Consider the following curve. y = 2x² - 6x + 1 Find the slope m of the tangent line at the point (4, 9). m = Find an equation of the tangent line to the curve at the point (4,9). y =
Consider the following curve. y = 2x² - 6x + 1 Find the slope m of the tangent line at the point (4, 9). m = Find an equation of the tangent line to the curve at the point (4,9). y =
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
I need help with this problem and an explanation of this problem.
![### Tangent Line to a Curve
**Consider the following curve:**
\[ y = 2x^2 - 6x + 1 \]
**Problem 1:**
Find the slope \( m \) of the tangent line at the point (4, 9).
\[ m = \_\_\_\_\_\_\_ \]
**Problem 2:**
Find an equation of the tangent line to the curve at the point (4, 9).
\[ y = \_\_\_\_\_\_\_ \]
### Solution Approach
1. **Finding the Slope \( m \) of the Tangent Line:**
- To find the slope of the tangent line at a specific point, we need to determine the derivative \( \frac{dy}{dx} \) of \( y = 2x^2 - 6x + 1 \).
2. **Derivative Calculation:**
- Differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2x^2 - 6x + 1) = 4x - 6 \]
3. **Slope at the Given Point (4, 9):**
- Substitute \( x = 4 \) into the derivative to find the slope \( m \):
\[ m = 4(4) - 6 = 16 - 6 = 10 \]
- Therefore, the slope \( m \) at the point (4, 9) is:
\[ \boxed{10} \]
4. **Equation of the Tangent Line:**
- Use the point-slope formula for the equation of the tangent line, which is \( y - y_1 = m(x - x_1) \).
- Here, \( (x_1, y_1) = (4, 9) \) and \( m = 10 \).
\[ y - 9 = 10(x - 4) \]
- Simplify to write the equation in slope-intercept form:
\[ y - 9 = 10x - 40 \]
\[ y = 10x - 40 + 9 \]
\[ y = 10x - 31 \]
- Therefore, the equation of the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e135d5c-f867-4afd-9f0d-b505f4f19664%2Ff3ac2e3e-f247-4603-980d-d25af891cf93%2F2tf3n7_processed.png&w=3840&q=75)
Transcribed Image Text:### Tangent Line to a Curve
**Consider the following curve:**
\[ y = 2x^2 - 6x + 1 \]
**Problem 1:**
Find the slope \( m \) of the tangent line at the point (4, 9).
\[ m = \_\_\_\_\_\_\_ \]
**Problem 2:**
Find an equation of the tangent line to the curve at the point (4, 9).
\[ y = \_\_\_\_\_\_\_ \]
### Solution Approach
1. **Finding the Slope \( m \) of the Tangent Line:**
- To find the slope of the tangent line at a specific point, we need to determine the derivative \( \frac{dy}{dx} \) of \( y = 2x^2 - 6x + 1 \).
2. **Derivative Calculation:**
- Differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(2x^2 - 6x + 1) = 4x - 6 \]
3. **Slope at the Given Point (4, 9):**
- Substitute \( x = 4 \) into the derivative to find the slope \( m \):
\[ m = 4(4) - 6 = 16 - 6 = 10 \]
- Therefore, the slope \( m \) at the point (4, 9) is:
\[ \boxed{10} \]
4. **Equation of the Tangent Line:**
- Use the point-slope formula for the equation of the tangent line, which is \( y - y_1 = m(x - x_1) \).
- Here, \( (x_1, y_1) = (4, 9) \) and \( m = 10 \).
\[ y - 9 = 10(x - 4) \]
- Simplify to write the equation in slope-intercept form:
\[ y - 9 = 10x - 40 \]
\[ y = 10x - 40 + 9 \]
\[ y = 10x - 31 \]
- Therefore, the equation of the
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning