CHAPTER 4

Analysis of Functions and their Graphs SECTION 4.1 1. Answer true or false. If f  (x) < 0 for all x on the interval I, then f (x) is concave down on the interval I. 2. Answer true or false. A point of inﬂection that has an x-coordinate where f  (x) = 0 is a point of inﬂection. 3. The largest interval over which f is increasing for f (x) = (x − 5)6 is A. [5, ∞)

B. [−5, ∞)

D. (−∞, −5].

C. (−∞, 5]

4. The largest interval over which f is increasing for f (x) = x3 − 2 is A. (−∞, −2]

B. [2, ∞)

C. (−∞, ∞)

5. The largest interval over which f is increasing for f (x) = A. [5, ∞)

√ 5

x − 5 is

C. (−∞, ∞)

B. (−∞, 5]

6. The largest open interval over which f is concave up for f (x) = A. (−∞, 7)

D. [−2, ∞).

B. (7, ∞)

√ 5

D. nowhere. x − 7 is

C. (−∞, ∞)

D. nowhere.

7. The largest open interval over which f is concave up for f (x) = 3x2 + 2x2 − x + 5 is A. (−∞, −0.623)

B. (−0.623, ∞)

C. (−∞, ∞)

D. nowhere.

8. The function f (x) = x5/7 has a point of inﬂection with an x-coordinate of A. 0

B.

5 7

C. −

5 7

D. None exist.

9. The function f (x) = x5 − 2 has a point of inﬂection with an x-coordinate of A. 2

B. −2

C. 0

D. None exist.

10. Use a graphing utility to determine where f (x) = cos x is decreasing on [0, 2π]. A. [0, π]

B. [π, 2π]

C. [π/2, 3π/2]

D. [0, 2π]

11. Answer true or false. tan x has a point of inﬂection on (−π/2, π/2). 12. Answer true or false. All functions of the form f (x) = axn , n odd and a = 0 have an inﬂection point. 13. f (x) = x4 − 8x2 − 2 is concave up on the interval I = A. (−∞, ∞)

B. [−1, ∞)

C. (−∞, −1)

1

D. [−1, 1].

2

True/False and Multiple Choice Questions

14. Answer true or false. If f  (−2) = −5 and f  (x) = 5, then there must be a point of inﬂection on (−2, 2). x2 has x2 − 9 points of inﬂection at x = −9 and x = 9 points of inﬂection at x = −3 and x = 3 a point of inﬂection at x = 0 no point of inﬂection.

15. The function f (x) = A. B. C. D.

Section 4.2

3

SECTION 4.2 1. Determine the x-coordinate of each stationary point of f (x) = 4x2 − 8x. A. x = 1 C. x = −1

B. x = 1 and x = 0 D. None exists.

2. Determine the x-coordinate of each critical point of f (x) = A. 0

B. 5

√ 5

x − 5.

C. −5

D. None exist.

3. Answer true or false. f (x) = x3/5 has a critical point. 4. Answer true or false. All relative extrema occur at critical points. 5. f (x) = x2 + 4x + 7 has a A. relative maximum at x = −2 C. relative maximum at x = 2

B. relative minimum at x = −2 D. relative minimum at x = 2.

6. f (x) = sin2 x on 0 < x < 2π has A. B. C. D.

both a relative maximum and a relative minimum a relative maximum only a relative minimum only neither a relative maximum nor a relative minimum.

7. f (x) = x4 − 8x3 has A. B. C. D.

a relative maximum at x = 0; no relative minimum no relative maximum; a relative minimum at x = 6 a relative maximum at x = 0; a relative minimum at x = 6 a relative maximum at x = 0; relative minimum at x = −6 and x = 6.

8. Answer true or false. f (x) = | tan2 x| has no relative extrema on (−π/2, π/2). 9. f (x) = |x − 2| has A. a relative maximum at x = 2 C. a relative maximum at x = 0

B. a relative minimum at x = 2 D. a relative minimum at x = 0.

10. f (x) = |x2 − 16| has A. B. C. D.

no relative maximum; a relative minimum at x = 4 a relative maximum at x = 4; no relative minimum relative minima at x = −4 and x = 4; a relative maximum at x = 0 relative maxima at x = −16 and x = 16; a relative minimum at x = 0.

