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Jse impiicit differentiation to find an equation of the tangent iin to the curve at the given point. x2/3 + y2/3 = 4 (-3/3, 1) (astroid)

Jse impiicit differentiation to find an equation of the tangent iin to the curve at the given point. x2/3 + y2/3 = 4 (-3/3, 1) (astroid)

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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Question
**Problem Statement:** Use implicit differentiation to find an equation of the tangent line to the curve at the given point. **Equation:** \[ x^{2/3} + y^{2/3} = 4 \] **Point:** \[ \left(-3\sqrt{3}, 1\right) \] **Curve Type:** Astroid **Explanation:** This problem involves finding the equation of the tangent line to a specific curve known as an astroid at the given point using implicit differentiation. An astroid is a type of hypocycloid with a distinctive star-like shape. The curve is defined by the equation \( x^{2/3} + y^{2/3} = 4 \). You are tasked with using implicit differentiation techniques to determine the slope of the tangent line at the specified point, \((-3\sqrt{3}, 1)\), and then formulating the equation of the tangent line.
Transcribed Image Text:**Problem Statement:** Use implicit differentiation to find an equation of the tangent line to the curve at the given point. **Equation:** \[ x^{2/3} + y^{2/3} = 4 \] **Point:** \[ \left(-3\sqrt{3}, 1\right) \] **Curve Type:** Astroid **Explanation:** This problem involves finding the equation of the tangent line to a specific curve known as an astroid at the given point using implicit differentiation. An astroid is a type of hypocycloid with a distinctive star-like shape. The curve is defined by the equation \( x^{2/3} + y^{2/3} = 4 \). You are tasked with using implicit differentiation techniques to determine the slope of the tangent line at the specified point, \((-3\sqrt{3}, 1)\), and then formulating the equation of the tangent line.
The image shows a graph of a rational function. The graph is a symmetrical curve that resembles an hourglass shape. The function is plotted on a Cartesian coordinate plane with the x-axis and y-axis intersecting at the origin (0,0). ### Key Features of the Graph: - **Axes Orientation**: The x-axis is labeled with numbers 0 and 8, with arrows indicating the positive direction of both the x-axis and y-axis. - **Curve Shape**: The graph consists of two curved sections: - The upper part of the curve is symmetrical across the y-axis and peaks at the origin. - The lower part mirrors the upper part downward, forming an open loop. - **Intercepts**: The curve crosses the x-axis at two points, including the origin (0,0). This graph is useful for analyzing the properties and behavior of rational functions, especially those involving symmetry and intercepts.
Transcribed Image Text:The image shows a graph of a rational function. The graph is a symmetrical curve that resembles an hourglass shape. The function is plotted on a Cartesian coordinate plane with the x-axis and y-axis intersecting at the origin (0,0). ### Key Features of the Graph: - **Axes Orientation**: The x-axis is labeled with numbers 0 and 8, with arrows indicating the positive direction of both the x-axis and y-axis. - **Curve Shape**: The graph consists of two curved sections: - The upper part of the curve is symmetrical across the y-axis and peaks at the origin. - The lower part mirrors the upper part downward, forming an open loop. - **Intercepts**: The curve crosses the x-axis at two points, including the origin (0,0). This graph is useful for analyzing the properties and behavior of rational functions, especially those involving symmetry and intercepts.
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