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Find an equation of the normal line to the parabola y = x² - 6x + 4 that is parallel to the line x - 4y = 6.

Find an equation of the normal line to the parabola y = x² - 6x + 4 that is parallel to the line x - 4y = 6.

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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**Problem Statement:** Find an equation of the normal line to the parabola \( y = x^2 - 6x + 4 \) that is parallel to the line \( x - 4y = 6 \). **Solution:** The equation of the line is denoted by \( y = \) followed by a box for the answer. An incorrect symbol (red X) indicates the solution entered was not correct. **Key Concepts:** 1. **Normal Line:** A line perpendicular to the tangent of a curve at a given point. 2. **Parallel Lines:** Lines with identical slopes but different y-intercepts. **Process Overview:** To solve for the normal line: 1. Find the derivative of the parabola to determine the slope of the tangent. 2. Use the negative reciprocal of the tangent slope for the normal line. 3. Ensure the normal line's slope matches the slope of the given line \( x - 4y = 6 \). **Final Objective:** Find the equation of the line \( y = \cdots \) such that it satisfies all given conditions.
Transcribed Image Text:**Problem Statement:** Find an equation of the normal line to the parabola \( y = x^2 - 6x + 4 \) that is parallel to the line \( x - 4y = 6 \). **Solution:** The equation of the line is denoted by \( y = \) followed by a box for the answer. An incorrect symbol (red X) indicates the solution entered was not correct. **Key Concepts:** 1. **Normal Line:** A line perpendicular to the tangent of a curve at a given point. 2. **Parallel Lines:** Lines with identical slopes but different y-intercepts. **Process Overview:** To solve for the normal line: 1. Find the derivative of the parabola to determine the slope of the tangent. 2. Use the negative reciprocal of the tangent slope for the normal line. 3. Ensure the normal line's slope matches the slope of the given line \( x - 4y = 6 \). **Final Objective:** Find the equation of the line \( y = \cdots \) such that it satisfies all given conditions.
**Question: Analyzing Horizontal Tangents** For what values of \( x \) does the graph of \( f(x) = 2x^3 + 5x^2 + 2x + 5 \) have a horizontal tangent? (Enter your answers as a comma-separated list. Round your answers to three decimal places.) x = [ ]✖️ **Explanation:** To find where the function \( f(x) = 2x^3 + 5x^2 + 2x + 5 \) has a horizontal tangent, we need to determine where its derivative \( f'(x) \) equals zero. A horizontal tangent occurs when the slope is zero, which is determined by setting the derivative to zero and solving for \( x \). **Steps to Solve:** 1. Differentiate \( f(x) \) to find \( f'(x) \). 2. Set \( f'(x) = 0 \) and solve for \( x \). **Solution:** 1. \( f'(x) = 6x^2 + 10x + 2 \). 2. Solve the equation \( 6x^2 + 10x + 2 = 0 \) for \( x \). After solving, provide x values rounded to three decimal places. **Note:** In the original content, a box is provided to input answers, symbolized by a red 'x' next to it, likely indicating an incorrect or pending answer entry.
Transcribed Image Text:**Question: Analyzing Horizontal Tangents** For what values of \( x \) does the graph of \( f(x) = 2x^3 + 5x^2 + 2x + 5 \) have a horizontal tangent? (Enter your answers as a comma-separated list. Round your answers to three decimal places.) x = [ ]✖️ **Explanation:** To find where the function \( f(x) = 2x^3 + 5x^2 + 2x + 5 \) has a horizontal tangent, we need to determine where its derivative \( f'(x) \) equals zero. A horizontal tangent occurs when the slope is zero, which is determined by setting the derivative to zero and solving for \( x \). **Steps to Solve:** 1. Differentiate \( f(x) \) to find \( f'(x) \). 2. Set \( f'(x) = 0 \) and solve for \( x \). **Solution:** 1. \( f'(x) = 6x^2 + 10x + 2 \). 2. Solve the equation \( 6x^2 + 10x + 2 = 0 \) for \( x \). After solving, provide x values rounded to three decimal places. **Note:** In the original content, a box is provided to input answers, symbolized by a red 'x' next to it, likely indicating an incorrect or pending answer entry.
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