11. f (x) = A. B. C. D.

x2 − 5 has

a relative maximum only a relative minimum only both a relative maximum and a relative minimum neither a relative maximum nor a relative minimum.

4

True/False and Multiple Choice Questions

12. On the interval (0, 2π), f (x) = | sin x cos(2x)| has A. B. C. D.

a relative maximum only a relative minimum both a relative maximum and a relative minimum no relative extrema.

13. Answer true or false. f (x) =

x has a relative maximum on (0, ∞).

14. Answer true or false. A graphing utility can be used to show f (x) = |x2 | has a relative maximum. 15. Answer true or false. A graphing utility can be used to show f (x) = |x2 | − 2 has two relative maxima on [−10, 10].

Section 4.3

5

SECTION 4.3 1. Answer true or false. If f  (−2) = −4 and f  (2) = 4, then there must be an inﬂection point on (−2, 2). 2. The polynomial function x2 − 4x + 7 has A. B. C. D.

one stationary point that is at x = 2 two stationary points, one at x = 0 and one at x = 2 one stationary point that is at x = −2 one stationary point that is at x = 0. 3 has x−2 a horizontal asymptote at y = 0 a horizontal asymptote at y = −2 horizontal asymptotes at x = −1 and x = 1 no horizontal asymptote.

3. The rational function A. B. C. D.

1 − x2 has x3 a stationary point at x = −2 a stationary point at x = 2 two stationary points, one at x = −1 and one at x = 1 three stationary points, one at x = −2, one at x = −1, and one at x = 1.

4. The rational function A. B. C. D.

5. Answer true or false. The rational function x4 −

1 has no vertical asymptote. x2

6. On a [−10, 10] by [−10, 10] window on a graphing utility the rational function f (x) = A. B. C. D.

x3 + 2 has x3 − 2

one horizontal asymptote; no vertical asymptote no horizontal asymptote; one vertical asymptote one horizontal asymptote; one vertical asymptote one horizontal asymptote; three vertical asymptotes.

7. Use a graphing utility to graph f (x) = x1/9 . How many points of inﬂection does the function have? A. 0

B. 1

C. 2

D. 3

8. Use a graphing utility to graph f (x) = x−1/9 . How many points of inﬂection are there? A. 0

B. 1

C. 2

D. 3

C. f (x) = x−1/4

D. f (x) = x1/3

9. Determine which function is graphed.

A. f (x) = x1/4

B. f (x) = x−1/3

6

True/False and Multiple Choice Questions

10. Use a graphing utility to generate the graph of f (x) = x4 − 2x3 + x2 − x + 1, then determine the x-coordinates of all relative extrema on (−10, 10) and identify them as relative maxima or relative minima. A. B. C. D.

There is a relative minimum at x = 1.26. There is a relative maximum at x = 1.26. There is a relative maximum at x = 0. There is a relative minimum at x = 0.

11. Answer true or false. Using a graphing utility it can be shown that f (x) = x4 sin x has a relative maximum on 0 < x < 2π. 12. Answer true or false. lim

√ 4

x→0+

13.

x=0

lim x5/2 =

x→+∞

A. 0 C. +∞

B. 1 D. It does not exist.

14. Answer true or false. A fence is to be used to enclose a rectangular plot of land. If there are 4900 feet of fencing, it can be shown that a 70 ft by 70 ft square is the rectangle that can be enclosed with the greatest area. (A square is considered a rectangle.) 15. Answer true or false. f (x) =

x2 has an oblique asymptote. x−1

Section 4.4

7

SECTION 4.4 1.

The graph represents a position function. Determine what is happening to the velocity at t = 0. A. It is positive. C. It is zero.

B. It is negative. D. There is insuﬃcient information to tell.

2.

The graph represents a position function. Determine what is happening to the acceleration at t = 2. A. It is positive. C. It is zero.

B. It is negative. D. There is insuﬃcient information to tell.

3.

The graph represents a velocity function. The acceleration at t = 4 is A. positive C. zero

B. negative D. There is insuﬃcient information to tell.

4.

Answer true or false. This can be the graph of a particle’s position if the particle is moving to the right at t = 0.

8

True/False and Multiple Choice Questions

5.

Answer true or false. For the position function graphed, the acceleration at t = 1 is positive. 6.

