Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
³√x+³√y⁴ = 2;(1,1)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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.
Recommended video:
05:13
Slopes of Tangent Lines
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.
Recommended video:
05:14Finding The Implicit Derivative
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.
Recommended video:
06:15Graphing The Derivative
Related Practice
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. d/dx(tan^−1 x) =sec² x
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Textbook Question
{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.
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Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x = e^y; (2, ln 2)
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Textbook Question
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?
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