Math 611b Assignment #6 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find a formula for the function graphed.1) 1)

A) f(x) = 5 + x, x < 2-3 x > 2

B) f(x) = 5 - x, x 2-3 x > 2

C) f(x) = 5 + x, x 2-3 x > 2

D) f(x) = 5 - x, x < 2-3 x 2

Solve the problem.2) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the

watermelon's average speed during the first 5 sec of fall.2)

A) 40 ft/sec B) 80 ft/sec C) 160 ft/sec D) 100 ft/sec

3) limx 7

x2 + 49x + 7

3)

A) 14 B) Does not exist C) 0 D) 7

Evaluate.

4) limx 0

x3 + 12x2 - 5x5x

4)

A) -1 B) 5 C) 0 D) Does not exist

1

Evaluate.5) lim f(x)

x -1/25)

A) -2 B) Does not exist C) 0 D) -1

6) limx 1-

f(x) 6)

A) Does not exist B) 12 C) 2 D) -1

Find the limit.7) Let lim

x 3f(x) = -4 and lim

x 3g(x) = -5. i) Find lim

x 3[f(x) - g(x)].

ii) Find limx 3

[f(x) · g(x)]. iii

iii) Find limx 3

f(x)g(x)

.

7)

A) -9, 20, 1.25 B) 1, 20, 0.8 C) -4, -20, 0.5 D) 3, -20, 1.5

2

Evaluate or determine that the limit does not exist for each of the limits (a) limx 1-

f(x), (b) limx 1+

f(x), and (c) limx 1

f(x)

for the given function f.8)

f(x) =

-2x - 2, for x < 1, 1, for x = 1,3x - 10, for x > 1

8)

A) (a) -4(b) -7(c) -11

B) (a) -7(b) -4(c) Does not exist

C) (a) -7(b) -4(c) -11

D) (a) -4(b) -7(c) Does not exist

Find the limit.

9) limx

7x3 - 3x2 + 3x-x3 - 2x + 5

9)

A) 32 B) C) 7 D) -7

10) limx -4-

x-3x2 +x-12

ii) Identify and name any points of discontinuity.

iii) Write an equation for the extended function that is continuous for all values of x 0

10)

A) 1 B) 35 C) 0 D)

3

Match the function with the graph of its end behavior model.

11) y =-4x3 + 5x2 + 1

x + 911)

A) B)

C) D)

Find a simple basic function as a right-end behavior model and a simple basic function as a left-end behavior model.12) y = ex + 2x 12)

A) y = ex; y = 2x B) y = -e-x; y = xC) y = -e-x; y = 2x D) y = ex; y = x

Use the definition f'(a) = limh->0

f(x + h) - f(x)h

to find the derivative of the given function for any value of x.

13) f(x) = 17 - 9x 13)A) -17 B) -9 C) 17 D) 8

Solve the problem.

14) Find ddx

(2x2 - 3). 14)

A) 4x2 - 3 B) 4x C) 4x - 3 D) 2x

4

The graph of a function is given. Choose the answer that represents the graph of its derivative.15) 15)

A) B)

C) D)

Solve the problem.16) Find the equation of the normal line to the curve y = 4x - 2x2 at the point (2, 0). 16)

A) x - 4y - 194 = 0 B) x + 12y - 2 = 0C) x - 4y - 2 = 0 D) x + 12y - 194 = 0

5

Find the values where the function is not differentiable.17) 17)

A) x = 0 B) x = 1 C) x = -1 D) x = 2

Solve the problem.18) The graph shows the yearly average interest rates for 30-year mortgages for years since 1988 (Year

0 corresponds to 1988). Sketch a graph of the rate of change of interest rates with respect to time.18)

A)

6

B)

C)

D)

Determine the values of x for which the function is differentiable.

19) y =1

x2 - 6419)

A) All reals except 64 B) All reals except -8 and 8C) All reals except 8 D) All reals

7

Find dy/dx.

20) y =8x - 7

x2 - 8x + 920)

A) 8x3 - 80x2 + 150x - 56(x2 - 8x + 9)2

B) -8x2 + 14x + 16(x2 - 8x + 9)2

C) 24x2 - 142x + 128(x2 - 8x + 9)2

D) 8x2 + 14x + 16x2 - 8x + 9

21) y = ln x8 21)

A) 1x B) 1

8x C) 8x D) 1

x- ln 8

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find thevalue of the indicated derivative.

