196 Chapter 4: Applications of Derivatives

Exercises 4.2

Checking the Mean Yalue Theoll!m

constant C. Since the gmph of I passes through the point (0, 2), the value of C is deter­mined from the condition that 1(0) = 2:

1(0) = -cos (0) + C = 2, so C = 3.

The function is/(x) = -cosx + 3. • Finding Velocity and Position from Acceleration

We can use Corollary 2 to find the velocity and position functions of an object moving along a vertical line. Assume the object or body is falling freely from rest with acceleration 9.8 m/sec2

• We assume the position 8(t) of the body is measured positive downward from the rest position (so the vertical coordinate line points downward, in the direction of the motion, with the rest position at 0).

We know that the velocity v(t) is some function whose derivative is 9.8. We also know that the derivative of g(t) = 9.8t is 9.8. By Corollary 2,

v(t) = 9.8t + C

for some constant C. Since the body falls from rest, v(O) = o. Thus

9.8(0) + C = 0, and C = o. The velocity function must be v(t) = 9.8t. What about the position function s(t)?

We know that 8(t) is some function whose derivative is 9.8t. We also know that the de­rivative of I(t) = 4.9t2 is 9.8t. By Corollary 2,

8(t) = 4.9t2 + C

for some constant C. Since s( 0) = 0,

4.9(0)' + C = 0, and C = o. The position function is s(t) = 4.9t2 until the body hits the ground

The ability to find functions from their mtes of change is one of the very powerful tools of calculus. As we will see, it lies at the heart of the mathematical developments in Chapter 5.

Find the value or values of c that satisfY the equatioo Which of the functioos in Exercises 7-12 satisfY the hypotheses of the Mean Value Theorem on the giveo interval, and which do not? Give reasons for your answers.

f(b) - f(a) ~ f'(c) b-a

in the conclusion of the Mean Value Theorem for the functions and in­terva1s in Exercises 1-<>.

1. f(x) ~ x2 + 2x - 1, [0, 1]

2. f(x) ~ x 2/3, [0, 1]

3. f(x) ~x+~, [t,2] 4. f(x) ~ v;=J, [1,3]

S.f(x)~x3_x2, [-1,2]

{

X3 6. g(x) ~ ;

x,

7. f(x) ~ x 2/3, [-1,8]

8. f(x) ~ x4/', [0, 1]

9. f(x) ~ v'x(1 - x), [0, 1]

{

SinX 10. f(x) ~ x'

0,

-'T1' ~x < 0

x~o

{

X2 - x 11. f(x) ~ 2x2 _ ~x - 3,

-2 ~ x::s; -1

-1 < x ~ 0

12. f(x) ~ {2x - 3, ° :5 X :5 2 6x - x2 - 7, 2 < x ::;; 3

13. The function

f(x) = {x' 0:5 X < I 0, x = 1

is zero at x = 0 and x = I and differentiable on (0, I), but its de· rivative on (0, 1) is never zero. How can this be? Doesn't Rolle's Theorem say tire derivative has to be zero somewhere in (0, I)? Give reasons for your answer.

14. For what values of a, m, and b does the function

{

3' f(x) = -x2 + 3x + a,

mx + b,

x=o

O<x<1

1~x~2

satisfY the hypotheses of the Mean Value Theorem on the interval [0,2]?

Roots (Zeros) 15. a. Plot the zeros of each polynomial on a line together with the

zeros of its first derivative.

i)y=x2_4

ii) y = x 2 + 8x + 15

iii) y = x 3 - 3x2 + 4 = (x + I)(x - 2)2

iv) y = x 3 - 33x2 + 216x = x(x - 9)(x - 24)

b. Use Rolle's Theorem to prove that between every two zeros of x" + Qn_tX"-1 + ... + QtX + ao there lies a zero of

nx"- 1 + (n - 1)all_tx"-2 + ... + al.

