Final Exam practice

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

1) The power P (in W) generated by a particular windmill is given by P = 0.015 V3 where V is the velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is 5.3 mph. Round your answer to the nearest tenth. A) 0.2 W/mph B) 1.3 W/mph C) 4.5 W/mph D) 2.8 W/mph 2) A state park charges $12 per day or fraction of a day to rent a tent site, plus a fixed $5 park maintenance fee. Let 3 T(x) represent the cost to stay in a tent site for x days. Find T 10 . 10 A) $137.00

B) $125.00

C) $120.00

D) $128.60

3) Recent research has shown that the population f(S) of cod in the North Sea next year as a function of this year's population S (measured in thousands of tons) can be described by the Shepherd model, aS f(S) = 1 + (S/b)c where a, b, and c are constants. The values of a, b, and c are 3.031, 249.48, and 3.24, respectively. Find the approximate value of this year's population that maximizes next year's population using this model. A) 4000 tons B) 195,000 tons C) 195 tons D) 160,000 tons

4) The population of coyotes in the northwestern portion of Alabama is given by the formula p(t) = (t2 + 100) ln(t + 2) , where t represents the time in years since 2000 (the year 2000 corresponds to t = 0). Find the rate of change of the coyote population in 2016 (t = 16). A) 92 coyotes/year B) 112 coyotes/year C) 20 coyotes/year

D) 216 coyotes/year

5) The information in the chart below gives the salary of a person for the stated years. Model the data with a linear function using the points (1, 24,300) and (3, 26,600). Then use this function to predict the salary for the year 2005. Year, x Salary, y 1990, 0 $23,500 1991, 1 $24,300 1992, 2 $25,200 1993, 3 $26,600 1994, 4 $27,200 A) $40,770

B) $40,750

C) $40,790

D) $40,730

6) The annual revenue and cost functions for a manufacturer of precision gauges are approximately R(x) = 520x - 0.03x2 and C(x) = 120x + 100,000, where x denotes the number of gauges made. What is the maximum annual profit? A) $1,433,333

B) $1,233,333

C) $1,333,333

D) $1,533,333

7) The force F (in N) exerted by a cam on a lever is given by F = x4 - 10x3 + 50x2 - 67x + 30, where x (1 x 5) is the distance (in cm) from the center of rotation of the cam to the edge of the cam in contact with the lever. Find the instantaneous rate of change of F with respect to x when x = 3 cm. A) 20 N/cm B) 44 N/cm C) 71 N/cm D) 90 N/cm

1

8) Exposure to ionizing radiation is known to increase the incidence of cancer. One thousand laboratory rats are exposed to identical doses of ionizing radiation, and the incidence of cancer is recorded during subsequent days. The researchers find that the total number of rats that have developed cancer t months after the initial exposure is modeled by N(t) = 1.25t2.3 for 0 t 10 months. Find the rate of growth of the number of cancer cases at the 7th month. Round your answer to the nearest tenth, if necessary. A) 40.1 cases/month B) 31.1 cases/month C) 252.6 cases/month

D) 36.1 cases/month

9) The correlation between respiratory rate and body mass in the first three years of life can be expressed by the function log R(w) = 1.88 - 0.35 log (w), where w is the body weight (in kg) and R(w) is the respiratory rate (in breaths per minute). Find R'(w) using implicit differentiation. A) R'(w) = 75.86w-1.35 B) R'(w) = -26.55w-1.35 C) R'(w) = -26.55w-0.35

D) R'(w) = -26.55w-0.65

10) In a lab experiment 6 grams of acid were produced in 16 minutes and 13 grams in 40 minutes. Let y be the grams produced in x minutes. Write a linear equation for grams produced. 7 4 24 4 7 4 7 4 x+ xxxA) y = B) y = C) y = D) y = 24 3 7 3 24 3 24 3 11) Find the point of diminishing returns (x, y) for the function R(x) = 6000 - x3 + 33x2 + 400x, 0 x 20, where R(x) represents revenue in thousands of dollars and x represents the amount spent on advertising in tens of thousands of dollars. A) (11 , 13,062) B) (13.2, 14,729.95) C) (14 , 15,324) D) (48.95, -12,637.83) 12) Given is a graph of a portion of the postage function, which depicts the cost (in cents) of mailing a letter, p, versus the weight (in ounces) of the letter, x. Find each limit, if it exists: x

lim p(x), lim p(x), lim p(x) x 3 3x 3+

A) 99; 77; does not exist C) 77; 77; 77

B) 77; 99; 77 D) 77; 99; does not exist

13) Suppose that a velocity function is given by v(t) = 9t3 . Find the position function s(t) if s(0) = 7. 9 9 A) s(t) = t4 B) s(t) = t4 + 7 C) s(t) = 27t2 + 7 D) s(t) = 9t4 + 7 4 4

2

14) The magnitude of an earthquake, measured on the Richter scale, is given by R(I) = log

I , where I is the I0

amplitude registered on a seismograph located 100 km from the epicenter of the earthquake, and I0 is the

amplitude of a certain small size earthquake. An earthquake measured 7.7 on the Richter scale. Express this reading in terms of I0 .