22) u(1) = 2, u (1) = -6, v(1) = 7, v (1) = -4.ddx

uv

at x = 1

22)

A) -347 B) -

5049 C) -

178 D) -

3449

Find the slope of the line tangent to the curve at the given value of x.23) y = x2 - 5x; x = 3 23)

A) -6 B) 6 C) 1 D) -9

Find the fourth derivative of the function.24) y = x3 - 3x-1 24)

A) 18x-4 B) 72x-5 C) -36x-5 D) -72x-5

Solve the problem.25) Find an equation of the tangent to the curve y = 2x2 - 2x + 1 that has slope 2. 25)

A) y = 2x B) y = 2x - 1 C) y = 2x + 1 D) y = 2x + 2

26) The population P, in thousands, of a small city is given by P(t) =900t

2t2 + 1, where t is the time, in

months. Find the growth rate, dPdt

.

26)

A) dPdt

=900(2t2 - 1)(2t2 + 1)2

B) dPdt

=900(1 + 6t2)(2t2 + 1)2

C) dPdt

=900(1 - 2t2)

2t2 + 1D) dP

dt=

900(1 - 2t2)(2t2 + 1)2

8

27) The function V = 3 r2 describes the volume of a right circular cylinder of height 3 feet and radius rfeet. Find the (instantaneous )rate of change of the volume with respect to the radius when r = 6.Leave answer in terms of .

27)

A) 36 ft3/ft B) 12 ft3/ft C) 6 ft3/ft D) 18 ft3/ft

28) The dollar profit from the expenditure of x thousand dollars on advertising is given by P(x) = 800+ 25x - 2x2. Find the marginal profit when the expenditure is x = 9.

28)

A) 225 thousand dollars B) 800 thousand dollarsC) 189 thousand dollars D) -11 thousand dollars

29) At time t, the position of a body moving along the s-axis is s = t3 - 27t2 + 240t m. Find the body'sacceleration each time the velocity is zero.

29)

A) a(20) = 120 m/sec2, a(16) = 20 m/sec2 B) a(10) = 6 m/sec2, a(8) = -6 m/sec2

C) a(10) = 0 m/sec2, a(8) = 0 m/sec2 D) a(10) = -6 m/sec2, a(8) = 6 m/sec2

Find dy/dx.30) y = 4 tan6x 30)

A) 24 tan7x B) 24 tan5x sec2x C) 24 tan5x D) 24 tan6x sec x

The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds).31) s = -2 + 3 cos t

Find the body's speed at time t = /3 sec.31)

A) -32

m/sec B) 32

m/sec C) 3 32

m/sec D) -3 3

2 m/sec

Find the indicated derivative.32) Find y if y = 4x sin x. 32)

A) y = 8 cos x - 4x sin x B) y = - 8 cos x + 4x sin xC) y = 4 cos x - 8x sin x D) y = - 4x sin x

Find dy/dx.33) y = sin6 x - cos 5x 33)

A) 30 sin5 x cos x sin 5x B) 6 sin6 x cos x - 5 sin 5xC) 6 sin5 x + sin 5x D) 6 sin5 x cos x + 5 sin 5x

Find dr/d .34) r = cot (5 - 8 ) 34)

A) -csc2(5 - 8 ) B) 8 csc (5 - 8 ) cot (5 - 8 )C) 8 csc2(5 - 8 ) D) -csc (5 - 8 ) cot (5 - 8 )

Solve the problem.35) The position of a particle moving along a coordinate line is s = 5 + 4t, with s in meters and t in

seconds. Find the particle's velocity at t = 1 sec.35)

A) 43

m/sec B) -13

m/sec C) 16

m/sec D) 23

m/sec

9

Find y .

36) y = 3 cot x10 36)

A) 6 csc2 x10

cot x10 B) 3

50csc2 x

10 cot x

10

C) -6 csc x10 D) -

310

csc2 x10

Find dy/dx.