16. Suppose that/" is continuous on [a, b] and that f has three zeros in tire interval. Show that/" has at least one zero in (a, b). Gener· alize this result.

17. Show that if /" > 0 throughout an interval [a, b], then!, has at most one zero in [a, b]. What if/,,< 0 throughout [a, b] instead?

18. Show that a cubic polynontial can have at most three real zeros.

Show that the functions in Exercises 19-26 have exactly one zero in tire given interval.

19. f(x) = x' + 3x + I, [-2, -I]

20. f(x) = x3 + ~ + 7, (-00,0) x

21. g(l) = vt + v'l+t - 4, (0,00)

22. g(l) = I ~ I + v'l+t - 3.1, (-I, I)

23. 7(0) = 0 + sin2 (~) - 8, (-00,00)

24.7(0) = 20 - oos2 0 + v2, (-00,00)

I 25. 7(0) = sec 0 - 03 + 5, (0, "'/2)

26. 7(0) = tanO - cotO - 0, (0, "'/2)

Find;ng Functions from Derivatives 27. Suppose that f( -I) = 3 and that !,(x) = 0 for all x. Must

f(x) = 3 for all x? Give reasons for your answer.

4.2 The Mean Value Theorem 197

28. Suppose that f(O) = 5 and that !,(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

29. Suppose that!'(x) = lx for allx. Findf(2) if

•• f(O) = 0 b. f(!) = 0 c. f(-2) = 3.

30. What can be said about functions whose derivatives are constant? Give reasons for your answer.

In Exercises 31-36, fmd all possible functions with the given derivative.

31 • •• y' = X b.y'=x2 c.y'=x3

32. a. y' = lx b.y'=lx-I c.y'=3x2 +lx-1

33 ,_ I I c.y'=s+---\-••• y --2 b.y'=I--x x 2 x

34. a. y' = ifx b. y' = _I_ I c. Y'=4x-Vx 2 x Vx

35. a. y' = sin21 b. y' = cos i 2

c. y' = 8in2t + cost

36. •• y' = sec' 0 b. y' = v8 c. y' = v8 - sec20

In Exercises 37-40, fmd the function with the given dctivative whose graph passes through the point P.

37. !'(x) = lx - I, P(O, 0)

38. g'(x) = 1, + lx, p(-I, I) x

39. 7'(0) = 8 - csc'O, (-;f, 0 )

40. 7'(1) = secltanl - I, P(O, 0)

F;nd;ng Po.;tion from Velodty or Acceleration Exercises 41-44 give the velocity v = dsldl and initial position ofa body moving along a coordinate line. Find the body's position at time t.

41. v = 9.81 + 5, .(0) = 10

42. v = 321 - 2, .(0.5) = 4

43. v = sin ",I, .(0) = 0

2 21 , 44. v = 7T cos W' s('7J'""") = 1

Exercises 45-48 give the acceleration a = d 2.ldI2, initial velocity,

and initial position of a body moving on a coordinste line. Find tire body's position at time I.

45. a = 32, v(O) = 20, .(0) = 5

46. a = 9.8, v(O) = -3, .(0) = 0

47. a = -4 sin 21, v(O) = 2, .(0) = -3

9 31 48. a = "" cos "" v(O) = 0, .(0) = -I

Applkations 49. Temperatnre cbange It took 14 sec for a mercury thermometer

to rise from -19'C to lOO'C when it was taken from a freezer and placed in boiling water. Show that sOO3ewhere along the way tire mercury was rising at the rate of 8.5°C/sec.

198 Chapter 4: Applications of Derivatives

50. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

51. Classical accounts tell us that a 170-oar trireme (aocient Greek or Roman WlIIship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme's speed exceeded 7.5 koots (sea miles per hour).

52. A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly II mph, assuming the initial and fmal speeds are zero.

53. Show that at some instaot during a 2-hour automobile 1rip the car's speedometer reading will equal the average speed for the 1rip.