A) 5,011,872 I0

B) 50,118,723 I0

C) 2207 I0

15) Find two numbers whose sum is 450 and whose product is as large as possible. A) 10 and 440 B) 224 and 226 C) 225 and 225

D) 39,810,717 I0

D) 1 and 449

16) The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t2 ). Find the growth rate at t = 13 months. A) 72 mice/month B) 144 mice/month C) 36 mice/month D) 172 mice/month 17) On a summer day, the surface water of a lake is at a temperature of 20° Celsius. What is this temperature in Fahrenheit? A) 36° B) 20° C) 68° D) 52° 18) A company knows that unit cost C and unit revenue R from the production and sale of x units are related by R2 + 8899. Find the rate of change of revenue per unit when the cost per unit is changing by $9 and the C= 258,000 revenue is $4000. A) $290.25

B) $180.00

C) $590.08

D) $889.90

19) The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.08t where k is a constant and t is the time in years. If the current population is 32,000, in how many years is the population expected to be 80,000? A) 5 yr B) 11 yr

C) 80 yr

D) 6 yr

20) Prairie dogs form an important part of the coyote's diet. As coyotes are hunting for prairie dogs, they must be careful to expend just the right amount of time at each burrow. If a coyote spends too little time at each burrow, it catches very few prairie dogs per kilocalorie of energy expended. Likewise, if the coyote spends too much time digging at a single burrow, it can expend a large amount of energy per prairie dog caught. The relation 1 20 + between energy expended and time spent at each burrow is approximated by E = t2 for t > 0.75 t t - 0.75 minutes, where t is in minutes and E is in kcal expended per prairie dog caught. How much time should a coyote spend at each burrow to minimize the energy expended per prairie dog caught. (Hint: pay close attention to the domain of the above function.) A) .8 minutes B) 10 minutes C) 1.5 minutes D) 2.0 minutes

21) An economist predicts that the buying power B(x) of a dollar x years from now will decrease according to the formula B(x) = 0.6x. How much will today's dollar be worth in 2 years? Round to the nearest cent. A) $0.36

B) $1.52

C) $1.20

D) $0.90

22) In the formula A(t) = A0 ekt, A(t) is the amount of radioactive material remaining from an initial amount A0 at a given time t and k is a negative constant determined by the nature of the material. An artifact is discovered at a certain site. If it has 78% of the carbon-14 it originally contained, what is the approximate age of the artifact, rounded to the nearest year? (carbon-14 decays at the rate of 0.0125% annually.) A) 863 yr B) 6240 yr C) 1988 yr D) 1760 yr

3

23) S(x) = -x3 - 9x2 + 165x + 1300, 5 x 20 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon. A) 6°C B) 5°C C) 19°C D) 20°C 24) The work W (in joules) done by a force F (in newtons) moving an object through a distance x (in meters) is given by W =

A) W =

F dx . Find a formula for W, if F = kx and k is a constant.

kx +C 2

B) W =

kx2 +C 2

C) W = kx2 + C

D) W = k + C

25) The total cost to produce x handcrafted wagons is C(x) = 110 + 5x - x2 + 3x3. Find the rate of change of cost with respect to the number of wagons produced (the marginal cost) when x = 7. A) $1125 per wagon B) $1015 per wagon C) $542 per wagon D) $432 per wagon 26) Southwest Dry Cleaners believes that it will need new equipment in 5 years. The equipment will cost $26,000. What lump sum should be invested today at 4% compounded semiannually, to yield $26,000? Round to the nearest cent. A) $21,329.06 B) $23,649.29 C) $24,611.99 D) $23,531.64 27) A college student invests $9000 in an account paying 6% per year compounded annually. In how many years will the amount at least quadruple? Round to the nearest tenth when necessary. A) 25.8 yr B) 29.3 yr C) 23.8 yr D) 27.6 yr 28) The volume of a sphere is increasing at a rate of 9 cm3 /s. Find the rate of change of its surface area when its 32 volume is cm3 3 A)