37) y =7

x3 37)

A) dydx

=37

x-4/7 B) dydx

=73

x4/7 C) dydx

=37

x4/7 D) dydx

=73

x-4/7

Use implicit differentiation to find dy/dx and d2y/dx2.38) xy - x + y = 4 38)

A) dydx

= -1 + y x + 1

; d2ydx2

=2y - 2

(x + 1)2B) dy

dx=

y + 1x + 1

; d2ydx2

=2y + 2

(x + 1)2

C) dydx

= -1 + y x + 1

; d2ydx2

=y + 1

(x + 1)2D) dy

dx=

1 - y1 + x

; d2ydx2

=2y - 2

(x + 1)2

Find where the slope of the curve is defined.39) x2y - 6xy2 = 10 39)

A) dydx

=2xy - 6y2

12y - x2; defined at every point except where y =

x212

B) dydx

=2xy - 6y2x(12y + x)

; defined at every point except where x = 0 or y = -x12

C) dydx

=2xy - 6y2x(12y - x)

; defined at every point except where x = 0 or y =x12

D) dydx

=6y2 + 12xy

x2; defined at every point except where x = 0

Solve the problem.40) Given (x - 3)2 + (y + 2)2 = 106, find dy/dx and the slope of the curve at the point (-6, 3). 40)

A) dydx

= -y + 2x - 3

; 59 B) dy

dx=

x - 3y + 2

; -95

C) dydx

= -x - 3y + 2

; 95 D) dy

dx=

y + 2x - 3

; -59

Find an equation for the tangent to the graph of y at the indicated point. Round to the nearest thousandth whennecessary.

41) y = tan-1 x, x = 6 41)

A) y =137

x + 1.243 B) y =136

x + 1.239

C) y =137

x + 1.406 D) y = 0.169x + 0.391

10

Find the derivative of y with respect to the appropriate variable.42) y = 2 sin-1 (5x3) 42)

A) 2

1 - 25x6B) 30x2

1 - 25x6C) 30x2

1 - 25x3D) 30x2

1 - 25x6

Find dy/dx.43) f(x) = 7e-3x 43)

A) 21e-3x B) -21e-3x C) -3e-3x D) 7e-3x

44) y = ln 3x2 44)

A) 2xx2 + 3

B) 12x + 3 C) 6

x D) 2x

45) y = x + 4 45)A) + 4)x + 3 B) x + 4 C) x + 4 D) + 3)x + 3

Use logarithmic differentiation to find dy/dx.46) y = (cos x)x 46)

A) (cos x)x (ln cos x - x tan x) B) ln x(cos x)x - 1

C) ln cos x - x tan x D) (cos x)x (ln cos x + x cot x)

Solve the problem.47) Suppose that the amount in grams of a radioactive substance present at time t (in years) is given by

A(t) = 530e-.26t. Find the rate of decay of the quantity present at the time when t = 3.47)

A) 2.2 grams per year B) -2.2 grams per yearC) 63.2 grams per year D) -63.2 grams per year

Find the extreme values of the function on the interval and where they occur. Identify any critical points that are notstationary points.

48) g(x) = -x2 + 12x - 32, 4 x 8 48)A) Local maximum at 7, 4 ; minimum value is 0 at x = 8 and at x = 4B) Local maximum at 6, 4 ; minimum value is 0 at x = 8 and at x = 4C) Local maximum at 6, 4 ; minimum value is -32 at x = 0D) Local maximum at 6, 68 ; minimum value is -32 at x = 0

Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema.49) y = 8x3 + 2 49)

A) Absolute maximum at (0, 2) B) Local minimum at (0, 0)C) None D) Local maximum at (0, 8)

Use the Concavity Test to find the intervals where the graph of the function is concave up.50) y = -3x2 + 18x + 4 50)

A) (- , 3) B) (- , ) C) (3, ) D) None

Find all points of inflection of the function.51) y = x3 - 12x2 + 2x + 15 51)

A) (4, -105) B) (-4, -48) C) (4, -46) D) (4, -252)

11

Use the graph of f to estimate where f' is 0, positive, and negative.52) 52)

A) Zero: x = ±1; positive: x = (- , -1) and (1, ); negative: x = (-1, 1)B) Zero: x = ±1; positive: x = (- , -1); negative: x = (-1, 1)C) Zero: x = ±1; positive: x = (- , -1) and (1, ); negative: x = (0, 1)D) Zero: x = ±1; positive: x = (1, ); negative: x = (-1, 1)

