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Find the equation of the tangent line to the curve (a lemniscate) 2 (x² + y²)² = 25 (x² - y²) at the point (3, 1).

Find the equation of the tangent line to the curve (a lemniscate) 2 (x² + y²)² = 25 (x² - y²) at the point (3, 1).

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 show all work and steps clearly. The format and steps have not always been clear. Thank you.

**Problem Statement:** Find the equation of the tangent line to the curve (a lemniscate) \[ 2(x^2 + y^2)^2 = 25(x^2 - y^2) \] at the point \((3, -1)\). **Solution Format:** The equation of this tangent line can be written in the form \( y = mx + b \) **Where:** \( m \) is: [Blank space for students to fill] \( b \) is: [Blank space for students to fill] **Instructions:** - Calculate the derivative of the given lemniscate equation to find the slope \( m \) at the point \((3, -1)\). - Use the point-slope form to determine the y-intercept \( b \) once the slope \( m \) is known. - Provide your final answer in the format \( y = mx + b \) filling in the blanks provided.
Transcribed Image Text:**Problem Statement:** Find the equation of the tangent line to the curve (a lemniscate) \[ 2(x^2 + y^2)^2 = 25(x^2 - y^2) \] at the point \((3, -1)\). **Solution Format:** The equation of this tangent line can be written in the form \( y = mx + b \) **Where:** \( m \) is: [Blank space for students to fill] \( b \) is: [Blank space for students to fill] **Instructions:** - Calculate the derivative of the given lemniscate equation to find the slope \( m \) at the point \((3, -1)\). - Use the point-slope form to determine the y-intercept \( b \) once the slope \( m \) is known. - Provide your final answer in the format \( y = mx + b \) filling in the blanks provided.
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