problem finds the derivative of a function at a point using the formal definition. The process is broken down into the following steps: 1 Let f(x) be the function x +9 a and b= 1 . f(9+h)-f(9) Then the limit lim h-0 h Evaluate the limit as h→ to calculate f'(9) 0 -1 h-0 ah+b can be simplified to lim - for:
problem finds the derivative of a function at a point using the formal definition. The process is broken down into the following steps: 1 Let f(x) be the function x +9 a and b= 1 . f(9+h)-f(9) Then the limit lim h-0 h Evaluate the limit as h→ to calculate f'(9) 0 -1 h-0 ah+b can be simplified to lim - for:
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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![This problem involves finding the derivative of a function at a point using the formal definition of a derivative. The process is broken down into the following steps:
Let \( f(x) \) be the function \( \frac{1}{x+9} \). Then the limit
\[
\lim_{{h \to 0}} \frac{f(9+h) - f(9)}{h}
\]
can be simplified to
\[
\lim_{{h \to 0}} \frac{-1}{ah + b}
\]
for:
\( a = \underline{\phantom{2}} \)
and
\( b = \underline{\phantom{2}} \).
Evaluate the limit as \( h \to \underline{\phantom{2}} \) to calculate \( f'(9) = \underline{\phantom{2}} \).
---
**Explanation:**
1. *Function Definition:* The function given is \( f(x) = \frac{1}{x + 9} \).
2. *Limit Expression:* The derivative at a point using the formal definition involves the limit of the difference quotient as \( h \) tends towards zero.
3. *Simplification:* The problem asks to simplify the limit expression \(\frac{f(9+h) - f(9)}{h}\) to a simpler form \(\frac{-1}{ah + b}\).
4. *Determining Constants:* The goal is to find the values of constants \( a \) and \( b \) that make this simplification possible and to calculate the derivative \( f'(9) \) by evaluating the limit.
This exercise is a standard approach to finding derivatives using first principles, often introduced in calculus courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b1d143-3544-4565-ae01-c2b2100e15ce%2F2c93ed4e-85d8-4383-a292-32188ba70799%2Fhuwnepc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This problem involves finding the derivative of a function at a point using the formal definition of a derivative. The process is broken down into the following steps:
Let \( f(x) \) be the function \( \frac{1}{x+9} \). Then the limit
\[
\lim_{{h \to 0}} \frac{f(9+h) - f(9)}{h}
\]
can be simplified to
\[
\lim_{{h \to 0}} \frac{-1}{ah + b}
\]
for:
\( a = \underline{\phantom{2}} \)
and
\( b = \underline{\phantom{2}} \).
Evaluate the limit as \( h \to \underline{\phantom{2}} \) to calculate \( f'(9) = \underline{\phantom{2}} \).
---
**Explanation:**
1. *Function Definition:* The function given is \( f(x) = \frac{1}{x + 9} \).
2. *Limit Expression:* The derivative at a point using the formal definition involves the limit of the difference quotient as \( h \) tends towards zero.
3. *Simplification:* The problem asks to simplify the limit expression \(\frac{f(9+h) - f(9)}{h}\) to a simpler form \(\frac{-1}{ah + b}\).
4. *Determining Constants:* The goal is to find the values of constants \( a \) and \( b \) that make this simplification possible and to calculate the derivative \( f'(9) \) by evaluating the limit.
This exercise is a standard approach to finding derivatives using first principles, often introduced in calculus courses.
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