Answer true or false. If the graph on the left is a position function, the graph on the right represents the corresponding velocity function. 7. Let s(t) = sin t be a position function of a particle. At t = A. positive

B. negative

π the particle’s velocity is 4

C. zero.

8. Let s(t) = t3 − t be a position function of a particle. At 3 the particle’s acceleration is A. positive

B. negative

C. zero.

9. s(t) = t − 2t2 , t ≥ 0. The velocity function is A. 1 − 2t

B. 1 − t

C. 1 − 4t

D. 1 − 4t3 .

C. 6t − 2

D. 3t2 .

10. s(t) = t3 − 2t, t ≥ 0. The acceleration function is A. 3t2 − 2

B. 6t

11. A projectile is dropped, and reaches the ground at 100 m/s. How long does it take the projectile to reach ground? A. 1,020 s

B. 510 s

C. 5 s

D. 10 s

12. Answer true or false. If a particle is dropped a distance of 200 m. It has a speed of 98.57 m/s (rounded to the nearest hundredth of a m/s) when it hits the ground. 13. s(t) = t5 − 2. Find t when a = 0. A. 12

B. −12

C. −2

D. 2

C. −5

D. −1

14. s(t) = t3 − 5, t ≥ 0. Find s when a = 0. A. 1

B. 5

15. Let s(t) = t6 − 5t be a position function. Fund v when t = 1. A. −1

B. 0

C. 2

D. 1

Section 4.4

9

16. Let s(t) = t4 − 2t + 1 be a position function. The acceleration function a(t) = A. 4t3 − 2

B. 12t2

C. 12t2 − 2

D. 4t3 .

C. 0

D. 1

17. Let s(t) = t5 . Find t when a = 0. A. 5

B. 20

10

True/False and Multiple Choice Questions

SECTION 4.5 1. f (x) = 3x2 − 4x + 2 has an absolute maximum on [−2, 2] of A. 16

B. 22

C. 12

D. 4.

C. 1

D. 5.

2. f (x) = |5 − 2x| has an absolute minimum of A. 0

B. 3

3. Answer true or false. f (x) = x3 − x2 + 2 has an absolute maximum and an absolute minimum. 4. Answer true or false. f (x) = x3 − 18x2 + 20x + 2 restricted to a domain of [0, 20] has an absolute maximum at x = 2 of −22, and an absolute minimum at x = 10 of −598. 5. f (x) =

x − 2 has an absolute minimum of

A. 0 at x = 2 6. f (x) =

B. 0 at x = 0

C. −2 at x = 0

D. 0 at x = −2.

x2 + 5 has an absolute maximum, if one exists, at

A. x = −5

B. x = 5

C. x = 0

7. Find the location of the absolute maximum of tan x on [0, π], if it exists. π A. 0 B. π C. 2

D. None exist.

D. None exist.

8. f (x) = x2 − 3x + 2 on (−∞, ∞) has A. B. C. D.

only an absolute maximum only an absolute minimum both an absolute maximum and an absolute minimum neither an absolute maximum nor an absolute minimum. 1 on [1, 3] has x2 an absolute maximum at x = 1 and an absolute minimum at x = 3 an absolute minimum at x = 1 and an absolute maximum at x = 3 no absolute extrema an absolute minimum at x = 2 and absolute maxima at x = 1 and x = 3.

9. f (x) = A. B. C. D.

10. Answer true or false. f (x) = sin x cos x on [0, π] has an absolute maximum at x =

π . 2

11. Use a graphing utility to assist in determining the location of the absolute maximum of f (x) = −(x2 − 3)2 on (−∞, ∞), if it exists. √ √ √ A. x = 3 and x = − 3 B. x = 3 only C. x = 0 D. None exist. 12. Answer true or false. If f (x) has an absolute minimum at x = 2, −f (x) also has an absolute minimum at x = 2. 13. Answer true or false. Every function has an absolute maximum and an absolute minimum if its domain is restricted to where f is deﬁned on an interval [−a, a], where a is ﬁnite.

Section 4.5

11

14. Use a graphing utility to locate the value of x where f (x) = x4 − 3x + 2 has an absolute minimum, if it exists.  3 3 A. 1 B. C. 0 D. None exist. 4 15. Use a graphing utility to estimate the absolute maximum of f (x) = (x − 5)2 on [0, 6], if it exists. A. 25

B. 0

16. The absolute minimum of f (x) = A. 0

B. −3

C. 1 √

D. None exist.

x − 3 occurs at x = C. 3

17. f (x) = cos(2x) has an absolute minimum on [0, π] at x = π C. 0 A. π B. 2

D. None exists.

D. None exists.

12

True/False and Multiple Choice Questions

SECTION 4.6 1. Express the number 60 as the sum of two nonnegative numbers whose product is as large as possible. A. 20, 40

B. 1, 59

C. 30, 30

D. 0, 60

2. A right triangle has a perimeter of 32. What are the lengths of each side if the area contained within the triangle is to be maximized? 32 32 32 , , 3 3 3 √ √ √ C. 32 − 16 2, 32 − 16 2, −32 + 32 2

A.

B. 10, 10, 12 D. 8, 10, 14

3. A rectangular sheet of cardboard 4 m by 2 m is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume? √ √ 3− 3 1 3+ 3 B. C. D. 1 A. 2 2 2 4. Suppose that the number of bacteria present in a culture bacteria at time t is given by N = 10000(x2 + x). Find the smallest number of bacteria in the culture during the time interval 0 ≤ t ≤ 50. A. 67

B. 10,000

C. 3,679

D. 73,891

5. An object moves a distance s away from the origin according to the equation s(t) = 4t3 − 2t + 1, where 0 ≤ t ≤ 10. At what time is the object farthest from the origin? A. 0

B. 2

C. 10

D.

1 4

6. An electrical generator produces a current in amperes starting at t = 0 s and running until t = 6π s that is given by cos(2t). Find the maximum current produced. A. 1 A

B. 0 A

C. 2 A

D.

1 A 2

7. A storm is passing with the wind speed in mph changing over time according to v(t) = −x2 + 14x + 55, for 0 ≤ t ≤ 10. Find the highest wind speed that occurs. A. 55 mph

B. 104 mph

C. 110 mph

D. 30 mph

8. A company has a cost of operation function given by C(t) = 0.01t2 − 2t + 1,000 for 0 ≤ t ≤ 500. Find the minimum cost of operation. A. 1,000

B. 900

C. 500

D. 0

9. Find the point on the curve x2 + y 2 = 25 closest to (0, 6). A. (0, 25)

B. (0, 5)

C. (5, 0)

D. (25, 0)

10. Answer true or false. The point on the parabola y = 3x2 closest to (0, 0.9) is (0, 0). 11. For a triangle with sides 6 m, 8 m, and 10 m, the smallest circle that contains the triangle has a diameter of A. 6 m

B. 8 m

C. 10 m

D. 12 m.

Section 4.6

13

12. Answer true or false. An object thrown upward reaches a height of s(t) = −16t2 + 32t. The object is highest at t = what, where t is in seconds? 13. Answer true or false. The rectangle with the largest area that can be drawn around a circle is a square. 14. Answer true or false. The rectangle with the largest area that can be drawn around a semi-circle is a square. 15. Answer true or false. An object that is thrown upward and reaches a height of s(t) = 50+120t−16t2 for 0 ≤ t ≤ 3. The object is highest at t = 3. 16. Two numbers sum to 50. Find the two numbers whose product is maximum. A. 25, 25

B. 50, 1

C. 20, 30

D. 10, 40

14

True/False and Multiple Choice Questions

SECTION 4.7 1. Approximate

15 by applying Newton’s Method to the equation x2 − 15 = 0.

A. 3.872983 2. Approximate

B. 3.872885 √ 3

C. 3.872990

D. 3.872995

10 by applying Newton’s Method to the equation x3 − 10 = 0.

