Textbook Question
Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).
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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 3.11.7a
The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.
Verified step by step guidance1
Start with the formula for the volume of a sphere: V = (4/3)πr^3.
Differentiate both sides of the equation with respect to time t to find the rate of change of volume with respect to time, dV/dt.
Apply the chain rule to differentiate the right side: dV/dt = d/dt[(4/3)πr^3] = (4/3)π * 3r^2 * (dr/dt).
Simplify the expression: dV/dt = 4πr^2 * (dr/dt).
This equation shows that the rate of change of the volume of the sphere with respect to time, dV/dt, is related to the rate of change of the radius with respect to time, dr/dt, by the factor 4πr^2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this context, we are interested in how the volume of a sphere changes with respect to time as the radius changes. By applying the chain rule, we can relate the rates of change of volume and radius.
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Intro To Related Rates
Volume of a Sphere
The volume V of a sphere is given by the formula V = (4/3)πr³, where r is the radius. This formula is essential for deriving the relationship between the volume and the radius. Understanding how volume depends on radius is crucial for applying calculus to find the rate of change of volume with respect to time.
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08:29Example 5: Packaging Design
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. In this problem, we apply the chain rule to express dV/dt in terms of dr/dt. This allows us to find how the volume's rate of change is influenced by the radius's rate of change, linking the two quantities effectively.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>
a. Determine the average velocity of the car during the first 45 minutes of the trip.
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Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
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Textbook Question
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x)=e^−x tan^−1 x on [0,∞)
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Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/ √x; a= 1/4
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Textbook Question
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
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