54. Free fall on the moon On our moon, the acceleration of gravity is 1.6 m/sec2 • Ifa rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

Theory and Examples 55. The geometric mean of a and b The ~metric mean of two

positive nombers a and b is the number Vab. Show that the value

of c in the conclusion of the Meao Valoe Theorem for I(x) ~ I/x on an interval of positive numbers [a, b j is c ~ v;;J,.

56. The arithmetic mean of Il and b The arithmetic mean of two numbers a and b is the number (a + b)/2. Show that the value of c in the conclusion of the Meao Value Theorem for I(x) ~ x2 on any interval [a,bj isc ~ (a + b)/2.

D 57. Graph the function

I(x) ~ sin x sin (x + 2) - sin2 (x + I).

What does the graph do? Why does the function behave this way? Give reasons for your answers.

58. RoUe's Theorem

L Construct a polynomial I(x) that has zeros atx ~ -2, -I, 0, I, and 2.

b. Graph I and its derivative f' together. How is what you see related to Rolle~ Theorem?

c. Do g(x) ~ sin x aod its derivative g' illustrate the same phenomenon as I and f'?

59. Unique solution Assume that I is continuous on [a, bj and dif­ferentiable on (a, b). Also assume that I(a) aod I(b) have opposite sigus and that f' # 0 between a aod b. Show that I(x) ~ 0 ex­actly once between a aod b.

60. Parallel tangents Assume that I and g are differentiable on [a, b j and that/(a) ~ g(a) aod/(b) ~ g(b). Show that there is at least one point between a and b where the taogents to the graphs of I and g are parallel or the same line. lllustrate with a sketch.

61. Suppose that f'(x):5 I for I :5 x :5 4. Show that 1(4)-1(1) :5 3.

62. Suppose that 0 < f'(x) < 1/2 for all x-values. Show that 1(-1) < 1(1) < 2 + 1(-1).

63. Show that I cou - II :5 Ix I for all x-values. (Hint: Consider I(t) ~ cos t on [0, xj.)

64. Show that for any numbers a and b, the sine inequality Isinb - sinal :5 Ib - al is true.

65. If the graphs of two differentishle functions I(x) andg(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

66. Ifl/(w) - I(x) I :5lw - xl for all values w and x and I is a dif­ferentiable function, show that -I :5 f' (x) :5 I for all x-values.

67. Assumethatl is differentiable ona :5 x:5 baodthat/(b) < I(a). Show that f' is negative at some point between a and b.

68. Let/be a function defmed on ao interval [a, bj. What conditions could ynu place on I to gusrantee that

. I' < I(b) - I(a) < I' mm - b-a _max,

where min f' and max f' refer to the minimum and maximum values off' on [a, b]? Give reasons for your aoswers.

D 69. Use the inequalities in Exercise 68 to estimate 1(0.1) if f'(x) ~ 1/(1 + x'cosx) forO:5 x:5 0.1 aod/(O) ~ 1.

D 70. Use the inequalities in Exercise 68 to estimate 1(0.1) if f'(x) ~ 1/(1 - x4) forO:5 x:5 0.1 aod/(O) ~ 2.

71. Let I be differentishle at every value of x and suppose that 1(1) ~ I, that f' < 0 on (-00, I), and thatf' > 0 on (1,00).

a. Show that I(x) '" I for all x.

b. Mustf'(I) ~ O?Explain.

72. Let I(x) ~ px2 + qx + r be a quadratic function defined on a closed interval [a, b j. Show that there is exactly one point c in (a, b) at which I satisfies the conclusion of the Mean Value Theorem.

4.3 Monotonic Functions and the First Derivative Test

In sketching the graph of a dllferentiable function it is useful to know where it increases (rises from left to right) and where it decreases (falls from left to right) over an interval. This section gives a test to determine where it increases and where it decreases. We also

show how to test the critical points of a function to identifY whether local extreme values are present.