(x)}{h}
\]
for each function.
**Given Function:**
\( f(x) = 3x^2 - 4x - 5 \)
Expert Solution
Step 1
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Given that,
f(x) =
We have to find the value of
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