60–62. {Use of Tech} Multiple tangent lines Complete the following steps. b. Graph the tangent lines on the given graph.x+y³−y=1; x=1

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.8.60b

Chapter 3, Problem 3.8.60b

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1

Verified step by step guidance

First, understand that the problem involves finding the tangent lines to the curve defined by the equation x + y³ - y = 1 at the point where x = 1.

To find the tangent line, we need to determine the derivative of y with respect to x, which involves implicit differentiation since y is not isolated.

Differentiate both sides of the equation x + y³ - y = 1 with respect to x. Remember that when differentiating y terms, you must apply the chain rule and multiply by dy/dx.

After differentiating, solve for dy/dx to find the slope of the tangent line at the point where x = 1.

Finally, use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope found from dy/dx, and (x₁, y₁) is the point on the curve, to graph the tangent lines on the given graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative. Understanding how to find and graph tangent lines is essential for analyzing the behavior of functions.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In the context of the given equation, x + y³ - y = 1, implicit differentiation allows us to find the derivative of y with respect to x, which is necessary for determining the slope of the tangent line at a specific point.

Graphing Techniques

Graphing techniques involve plotting points, lines, and curves on a coordinate plane to visually represent mathematical relationships. For the problem at hand, understanding how to accurately graph the original equation and the tangent lines is crucial for visualizing their interactions and confirming the correctness of the tangent line calculations.

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