Use the graph of f to estimate where f'' is 0, positive, and negative.53) 53)

A) Zero: x = ±1; positive: (- , 0) ); negative: (0, )B) Zero: x = 0; positive: (0, ); negative: (- , 0)C) Zero: x = 0; positive: (- , 0) ); negative: (0, )D) Zero: x = ±1; positive: (0, ); negative: (- , 0)

Use the Second Derivative Test to find the local extrema for the function.54) y = x2 + 3x - 1 54)

A) Local minimum: -32

, -134 B) Local maximum: -

32

, -134

C) Local minimum: 32

, -54 D) Local maximum: 3

2, 13

4

12

Solve the problem.55) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the

points (- 2,1) , -6

3,59

, (0,0), 63

,59

and ( 2,1), and whose first two derivatives have the

following sign patterns.

y : + - + -

- 2 0 2

y :

- + -

-6

36

3

55)

A) B)

C) D)

56) From a thin piece of cardboard 30 in. by 30 in., square corners are cut out so that the sides can befolded up to make a box. What dimensions will yield a box of maximum volume? What is themaximum volume? Round to the nearest tenth, if necessary.

56)

A) 10 in. × 10 in. × 10 in.; 1000 in.3 B) 15 in. × 15 in. × 7.5 in.; 1687.5 in.3

C) 20 in. × 20 in. × 10 in.; 4000 in.3 D) 20 in. × 20 in. × 5 in.; 2000 in.3

57) Find the number of units that must be produced and sold in order to yield the maximum profit,given the following equations for revenue and cost:R(x) = 40x - 0.5x2C(x) = 4x + 2.

57)

A) 38 units B) 37 units C) 44 units D) 36 units

13

58) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find thedimensions of the stiffest beam than can be cut from a 10-in.-diameter cylindrical log. (Roundanswers to the nearest tenth.)

10"

58)

A) w = 6.0; d = 9.7 B) w = 6.0; d = 7.7 C) w = 4.0; d = 9.7 D) w = 5.0; d = 8.7

59) Water is falling on a surface, wetting a circular area that is expanding at a rate of 5 mm2/s. Howfast is the radius of the wetted area expanding when the radius is 145 mm? (Roundapproximations to four decimal places.)

59)

A) 182.2122 mm/s B) 0.0345 mm/s C) 0.0055 mm/s D) 0.0110 mm/s

60) One airplane is approaching an airport from the north at 203 km/hr. A second airplane approachesfrom the east at 300 km/hr. Find the rate at which the distance between the planes changes whenthe southbound plane is 32 km away from the airport and the westbound plane is 16 km from theairport.

60)

A) 2714 km/hr B) 2979 km/hr C) 96 km/hr D) 499 km/hr

61) Suppose that5

3f(x) dx = -4. Find

5

5f(x) dx and

3

5f(x) dx . 61)

A) -4; 4 B) 0; -4 C) 0; 4 D) 5; -4

62) Suppose that h is continuous and that2

-2h(x) dx = 6 and

7

2h(x) dx = -9. Find

7

-2h(t) dt and

-2

7h(t) dt .

62)

A) -15; 15 B) -3; 3 C) 3; -3 D) 15; -15

Interpret the integrand as the rate of change of a quantity and evaluate the integral using the antiderivative of thequantity.

63)0

9 sin x dx 63)

A) 18 B) 9 C) 162 D) 2

Evaluate the integral.

64)8

1/22 -

1x

dx 64)

A) 16 - ln 16 B) 15 - ln 4 C) 15 - ln 0.25 D) 15 - ln 16

14

Find the total area of the region between the curve and the x-axis.65) y = x2 - 6x + 9; 2 x 4 65)

A) 43 B) 2

3 C) 73 D) 1

3

Find the area of the shaded region.66) f(x) = x3 + x2 - 6x

g(x) = 6x

66)

A) 8112 B) 343

12 C) 1603 D) 768

12

Find the area of the regions enclosed by the lines and curves.67) y = 3x + 4, y = x2 + 4 67)

A) 932 B) 9 C) 9

2 D) 18

15