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Find the equation of the line, in slope-intercept form, that TANGENT to f(x) = 3x² − 10x +1 at x = 3. A. y 8x26 = = 8x - 22 B. y = C. y=-2x - 8 D. y = -2x 4 E. NO correct choices

Find the equation of the line, in slope-intercept form, that TANGENT to f(x) = 3x² − 10x +1 at x = 3. A. y 8x26 = = 8x - 22 B. y = C. y=-2x - 8 D. y = -2x 4 E. NO correct choices

Calculus: Early Transcendentals
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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### Tangent Line Problem **Question:** Find the equation of the line, in slope-intercept form, that is tangent to \( f(x) = 3x^2 - 10x + 1 \) at \( x = 3 \). **Choices:** A. \( y = 8x - 26 \) B. \( y = 8x - 22 \) C. \( y = -2x - 8 \) D. \( y = -2x - 4 \) E. NO correct choices **Selected Answer:** E (NO correct choices) ### Detailed Solution: To find the equation of the tangent line, follow these steps: 1. **Find the derivative of \( f(x) \), which gives the slope of the tangent line:** \( f(x) = 3x^2 - 10x + 1 \) Using the power rule, the derivative \( f'(x) = 6x - 10 \). 2. **Evaluate the derivative at \( x = 3 \):** \( f'(3) = 6(3) - 10 = 18 - 10 = 8 \) So, the slope of the tangent line at \( x = 3 \) is 8. 3. **Find the y-coordinate of the point on \( f(x) \) at \( x = 3 \):** \( f(3) = 3(3)^2 - 10(3) + 1 = 3(9) - 30 + 1 = 27 - 30 + 1 = -2 \) Therefore, the point of tangency is \( (3, -2) \). 4. **Use the point-slope form of the equation of a line to write the equation of the tangent line:** The point-slope form is: \( y - y_1 = m(x - x_1) \) Where \( m \) is the slope, and \( (x_1, y_1) \) is the point. Substituting \( m = 8 \), \( x_1 = 3 \), and \( y_1 = -2 \): \( y - (-2) = 8(x - 3) \) Simplify to the slope-intercept form
Transcribed Image Text:### Tangent Line Problem **Question:** Find the equation of the line, in slope-intercept form, that is tangent to \( f(x) = 3x^2 - 10x + 1 \) at \( x = 3 \). **Choices:** A. \( y = 8x - 26 \) B. \( y = 8x - 22 \) C. \( y = -2x - 8 \) D. \( y = -2x - 4 \) E. NO correct choices **Selected Answer:** E (NO correct choices) ### Detailed Solution: To find the equation of the tangent line, follow these steps: 1. **Find the derivative of \( f(x) \), which gives the slope of the tangent line:** \( f(x) = 3x^2 - 10x + 1 \) Using the power rule, the derivative \( f'(x) = 6x - 10 \). 2. **Evaluate the derivative at \( x = 3 \):** \( f'(3) = 6(3) - 10 = 18 - 10 = 8 \) So, the slope of the tangent line at \( x = 3 \) is 8. 3. **Find the y-coordinate of the point on \( f(x) \) at \( x = 3 \):** \( f(3) = 3(3)^2 - 10(3) + 1 = 3(9) - 30 + 1 = 27 - 30 + 1 = -2 \) Therefore, the point of tangency is \( (3, -2) \). 4. **Use the point-slope form of the equation of a line to write the equation of the tangent line:** The point-slope form is: \( y - y_1 = m(x - x_1) \) Where \( m \) is the slope, and \( (x_1, y_1) \) is the point. Substituting \( m = 8 \), \( x_1 = 3 \), and \( y_1 = -2 \): \( y - (-2) = 8(x - 3) \) Simplify to the slope-intercept form
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