A. 2.1544

B. 3.1623

C. 1.5849

D. 1.9953

3. Use Newton’s Method to approximate the solutions of x4 − 15 = 0. A. −1, 9680, 0, 1.9680 C. −1.7321, 0, 1.7321

B. −1.9680, 1.9680 D. −1.7321, 1.7321

4. Use Newton’s Method to ﬁnd the largest positive solution of x4 − 5x2 − 14 = 0. A. 1.4142

B. 1.9343

C. 3.7417

D. 2.6458

5. Use Newton’s Method to ﬁnd the largest positive solution of x4 + x3 − 6x2 − 7x − 7 = 0. A. 1.7325

B. 2.646

C. 1.7319

D. 1.7316

6. Use Newton’s Method to ﬁnd the largest positive solution of x4 − x2 − 30 = 0. A. 2.3403

B. 1.5651

C. 2.4495

D. 5.4772

7. Use Newton’s Method to ﬁnd the largest positive solution of x3 + x2 − 3x − 3 = 0. A. 1.732

B. 1.000

C. 0.500

D. 1.316

8. Use Newton’s Method to ﬁnd the largest positive solution of x4 + 3x2 − 40 = 0. A. 2.545

B. 6.325

C. 1.495

D. 2.236

9. Use Newton’s Method to ﬁnd the largest positive solution of x4 + x3 − 2x − 2 = 0. A. 1.260

B. 1.414

C. 1.587

D. 2.000

10. Use Newton’s Method to ﬁnd the largest positive solution of x5 + 5x3 − 6x2 − 30 = 0. A. 3.162

B. 2.340

C. 5.477

D. 1.817

11. Use Newton’s Method to ﬁnd the largest positive solution of x4 + x3 − 7x2 − 8x − 8 = 0. A. 2.236

B. 1.380

C. 1.710

D. 2.828

12. Use Newton’s Method to ﬁnd the x-coordinate of the intersection of y = x4 + x3 and y = 7x2 + 8x + 8. A. 3.742

B. 2.410

C. 2.828

D. 1.260

13. Use Newton’s Method to approximate the greatest x-coordinate of the intersection of y = x3 − x and y = x4 + x − 4. A. 2.236

B. 2.410

C. 1.414

D. 1.260

Section 4.7

15

14. Use Newton’s Method to approximate the x-coordinate intersection of y = 2x5 + 2x3 and y = −2x4 + 10x2 + 20x. A. 2.236

B. 1.380

C. 1.716

D. 1.627

15. Use Newton’s Method to ﬁnd the greatest x-coordinate of the intersection of y = 3x4 − 21x2 and y = 18x2 − 90. A. 3.162

B. 2.340

C. 5.477

D. 1.732

16. Use Newton’s Method to ﬁnd the largest solution of x5 − 2x4 + 3x3 − x2 + 4x + 7 = 0. A. 7

B. 0

C. −0.826

D. 1.427

17. Use Newton’s Method to ﬁnd the largest solution of x7 − 4x3 + 2x2 − 1 = 0. A. −1.498

B. −1

C. 1

D. 2.502

16

True/False and Multiple Choice Questions

SECTION 4.8 1. Answer true or false. f (x) =

1 on [−1, 1] satisﬁes the hypotheses of Rolle’s Theorem. x3

2. Find the value c such that the conclusion of Rolle’s Theorem are satisﬁed for f (x) = 2x2 − 8 on [−2, 2]. B. −1

A. 0

C. 1

D. 0.5

3. Answer true or false. Rolle’s Theorem is used to ﬁnd the zeros of a function. 4. Answer true or false. The Mean-Value Theorem can be used on f (x) = |x − 1| on [−2, 1]. 5. Answer true or false. The Mean-Value Theorem guarantees there is at least one c on [0, 1] such that f  (x) = 0.5 when f (x) = x. 6. If f (x) =

√ 5

A. 1

x on [0, 1], ﬁnd the value c that satisﬁes the Mean-Value Theorem. B.

1 3

C.

1 243

D.

1 9

7. Answer true or false. The hypotheses of the Mean-Value Theorem are satisﬁed for f (x) = on [−1, 1].

 5

|x|

8. Answer true or false. The hypotheses of the Mean-Value Theorem are satisﬁed for f (x) = cos x on [0, 4π]. 9. Answer true or false. The hypotheses of the Mean-Value Theorem are satisﬁed for f (x) = on [0, 4π].

1 cos x

10. Find the value for which f (x) = x2 + 7 on [1, 3] satisﬁes the Mean-Value Theorem. A. 2

B.

9 4

C.

7 3

D.

11 4

11. Find the value for which f (x) = x3 − 5 on [2, 3] satisﬁes the Mean-Value Theorem. A. 2.5166

B. 2.5000

C. 2.2500

D. 2.1250

12. Answer true or false. A graphing utility can be used to show that Rolle’s Theorem can be applied to show that f (x) = (x − 5)2 has a point where f  (x) = 0. 13. Answer true or false. According to Rolle’s Theorem if a function’s derivative is 0, the graph of the function must cross the x-axis. 14. Find the value c that satisﬁes Rolle’s Theorem for f (x) = cos x on [π/2, 3π/2]. A.

