120 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE. Figure 1.30: The graph of g(x) =x 2/3.

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1 120 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE Figure 1.30: The graph of g(x) =x 2/3. We shall return to local extrema and monotonic functions, and look at them in more depth in Section Exercises Throughout these exercises, there are problems for which the calculation of the derivative, from the definition, is di cult. You may wish to revisit these exercises after you have read the first few sections of Chapter 2, and have some easy rules for calculating derivatives. In each of Exercises 1 through 7, find the critical points and associated critical values, if any, of the given function. 1. f(x) = 1 3 x1/3 x. 2. g(x) = 1 1+x h(x) =x 3 +4x 2 x s(x) =2 p x 4x. 5. g(x) =(x h) 2 + k. 6. j(x) = p 1+x p(x) = p 1 x 2.

2 1.5. EXTREMA AND THE MEAN VALUE THEOREM 121 In each of Exercises 8 through 12, you are given the derivative of a function f. Find the intervals (in the natural domain of f 0 ) on which f is increasing and those on which f is decreasing. You are GIVEN the derivative; you do not need to calculate it. 8 >< x +5 ifx<0; 8. f 0 (x) = x >: 2 +7x 12 if 0 <x<10; 0 if x> f 0 (x) = x 1 x f 0 (x) = p x f 0 (x) = 3p x f 0 (x) = x 3 +3x 2 4x 12 2 x In each of Exercises 13 through 16, you are given a function f and an interval [a, b]. Find a c as is guaranteed to exist by the Mean Value Theorem, i.e., such that a<c<band the slope of the secant line between a and b is the same as the slope of the tangent line at c. 13. f(x) = x 2 + 9, on the interval [0, 3]. 14. f(x) = p x + 1, on the interval [0, 15]. 15. f(x) =x 3 + x 2 6x + 1, on the interval [ 3, 2]. 16. f(x) =3x + 7, on the interval [5, 9]. 17. A man s normal systolic pressure S at age t years is modeled by the formula S =0.006t t Pressure is measured in millimeters of mercury, or mm HG. a. What systolic pressure is predicted for a newborn? b. Use the Mean Value Theorem to show that the model predicts an age c in the interval (20, 30) such that S 0 (c) =0.28. c. Find the c, from part (b), explicitly, by calculating S 0 (t). In Exercises 18 through 21, show that f(x) is a solution to the initial value problem df/dx = p(x), wheref(a) =b.

3 122 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE 18. p(x) =3x 7, f(x) =1.5x 2 7x + 4, f(3) = p(x) = 2 3 x( 1/3) x 2, f(x) =x 2/3 + x 1, f(8) = p(x) =x(x 2 1) 1/2 + 2, f(x) = p x 2 1+2x + 7, f( 1) = p(x) =gx + v 0, f(x) = 1 2 gx2 + v 0 x + h 0, f(0) = h Find all critical points of the function h(x) =x 4 4x 2 on the specified interval. a. ( 3, 1). b. [ 3, 1). c. [ 1, 3]. d. [3, 6]. In each of Exercises 23 through 26, show that the function has exactly one zero in the interval. Accomplish this by using the Intermediate Value Theorem to show there is at least one zero, and Corollary to show the root is unique. 23. a(x) =6x 3 +9x 10, I =( 1, 1). 24. b(x) =x 3 +6x 2 3x 18, I =[1, 2]. 25. c(x) = p 2x x 2, I =[1, 2]. 26. d(x) = 1 3x x, I =[0.25, 2]. 27. Consider the following logic. f(x) is continuous on [a, b], di erentiable on (a, b) and f(a) = f(b). By Rolle s Theorem, there is exactly one value c in (a, b) such that f 0 (c) = 0. Is this statement true? If not, why not? 28. A toy company finds that the number, D, of action figures purchased (the demand), as a function of the price charged p (in dollars) is D(p) = 10p 2 200p , for 0 apple p apple 10. a. Write the function R(p) giving the net revenue the total amount of money earned from sales of the action figures as a function of the price charged. b. Find the intervals on which R(p) is increasing/decreasing. 29. As we will derive (in a slightly more general context) in Example , the function describing the height of an object, above ground-level, in free-fall can be modeled by p(t) = 1 2 gt2 + v 0 t + p 0,

