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Find the equations for all the lines tangent to x + y° – y= 1 at x = 1. r+ y³ – y = 1 2

Find the equations for all the lines tangent to x + y° – y= 1 at x = 1. r+ y³ – y = 1 2

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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### Calculus and Analytical Geometry #### Problem 8: Tangent Lines to a Given Curve **Problem Statement:** Find the equations for all the lines tangent to \( x + y^3 - y = 1 \) at \( x = 1 \). **Graphical Representation:** There is a graph provided that depicts the curve defined by the equation \( x + y^3 - y = 1 \). The graph includes both the x-axis and y-axis, with the curve intersecting these axes. Notable features include: - The x-axis is labeled and marked at intervals of 1 unit (1, 2, etc.). - The y-axis is similarly labeled and marked at intervals of 1 unit (-1, 1, etc.). - The curve seems to have an inflection point around \( x = 1 \). The task involves calculating the tangent lines to the curve \( x + y^3 - y = 1 \) specifically at \( x = 1 \). **Approach to Solution:** To find the tangent lines, we must follow these steps: 1. Determine the points on the curve where \( x = 1 \) by solving the equation \( 1 + y^3 - y = 1 \). 2. After finding the relevant \( y \)-values, calculate the derivatives \( \frac{dy}{dx} \) at these points to find the slope of the tangent lines. 3. Use the point-slope form of the equation of a line to write the equations of the tangent lines. This problem will require knowledge of differential calculus, particularly implicit differentiation, as well as algebraic manipulation to solve for the points of tangency and to construct the tangent line equations.
Transcribed Image Text:### Calculus and Analytical Geometry #### Problem 8: Tangent Lines to a Given Curve **Problem Statement:** Find the equations for all the lines tangent to \( x + y^3 - y = 1 \) at \( x = 1 \). **Graphical Representation:** There is a graph provided that depicts the curve defined by the equation \( x + y^3 - y = 1 \). The graph includes both the x-axis and y-axis, with the curve intersecting these axes. Notable features include: - The x-axis is labeled and marked at intervals of 1 unit (1, 2, etc.). - The y-axis is similarly labeled and marked at intervals of 1 unit (-1, 1, etc.). - The curve seems to have an inflection point around \( x = 1 \). The task involves calculating the tangent lines to the curve \( x + y^3 - y = 1 \) specifically at \( x = 1 \). **Approach to Solution:** To find the tangent lines, we must follow these steps: 1. Determine the points on the curve where \( x = 1 \) by solving the equation \( 1 + y^3 - y = 1 \). 2. After finding the relevant \( y \)-values, calculate the derivatives \( \frac{dy}{dx} \) at these points to find the slope of the tangent lines. 3. Use the point-slope form of the equation of a line to write the equations of the tangent lines. This problem will require knowledge of differential calculus, particularly implicit differentiation, as well as algebraic manipulation to solve for the points of tangency and to construct the tangent line equations.
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