π 4

B.

π 2

C.

3π 4

D. π

15. Find the value c that satisﬁes the Mean-Value Theorem for f (x) = x3 = 0 on [0, 1]. √ √ √ √ 3 3 2 2 A. B. C. D. 3 2 2 3

Section 4.8

17

16. Find the value c that satisﬁes Rolle’s Theorem for f (x) = x3 − x on [−1, 1]. A. −1.0

B. 0.0

C. 1.0

D. 0.5

 π π 17. Find the value c that satisﬁes Rolle’s Theorem for f (x) = 5 cos x on − , . 2 2 π π A. − B. C. 0 D. 2 2

π 4

18

True/False and Multiple Choice Questions

CHAPTER 4 TEST 1. The largest interval on which f (x) = x2 + 4x + 2 is increasing is A. [0, ∞)

C. [−2, ∞)

B. (−∞, 0]

2. Answer true or false. The function f (x) =

D. (−∞, −2].

x − 6 is concave down on its entire domain.

3. The function f (x) = x5 − 1 is concave down on A. (−∞, 2)

B. (2, ∞)

C. (−∞, 0)

D. (0, ∞).

4. Answer true or false. f (x) = x5 + 2 has a point of inﬂection. 5. f (x) = |x2 − 9| is concave down on A. (−∞, −3) ∪ (3, ∞) C. (−3, 3)

B. (−∞, −9) ∪ (9, ∞) D. (−9, 9).

6. f (x) = 2x2 + 2 has an absolute minimum on [−3, 3] of A. 2

B. −2

C. 52

D. −52.

7. Use a graphing utility to determine where f (x) = cos x is increasing on [0, 2π]. A. [0, π]

B. [π, 2π]

C. [π/2, 3π/2]

D. [0, π/2] ∪ [3π/2, 2π]

8. Answer true or false. f (x) = x5 − 2x3 + x has a point of inﬂection. 9. f (x) = −x4 − 6x2 is concave up on A. (−∞, ∞)

B. (−∞, −81)

C. (−∞, −9)

D. nowhere.

10. Answer true or false. If f  (−1) = 6 and f  (1) = 6, and if f is continuous on [−1, 1], then there is a point of inﬂection on (−1, 1). 11. Determine the x-coordinate of each stationary point of f (x) = 2x4 − 8. A. −1

B. 0

C. 16

D. 1

12. Answer true or false. f (x) = x3/11 has a critical point at x = 0. 13. f (x) = x2 − 8x + 7 has A. a relative maximum at x = 4

B. a relative minimum at x = 4

C. a relative maximum at x = −4

D. a relative minimum at x = −4.

14. f (x) = x3 + 1 has an absolute maximum on [−1, 1] of A. 0

B. 6

C. 11

D. 2.

Chapter 4 Test

19

15. f (x) = |6x4 | has A. B. C. D.

no relative maximum; a relative minimum at x = 4 a relative maximum at x = 4; no relative minimum a relative maximum at x = 0; relative minima at x = −4 and x = 4 no relative maximum; a relative minimum at x = 0.

16. Answer true or false. f (x) = x3 − 7 has a relative minimum on (0, ∞). 3 has x−5 a horizontal asymptote at y = 0 a horizontal asymptote at y = 5 a horizontal asymptote at y = 4 no horizontal asymptote.

17. The rational function A. B. C. D.

18. Answer true or false. f (x) =

3 has a vertical asymptote. x−7

19.

This is the graph that would appear on a graphing utility if the function that is graphed is A. f (x) = x1/5

B. f (x) = x1/4

20. Answer true or false. lim

√ 5

x→0+

C. f (x) = x−1/5

D. f (x) = x−1/4 .

x ln x = ∞

21. A weekly proﬁt function for a company is P (x) = −0.01x2 + 3x − 50,000, where x is the number of the company’s only product that is made and sold. How many individual items of the product must the company make and sell weekly to maximize the proﬁt? A. 300

