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Consider the curve x²y + y²x = 6 A) Find in terms of x and y dy dx B) Write the equation for the tangent line where x = 2 and y = 1. C) Find the coordinates of the point (x, y) where the curve has a horizontal tangent line.

Consider the curve x²y + y²x = 6 A) Find in terms of x and y dy dx B) Write the equation for the tangent line where x = 2 and y = 1. C) Find the coordinates of the point (x, y) where the curve has a horizontal tangent line.

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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Please answer A, B, and C!

**Problem Statement:** Consider the curve \( x^2y + y^2x = 6 \). A) Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). B) Write the equation for the tangent line where \( x = 2 \) and \( y = 1 \). C) Find the coordinates of the point \((x, y)\) where the curve has a horizontal tangent line. **Instructions:** 1. **Part A:** Use implicit differentiation to find the derivative \(\frac{dy}{dx}\) of the given curve. 2. **Part B:** Using the derivative from Part A, compute its value at the point where \(x = 2\) and \(y = 1\) to find the slope of the tangent line at this point. Then, use the point-slope form to write the equation of the tangent line. 3. **Part C:** Identify the conditions for a horizontal tangent line (i.e., where \(\frac{dy}{dx} = 0\)) and solve for the coordinates \((x, y)\) that satisfy this condition on the curve.
Transcribed Image Text:**Problem Statement:** Consider the curve \( x^2y + y^2x = 6 \). A) Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). B) Write the equation for the tangent line where \( x = 2 \) and \( y = 1 \). C) Find the coordinates of the point \((x, y)\) where the curve has a horizontal tangent line. **Instructions:** 1. **Part A:** Use implicit differentiation to find the derivative \(\frac{dy}{dx}\) of the given curve. 2. **Part B:** Using the derivative from Part A, compute its value at the point where \(x = 2\) and \(y = 1\) to find the slope of the tangent line at this point. Then, use the point-slope form to write the equation of the tangent line. 3. **Part C:** Identify the conditions for a horizontal tangent line (i.e., where \(\frac{dy}{dx} = 0\)) and solve for the coordinates \((x, y)\) that satisfy this condition on the curve.
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