10 points) Let f(x) = x² – 1. Find the equation of the line that is tangent to the graph of f t the point where x = 1. [Note: you may use derivative rules to find f'(1)]
10 points) Let f(x) = x² – 1. Find the equation of the line that is tangent to the graph of f t the point where x = 1. [Note: you may use derivative rules to find f'(1)]
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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![**Problem:** (10 points) Let \( f(x) = x^2 - 1 \). Find the equation of the line that is tangent to the graph of \( f \) at the point where \( x = 1 \). [Note: you may use derivative rules to find \( f'(1) \)]
**Solution Approach:**
To find the equation of the tangent line, follow these steps:
1. **Find the derivative of \( f(x) \):**
\[
f'(x) = \frac{d}{dx}(x^2 - 1) = 2x
\]
2. **Evaluate the derivative at \( x = 1 \):**
\[
f'(1) = 2 \times 1 = 2
\]
3. **Find \( f(1) \) to get the y-coordinate of the tangent point:**
\[
f(1) = 1^2 - 1 = 0
\]
4. **Use the point-slope form of the line equation:**
\[
y - y_1 = m(x - x_1)
\]
Here, \( m = 2 \), \( x_1 = 1 \), and \( y_1 = 0 \).
5. **Substitute the known values into the formula:**
\[
y - 0 = 2(x - 1)
\]
\[
y = 2x - 2
\]
**Conclusion:**
The equation of the tangent line to the graph of \( f(x) = x^2 - 1 \) at the point where \( x = 1 \) is \( y = 2x - 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1bff4cd-6298-4ffa-baed-93dcf4f2093d%2Fd49d9c8c-f441-411f-af16-55f4fdba1d4c%2Fbgij3ti_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:** (10 points) Let \( f(x) = x^2 - 1 \). Find the equation of the line that is tangent to the graph of \( f \) at the point where \( x = 1 \). [Note: you may use derivative rules to find \( f'(1) \)]
**Solution Approach:**
To find the equation of the tangent line, follow these steps:
1. **Find the derivative of \( f(x) \):**
\[
f'(x) = \frac{d}{dx}(x^2 - 1) = 2x
\]
2. **Evaluate the derivative at \( x = 1 \):**
\[
f'(1) = 2 \times 1 = 2
\]
3. **Find \( f(1) \) to get the y-coordinate of the tangent point:**
\[
f(1) = 1^2 - 1 = 0
\]
4. **Use the point-slope form of the line equation:**
\[
y - y_1 = m(x - x_1)
\]
Here, \( m = 2 \), \( x_1 = 1 \), and \( y_1 = 0 \).
5. **Substitute the known values into the formula:**
\[
y - 0 = 2(x - 1)
\]
\[
y = 2x - 2
\]
**Conclusion:**
The equation of the tangent line to the graph of \( f(x) = x^2 - 1 \) at the point where \( x = 1 \) is \( y = 2x - 2 \).
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