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 Statement:**
Find the limit algebraically.
\[
\lim_{{h \to 0}} \frac{{f(-6 + h) - f(-6)}}{h}
\]
where
\[
f(x) = -8x^2 + 8x - 7
\]
**Explanation:**
This problem involves finding the derivative of the function \( f(x) = -8x^2 + 8x - 7 \) at the point \( x = -6 \) using the definition of the derivative. The expression given is the formula for the derivative of a function at a specific point, known as the difference quotient.
**Steps to Solve:**
1. **Substitute \( x = -6 + h \) into the function:**
\[
f(-6 + h) = -8(-6 + h)^2 + 8(-6 + h) - 7
\]
2. **Simplify \( f(-6 + h) \).**
3. **Calculate \( f(-6) \).**
\[
f(-6) = -8(-6)^2 + 8(-6) - 7
\]
4. **Substitute these values into the difference quotient:**
\[
\frac{{f(-6 + h) - f(-6)}}{h}
\]
5. **Simplify the expression and take the limit as \( h \to 0 \).**
This process will yield the derivative of the function at \( x = -6 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fa35088-fa5f-472d-877a-bca1a109875f%2Fedc0f48b-32ca-450b-b262-f4d93158ead6%2Fdr4kihy_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the limit algebraically.
\[
\lim_{{h \to 0}} \frac{{f(-6 + h) - f(-6)}}{h}
\]
where
\[
f(x) = -8x^2 + 8x - 7
\]
**Explanation:**
This problem involves finding the derivative of the function \( f(x) = -8x^2 + 8x - 7 \) at the point \( x = -6 \) using the definition of the derivative. The expression given is the formula for the derivative of a function at a specific point, known as the difference quotient.
**Steps to Solve:**
1. **Substitute \( x = -6 + h \) into the function:**
\[
f(-6 + h) = -8(-6 + h)^2 + 8(-6 + h) - 7
\]
2. **Simplify \( f(-6 + h) \).**
3. **Calculate \( f(-6) \).**
\[
f(-6) = -8(-6)^2 + 8(-6) - 7
\]
4. **Substitute these values into the difference quotient:**
\[
\frac{{f(-6 + h) - f(-6)}}{h}
\]
5. **Simplify the expression and take the limit as \( h \to 0 \).**
This process will yield the derivative of the function at \( x = -6 \).
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