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1. The volume of a sphere is changing at 10 inches per second. How fast is the radius changing at the moment the radius is 7 inches?

1. The volume of a sphere is changing at 10 inches per second. How fast is the radius changing at the moment the radius is 7 inches?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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Decide if each of the following problems is a linearization problem, an optimization prob- lem, a related rates problem, a graphing problem, or a Newton's method problem. Identify. Label the problem as one of the five types or explain that it is not exactly one of those types (and how you know). Begin solution. For each problem, start the solution. Linearization. State what function you would linearize and where you would linearize it. Optimization. Draw a diagram to define your variables. Write the function you would optimize, including the domain of the function. Related Rates. Draw a diagram to define your variables. Write an equation that relates your variables. Graphing. State the function, its domain, and any critical points in the domain. Newton's method. State the function to which you would apply Newton's method, your initial point, and the Newton iteration. Other. Explain how you might use calculus to solve this problem. 1. The volume of a sphere is changing at 10 inches per second. How fast is the radius changing at the moment the radius is 7 inches? 2. Estimate a solution to In(x? + 5) = 1 using the tools of calculus. 3. Estimate arctan(0.998) using tools of calculus. 4. When you study, you add knowledge to your mind, but the rate at which it accumulates varies according to the conditions – in fact, some conditions can make this accumulation rate negative! Here's what happens when you study all night: the rate of knowledge accumulation at time t (measured in hours after 9: 00 PM) is given by the formula j(t) = 흡?-+ 2 nuggets/hour (nuggets of wisdom is the standard unit for knowledge). At what time are you accu- mulating knowledge at the greatest rate?
Transcribed Image Text:Decide if each of the following problems is a linearization problem, an optimization prob- lem, a related rates problem, a graphing problem, or a Newton's method problem. Identify. Label the problem as one of the five types or explain that it is not exactly one of those types (and how you know). Begin solution. For each problem, start the solution. Linearization. State what function you would linearize and where you would linearize it. Optimization. Draw a diagram to define your variables. Write the function you would optimize, including the domain of the function. Related Rates. Draw a diagram to define your variables. Write an equation that relates your variables. Graphing. State the function, its domain, and any critical points in the domain. Newton's method. State the function to which you would apply Newton's method, your initial point, and the Newton iteration. Other. Explain how you might use calculus to solve this problem. 1. The volume of a sphere is changing at 10 inches per second. How fast is the radius changing at the moment the radius is 7 inches? 2. Estimate a solution to In(x? + 5) = 1 using the tools of calculus. 3. Estimate arctan(0.998) using tools of calculus. 4. When you study, you add knowledge to your mind, but the rate at which it accumulates varies according to the conditions – in fact, some conditions can make this accumulation rate negative! Here's what happens when you study all night: the rate of knowledge accumulation at time t (measured in hours after 9: 00 PM) is given by the formula j(t) = 흡?-+ 2 nuggets/hour (nuggets of wisdom is the standard unit for knowledge). At what time are you accu- mulating knowledge at the greatest rate?
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