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Chapter 3: Q 40. (page 299)
suppose the sides of a cube are expanding at a rate of inches per minute.
How fast is the volume of the cube changing at the moment that the cube has a side length of inches?
The rate of change in volume is.
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Determine whether or not each function satisfies the hypotheses of the Mean Value Theorem on the given interval . For those that do, use derivatives and algebra to find the exact values of all that satisfy the conclusion of the Mean Value Theorem.
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Use the second-derivative test to determine the local extrema of each function in Exercises . If the second-derivative test fails, you may use the first-derivative test. Then verify your algebraic answers with graphs from a calculator or graphing utility. (Note: These are the same functions that you examined with the first-derivative test in Exercises of Section .)
For each set of sign charts in Exercises 53–62, sketch a possible graph of f.
Use a sign chart for to determine the intervals on which each function is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.
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It took Alina half an hour to drive to the grocery store that is 20 miles from her house.
(a) Use the Mean Value Theorem to show that, at some point during her trip, Alina must have been traveling exactly 40 miles per hour.
(b) Why does what you have shown in part (a) make sense in real-world terms?
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