8 cm2 /s 3

B) 3 cm2 /s

C) 6 cm2 /s

D) 9 cm2 /s

29) The population of a city, in millions, since 1990 has grown at a rate of P (t) = 0.46e0.023t million people per year, where t is the number of years after 1990. If there were 1.52 million people in 2000, estimate (to two decimal places) the population in 2006. A) P(16) 28.90 million B) P(16) -23.65 million C) P(16) 5.24 million D) P(16) 52.55 million 30) The cost of a computer system increases with increased processor speeds. The cost C of a system as a function of processor speed is estimated as C(s) = 13s2 - 6s + 1000, where s is the processor speed in MHz. Determine the intervals where the cost function C(s) is decreasing. A) (0.2, ) B) Nowhere

C) Everywhere

D) (

, 0.2)

31) Let the demand and supply functions be represented by D(p) and S(p), where p is the price in dollars. Find the equilibrium price and equilibrium quantity for the given functions. D(p) = 163,200 - 260p S(p) = 540p A) $204; 110,160 B) $280; 110,160 C) $280; 90,400 D) $302; 84,680

4

32) The graph gives the profit P(x) as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.

A) 3 units

B) 6 units

C) 5 units

D) 4 units

33) When an amount of heat Q (in kcal) is added to a unit mass (in kg) of a substance, the temperature rises by an amount T (in degrees Celsius). The quantity dQ/dT, called the specific heat, is 0.18 for glass. If dQ/dt = 11.1 kcal/min for a 1 kg sample of glass at 20.0°C, find dT/dt for this same sample. A) 64.7 kcal/min B) 1.998 kcal/min C) 29.8 kcal/min D) 61.7 kcal/min 34) Northwest Molded molds plastic handles which cost $1.00 per handle to mold. The fixed cost to run the molding machine is $5505 per week. If the company sells the handles for $4.00 each, how many handles must be molded weekly to break even? A) 1223 handles B) 5505 handles C) 1101 handles D) 1835 handles 35) The population of a small country increases according to the function B = 1,800,000e0.04t, where t is measured in years. How many people will the country have after 4 years? A) 1,432,584 people B) 2,601,792 people C) 3,298,647 people D) 2,112,320 people 36) When a radioactive substance decays, the number N of grams remaining from an initial mass N given by N = N

(in grams) is 0 (1/2)n , where n is the number of half-lives for which the substance has decayed. Given that

0 the half-life for tritium is 13 years, find the rate in (grams/half-life) at which a 88-gram initial mass of radioactive tritium decays after 41.6 years. A) -3.14 g/half-life B) -9.57 g/half-life C) -6.64 g/half-life D) -60.99 g/half-life

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37) Suppose that the cost, C, of producing x units of a product can be illustrated by the given graph. Is C(x) continuous at x = 50? x = 100? x = 150?

A) Yes; no; yes

B) Yes; no; no

C) No; no; no

D) Yes; yes; yes

38) A hotel has 220 units. All rooms are occupied when the hotel charges $80 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $28 per day to service and maintain. What should the hotel charge per day in order to maximize daily profit? A) $174 B) $84 C) $164 D) $150 39) A swimming pool has a leak, and the leak is getting worse. The table below gives the leakage rate every 6 hours. Use right endpoints to estimate the number of gallons lost in 48 hours. Time Leakage (hr) (gal/hr) 0 0 6 0.6 12 1.3 18 1.9 24 3.0 30 4.5 36 5.9 42 7.0 48 8.5 A) 32.7 gal

B) 256.2 gal

C) 145.2 gal

D) 196.2 gal

40) A shoe company will make a new type of shoe. The fixed cost for the production will be $24,000. The variable cost will be $30 per pair of shoes. The shoes will sell for $102 for each pair. What is the profit if 600 pairs are sold? A) $19,200 B) $55,200 C) $67,200 D) $43,200 41) Find the present value of the deposit. $9000 at 8% compounded quarterly for 10 years. Round to the nearest cent. A) $19,792.36 B) $4076.01 C) $4156.01 D) $19,872.36

6

42) After a new firm starts in business, it finds that its rate of profit (in hundreds of dollars) after t years of operation is given by P'(t) = 3t2 + 12t + 7. Find the profit in year 3 of the operation. A) $7400 B) $6600

C) $4600

D) $5600

43) Rats are not native to the islands off the western coast of South America. However, rats are often introduced accidentally to an island by visiting ships. The population of introduced rats follows the logistic function with k = 0.00023 and t in months. Assume that there are 8 rats initially and that the maximum population size is 10,000. Find the rate of growth of the population after 4 months. A) 2308 rats/month B) 2280 rats/month C) 2289 rats/month D) 2416 rats/month Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema. 44) f(x) = x4/3 - x2/3 A) Relative maximum of 0 at 0; Relative minimum of -

