Exploratory Problems for Chapter 16 531
(c) Find . 3
(d) Let 2 2. What is ? 2
(e) Let 2. Find 2.
9. Below is a graph of on the interval [2, 3]. # is given by #ln on the domain [2, 3].
—2 —1 1 2 3 x
(a) How many zeros does # have? (b) Find # .
(c) Approximate the critical points of #.
(d) Identify the local extrema of #. (Estimate the positions.)
(e) Where does # attain its absolute maximum value? Its absolute minimum value?
(f) Which function has a higher absolute maximum value, or #? Which function has a lower absolute minimum value, or #? Explain.
10. is a continuous function with exactly two zeros, one at 1 and the other at
4. has a local minimum at 3 and a local maximum at 7. These are the only local extrema of . Let 4.
(a) Find in terms of and its derivatives.
(b) Can we determine (deÞnitively) whether has an absolute minimum value on , ? If we can, where is that absolute minimum value attained? Can we determine (deÞnitively) whether has an absolute maximum value? If we can, where is that absolute maximum value attained?
(c) What are the critical points of ?
(d) On what intervals is the graph of increasing? On what intervals is it decreasing? (e) Identify the local maximum and minimum points of .
(f) Can we determine (deÞnitively) whether has an absolute minimum value? If so, can we determine what that value is?
If you havenÕt already done so, step back, take a good look at the problem (a birdÕs-eye view) and make sure your answers make sense.
11. Assume that , , and ! are differentiable. Differentiate * where (a) * !. (Hint: Use the Product Rule twice.)
(b) * ln .
532 CHAPTER 16 Taking the Derivative of Composite Functions
12. Let be the function whose graph is drawn on the axes below.
—6 —5 —4 —3 —2 —1 1 2 3 4 5
(—5, —1) —2
Let ’2, & 2, and +2.
(a) On three separate sets of axes, draw the graphs of the function ’ 2, & 2, and + 2, labeling the -intercepts, the -intercept, and the - and -coordinates of the local extrema.
(b) Suppose we know that
412, 232, 00. Find the following, explaining your reasoning brießy.
i. ’ 2 ii. & 2 iii. + 2
In Problems 13 through 18, Þnd ! . Assume that and are differentiable on , . Your answers may be in terms of , , , and .
13. !ln ln 14. !
17. ! 2 2
18. !3 2
In Problems 19 through 22, Þnd . Take the time to prepare the expression so that it is as simple as possible to differentiate.
19. 3 ln 2
Exploratory Problems for Chapter 16 533
20. 2 35
22. 5 ln 21
In Problems 23 through 29, differentiate. In Problems 23 through 25, assume is differentiable. Your answers may be in terms of and .
23. ln 2
24. 2 2
25. ln 3 2
29. Let .
(a) Use numerical methods to approximate 2.
(b) Refer to your answer to part (a) to show that 1. What is it about that makes it not a power function?
(c) Refer to your answer to part (a) to show that ln . What is it about that makes it not an exponential function?
(d) Challenge: Figure out how to rewrite so you can use the Chain Rule to differ-entiate it.
30. Consider the function 22 1. You must give exact answers for all of the following questions. Show your work. Your work must stand independent of your calculator.
(a) Find all the -intercepts.
(b) Identify the local extrema of .
(c) Sketch a graph of , labeling the -coordinates of all local and global extrema.
(d) Now consider the function . i. What are the critical points of ?
ii. Classify the critical points of .
31. A craftsman is making a mobile consisting of hanging circles each with an inscribed triangle of stained glass. Each piece of stained glass will be an isosceles triangle. Show thatifshewantstomaximizetheamountofstainedglassused,theglasstrianglesshould be equilateral. In other words, show that the isosceles triangle of maximum area that can be inscribed in a circle of radius is an equilateral triangle.
534 CHAPTER 16 Taking the Derivative of Composite Functions
32. A holiday ornament is being constructed by inscribing a right circular cone of brightly coloredmaterialinatransparentsphericalballofradius2inches.Whatisthemaximum possible volume of such an inscribed cone?
33. A craftsman is making a ribbon ornament by inscribing an open hollow cylinder of coloredribboninatransparentsphericalballofradius.Whatisthemaximumsurface area of such a cylinder?
34. Phone cables are to be run from an island to a town on the shore. The island is 5 miles from the shore and 13 miles from the town. The cable will be run in a straight line from the island to the shore and then in another straight line along the shoreline. If it costs 60% more to run the cable under water than it does to run it under the ground, how far should the cable be run along the shore?
35. The volume of a cylindrical tree trunk varies with time. Let give the radius of the trunk at time and let ! give the height of time .
(a) Express the rate of change of ,, the cross-sectional area, with respect to time in terms of and .
(b) Express the rate of change of volume with respect to time in terms of , , !, and ! .
36. Supposetheamountofpowergeneratedbyanenergygeneratingsystemisafunctionof -, thevolumeofwaterßowingthroughthesystem.Thefunctionisgivenby -. The volume of water in the sytem is determined by , the radius of an adjustable valve; - -. The radius varies with time: .
(a) Express , the rate of change of the power with respect to a change in the valveÕs radius, in terms of the functions - and - and their derivatives.
(b) Express , the rate of change of the power with respect to time, in terms of the functions -, -, and and their derivatives.
C H A P T E R
Implicit Differentiation and its Applications
17.1 INTRODUCTORY EXAMPLE
In this section you will add sophistication to your differentiation skills by applying two fundamental principles.
Principle (i): If two expressions are equal, their derivatives are equal.
Principle (ii): Whether is given implicitly or explicitly in terms of ,1 we can differentiate an expression in with respect to by treating the same way we would treat and using the Chain Rule.
Try Example 17.1 below. Do the problem on your own, writing down your solution so you can compare it with the discussion that follows.
Let 1, where 0.
(a) Approximate 2 numerically.
(b) Using appropriate rules of differentiation, Þnd 2 exactly.
(c) Compare your answers to parts (a) and (b). If they are not very close to one another, identify your error.
(a) To approximate the slope of the tangent line to at 2, we Þnd the slope of the secant line through 2, 2 and a nearby point on the graph of , say 2.0001, 2.0001.
1 For example, 3 1 is an equation that gives implicitly in terms of ; 2 3 is an equation that gives explicitly in terms of .