4 1.5. EXTREMA AND THE MEAN VALUE THEOREM 123 where g =9.8, in meters per second per second, v 0 is the initial velocity of the object, in meters per second, and p 0 is the initial height of the object, in meters. Note that, immediately after the object hits the ground, it is no longer in free-fall, and its height is no longer given by p(t). a. A stick is thrown upwards starting at a height of 10 meters, with an initial velocity of 5 m/s. Write the position as a function of time for the stick (for t until the stick hits the ground). 0, and up b. When does the stick hit the ground (i.e., when does it attain its minimum height)? c. Find the intervals on which the height is increasing/decreasing. d. Given this information, when does the stick reach its maximum height? 30. The population of a certain( ant colony in terms of time t (in years) is given by the following t t 2 if 0 <tapple7; piecewise function P (t) = t t if t>7. a. Show that this function is continuous for t>0. b. Find the critical values of the function. c. Describe the intervals on which the function is increasing/decreasing. 31. Suppose that f(x) =sin 2 (x) and g(x) = cos 2 (x). It s a fact that f 0 (x) =g 0 (x), for all x (we ll prove this in later sections). What implication does this have for the relationship between sin 2 (x) and cos 2 (x)? 32. Let f(x) = ( x 1/3 if x<0; x 2 if x 0;. Take I =( 1, 1). a. What conclusion, if any, can be made about f(x) using Corollary ? b. What conclusion, if any, can be made about f(x) using Corollary ? 33. Show by example that there exists a function f that is continuous and strictly increasing on an interval, and that the set of x such that the condition f 0 (x) > 0doesnot hold is infinite. This suggests that Corollary may be generalized further. 34. Suppose that the position function of a particle at time t is given by p(t) = ( t if 0 apple t apple 1; at 2. + b if t>1;

5 124 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE What are a and b if t =1isnot a critical point? 35. Let f 1 (x) = x, f 2 (x) =0.5x , and f 3 (x) =x 2. a. If h(x) =min(f 1 (x),f 2 (x)), what are the critical points of h? b. If j(x) =min(f 1 (x),f 3 (x), what are the critical points of j? 36. Suppose that the position function of a particle at time t is given by p(t) = ( (t 1) 2 if 0 apple t apple 1; at n. if t>1 What must a be if t = 1 is non-critical? 37. Show by example that the Mean Value Theorem does not hold if the hypothesis f is continuous on the closed interval [a, b] is changed to f is continuous on the open interval (a, b). 38. If f is di erentiable on ( 1, 1) and obtains a global extreme value at x, thenf 0 (x) = 0. Is this statement true? Why or why not? 39. A typical liability insurance policy has a deductible, D, and a limit, L. If the total loss costs for a claim are x, then the policyholder must pay the first D dollars. The insurer will then pay the rest of the costs up to the limit L. Responsibility reverts back to the policyholder if the claim payment exceeds D + L. For example, suppose a policy has a deductible of $500 and a limit of $10,000. If a policyholder is in an accident with total loss costs of $12,000, then the policyholder pays a $500 deductible, the insurer pays $10,000, and the policyholder pays the additional $1500. a. Given a policy with deductible D and limit L, letp (x) be the amount of a loss that a policyholder must pay. Write a formula for P (x). b. Let I(x) be the amount of a loss an insurer must pay. Write a formula for I(x). c. Identify the critical points of P (x). Assume a domain of [0, 1). 40. a. Suppose f and g are both continuous on [a, b] and di erentiable on (a, b). Show there exists a number c in (a, b) such that [f(b) f(a)]g 0 (c) =[g(b) g(a)]f 0 (c). This statement is often called the Generalized Mean Value Theorem. Hint: Consider the function h(x) =[f(b) f(a)]g(x) [g(b) g(a)]f(x).

6 1.5. EXTREMA AND THE MEAN VALUE THEOREM 125 b. Show that the Mean Value Theorem is a consequence of the Generalized Mean Value Theorem by setting g(x) = x. 41. Suppose f is continuous on [a, b], di erentiable on (a, b), and that there is some positive number M such that f 0 (x) applem f or all x 2 (a, b). Prove that for all x and y in (a, b), f(x) f(y) applem x y. This statement can be interpreted geometrically by saying the slope of the secant line between two points in (a, b) is uniformly bounded. In general, if a function f has the property that there exists a constant K such that f(x) f(y) applek x y for all x and y in a given domain, we say f is Lipschitz continuous. In the special case that f :[a, b]! [a, b] and K<1, f is called a contraction mapping. 42. Suppose that f and g are continuous on [a, b], di erentiable on (a, b), that f(a) =g(a) and that f 0 (x) <g 0 (x) for all x in (a, b). Prove that g(b) >f(b). Hint: Consider the function h = g f. 43. Prove that p 1+x<1+ x 2 for x>0. Hint: use the previous problem. 44. In this chapter, we used Rolle s Theorem to prove the Mean Value Theorem. Prove Rolle s Theorem using the Mean Value Theorem. 45. Suppose that the average temperature at a certain location is given by T (t) degrees Celsius, where t is measured in years since 1990, and the domain of T is [0, 12]. Find the open intervals on which T is increasing, if T 0 (t) =t 2 8t Suppose that f is di erentiable on an open interval I, and that, for all x in I, f 0 (x) 0 (respectively, f 0 (x) apple 0). Suppose further that there is no subinterval [a, b] of I such that a < band f is constant on I. Prove that f is strictly increasing (respectively, decreasing) on I.