Finding the derivative and finding the equation of the tangent line Finding f'(x) from the Definition of Derivative The four steps used to find the derivative f'(x) for a function y = f(x) are summarized here. 1. Find and simplify f(x + h). 2. Find and simplify f(x + h) − f(x). 3. Find and simplify the quotient Part-b f(x +h)-f(x) h 4. Find the limit as h approaches 0; f'(x) = lim Part-a. f(x) = 2x² Find the f'(x) using the definition f'(x) = Limo f(x+h)-f(x) h After find in the f (x) in Part-a f(x +h)-f(x) h a. find f'(2) b. find f(2) c. Find the equation of the tangent line at x= 2 if this limit exists. (Use the steps given above and the lesson posted)
Finding the derivative and finding the equation of the tangent line Finding f'(x) from the Definition of Derivative The four steps used to find the derivative f'(x) for a function y = f(x) are summarized here. 1. Find and simplify f(x + h). 2. Find and simplify f(x + h) − f(x). 3. Find and simplify the quotient Part-b f(x +h)-f(x) h 4. Find the limit as h approaches 0; f'(x) = lim Part-a. f(x) = 2x² Find the f'(x) using the definition f'(x) = Limo f(x+h)-f(x) h After find in the f (x) in Part-a f(x +h)-f(x) h a. find f'(2) b. find f(2) c. Find the equation of the tangent line at x= 2 if this limit exists. (Use the steps given above and the lesson posted)
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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![## Finding the Derivative and Finding the Equation of the Tangent Line
### Finding \( f'(x) \) from the Definition of Derivative
The four steps used to find the derivative \( f'(x) \) for a function \( y = f(x) \) are summarized as follows:
1. **Find and simplify** \( f(x + h) \).
2. **Find and simplify** \( f(x + h) - f(x) \).
3. **Find and simplify the quotient** \( \frac{f(x + h) - f(x)}{h} \).
4. **Find the limit as \( h \) approaches 0**:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}, \text{ if this limit exists.}
\]
(Refer to the lesson and steps given above for detailed explanation and examples.)
### Part -a
Given the function:
\[
f(x) = 2x^2
\]
#### Task:
Find the derivative \( f'(x) \) using the definition:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \quad \text{(Use the steps given above and the lesson posted)}
\]
### Part -b
After finding \( f'(x) \) in Part-a, complete the following tasks:
a. Find \( f'(2) \).
b. Find \( f(2) \).
c. Find the equation of the tangent line at \( x = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf434c73-38f1-4a89-b981-edd99d760c82%2Ffce8f79c-f11e-4b45-bc1a-79eac65e7a53%2Fvq2ffak_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Finding the Derivative and Finding the Equation of the Tangent Line
### Finding \( f'(x) \) from the Definition of Derivative
The four steps used to find the derivative \( f'(x) \) for a function \( y = f(x) \) are summarized as follows:
1. **Find and simplify** \( f(x + h) \).
2. **Find and simplify** \( f(x + h) - f(x) \).
3. **Find and simplify the quotient** \( \frac{f(x + h) - f(x)}{h} \).
4. **Find the limit as \( h \) approaches 0**:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}, \text{ if this limit exists.}
\]
(Refer to the lesson and steps given above for detailed explanation and examples.)
### Part -a
Given the function:
\[
f(x) = 2x^2
\]
#### Task:
Find the derivative \( f'(x) \) using the definition:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \quad \text{(Use the steps given above and the lesson posted)}
\]
### Part -b
After finding \( f'(x) \) in Part-a, complete the following tasks:
a. Find \( f'(2) \).
b. Find \( f(2) \).
c. Find the equation of the tangent line at \( x = 2 \).
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