B. 150

C. 600

D. 60

C. 0

2 D. − . 3

C. 9

D. None exist

22. f (x) = 3 sin(x2 ) has an absolute minimum of A. −5

B. −3

1 has an absolute maximum on [1, 3] of x 1 A. 1 B. 9

23. f (x) =

24. Answer true or false. f (x) =

1 has an absolute maximum of 1 on [−1, 1]. x3

20

True/False and Multiple Choice Questions

25. Express the number 60 as the sum of two nonnegative numbers whose product is as large as possible. A. 5, 55

B. 10, 50

C. 30, 30

D. 1, 59

26. An object moves a distance s away from the origin as given by s(t) = t2 + 2, 0 ≤ t ≤ 10. At what time is the object farthest from the origin? A. 0

B. 2

C. 8

D. 10

27. Find the point on the curve x2 + y 2 = 16 closest to (0, 3). A. (0, 4)

B. (4, 0)

C. (−4, 0)

D. (0, −4)

28.

The graph represents a position function. Determine what is happening to the velocity at t = 3. A. It is increasing. C. It is constant.

29.

The graph represents a position function. Determine what is happening to the acceleration at t = 1. A. It is positive. C. It is zero.

30. Let s(t) = t2 − 2 be a position function particle. The particle’s acceleration for t > 0 is A. positive C. zero

31. Let s(t) = 4 − t3 be a position function. The particle’s velocity for t > 0 is A. positive

B. negative

C. zero.

32. s(t) = 4t2 − 12. a = 0 when t = A. 0 33. Approximate A. 3.8724

B. 8 √

C. 2

D. nowhere.

C. 3.8751

D. 3.8730

15 using Newton’s Method. B. 3.8715

Chapter 4 Test

21

34. Use Newton’s Method to approximate the greatest x-coordinate of the solution of x3 + x2 − 7x − 7 = 0. A. 4.000

B. 2.646

C. 5.292

D. 3.037

35. Use Newton’s Method to approximate the greatest x-coordinate where the graphs of y = x3 −6x−5 and y = −x2 + x + 2 cross. A. 4.000

B. 2.646

C. 5.292

D. 3.037

36. Answer true or false. The hypotheses of Rolle’s Theorem are satisﬁed for f (x) =

1 − 1 on [−1, 1]. x8

37. Answer true or false. Given f (x) = x2 − 9 on [−3, 3], the value c that satisﬁes Rolle’s Theorem is 0. 38. Answer true or false. f (x) = x3 on [−1, 1]. The value c that satisﬁes the Mean-Value Theorem is 0.

SOLUTIONS

SECTION 4.1 1. F 2. F 3. A 4. C 13. A 14. F 15. D

5. D

6. A

7. B

8. A

9. C

10. A

11. T

12. T

5. B

6. A

7. B

8. F

9. C

10. C

11. B

12. C

5. F

6. C

7. B

8. A

9. C

10. A

11. T

12. F

2. A 3. B 4. T 5. F 6. T 14. C 15. D 16. B 17. C

7. A

8. A

9. C

10. B

11. D

12. F

7. D

8. A

9. A

10. F

11. A

12. F

6. A

7. B

8. B

9. B

10. F

11. C

12. B

2. A 3. D 4. D 5. B 6. C 14. C 15. A 16. C 17. A

7. A

8. D

9. A

10. D

11. D

12. C

7. F

8. T

9. F

10. A

11. A

12. F

5. C 6. A 7. B 8. T F 17. A 18. T 19. B A 28. A 29. A 30. A F

9. D 20. F 31. B

10. F 21. A 32. A

11. B 22. B 33. D

12. T 23. A 34. B

SECTION 4.2 1. A 13. F

2. C 3. T 4. T 14. F 15. T

SECTION 4.3 1. F 13. C

2. A 3. D 4. C 14. T 15. T

SECTION 4.4 1. A 13. C

SECTION 4.5 1. A 13. T

2. A 3. F 4. F 5. A 6. D 14. B 15. A 16. C 17. B

SECTION 4.6 1. C 13. T

2. C 3. B 4. C 5. C 14. F 15. F 16. A

SECTION 4.7 1. A 13. A

SECTION 4.8 1. F 13. F

2. A 3. F 4. F 5. F 6. C 14. D 15. A 16. B 17. C

CHAPTER 4 TEST 1. C 13. B 24. F 35. B

2. T 3. C 4. 14. A 15. D 25. C 26. D 36. F 37. T

T 16. 27. 38.

22

m4
m4