1 2 2 at and 4 4 4

B) Relative maximum of 0 at 0; Relative maximum of -

1 2 at 4 4

C) No relative extrema. D) Relative minimum of of -

1 2 at 4 4

45) f(x) = ln x - x, x > 0 A) (1, -1), relative maximum C) (-1, 0), relative minimum

B) (1, 0), relative minimum D) (-1, -1) relative maximum

Decide whether the graph represents a function. 46)

A) Function

B) Not a function

Find the integral. 47)

x4 dx

A) 4x3 + C

B) 5x5 + C

C)

7

x5 +C 5

D)

x3 +C 4

48)

1

x(ln x)20

A) -

49)

dx

1

19(ln x)19

+C

B)

1

x(ln x)21

+C

C) -

C)

B) 5 ln x + 2xex-1 + C

10 + 2xex-1 + C x2

D)

10 + 2ex + C x2

1 (6x - 7)3/2 + C 2

B)

1 (6x - 7)3/2 + C 6

C)

1 (6x - 7)3/2 + C 9

D)

1 (6x - 7)3/2 + C 3

4 3 3 2 x - x +C 3 2

4 3 3 2 x - x +C 3 2

C)

4 2 3 x + x+C 3 2

D)

4 3 x +C 3

B) -

e-1/t7 +C 7

B) - e1/t7 + C

C) -

e1/t7 +C 7t7

D) -

e1/t7 +C 7

(x6 - 2x5 )7 (6x5 - 10x4 ) dx

A)

54)

+C

e1/t7 dt t8

A)

53)

1

19x(ln x)19

(4x2 - 3x) dx

A)

52)

D) -

6x - 7 dx

A)

51)

+C

5 + 2ex dx x

A) 5 ln x + 2ex + C

50)

1

21(ln x)21

1 6 (x - 2x5 )7 + C 7

B)

1 6 (x - 2x5 )8 + C 8

C) (x6 - 2x5 )8 + C

B)

6 5x +C 5 ln 6

C)

D) 6x5 - 10x4 + C

6 5x dx

A)

6 5x +C ln 6

8

6 5x +C 5

D)

6 6x +C 6

Give the domain and range of the function. 55)

A) Domain (-7, 7) ; Range {-3} C) Domain {-3} ; Range (-7, 7)

B) Domain [-7, 7] ; Range {-3} D) Domain ( , ) ; Range {-3}

Identify the open intervals where the function is changing as requested. 56) Increasing

A) (3, )

B) (3, 6)

C) (-2, )

D) (-2, 0)

B) (x + 1)-2

C) (x + 1)-3

D) -2(x + 1)-2

C) 31,536

D) 31,584

Find f"(x) for the function. x 57) f(x) = x+1 A) -2(x + 1)-3

Find the derivative. 58) g(x) = 5x5 + x4 - 4x2 + 7, find g'(-6) A) 32,448 B) -816 59) y = (x + 3)4 e-4x A) -(x + 3)3 (4x + 8) e-5x

B) -16(x + 3)3 e-4x D) -(x + 3)3 (4x + 8) e-4x

C) (x + 3)3 (x + 7) e-4x

9

60) y = ex ln x, x > 0 ex A) x

B)

ex(x ln x + 1) x

C) ex ln x

D)

ex(ln x + x) x

D)

ln 7 (7 x) x

61) y = 2(7 x) A) 2 ln 7 (7 x) ( x)

62) f(x) =

B) ln 7 (7 x) ( x)

C)

2 ln 7 (7 x) x

5

(2x - 3)4

A) f'(x) =

5

4(2x - 3)3

B) f'(x) =

5

C) f'(x) =

8(2x - 3)5

-40

(2x - 3)3

D) f'(x) =

-40

(2x - 3)5

Find the absolute extrema if they exist as well as where they occur. 63) f(x) = -3x4 + 16x3 - 18x2 + 3

A) Absolute maximum of 11 at x = 2; no absolute minima B) No absolute extrema C) Absolute maximum of -2 at x = 1; no absolute minima D) Absolute maximum of 30 at x = 3; no absolute minima

Give the range for the function if the domain is {-2, -1, 0, 1, 2}. 64) 3x + y = 11 A) {-5, -8, -11, -14, -17} C) {17, 14, 11, 8, 5}

B) {-5, -7, -9, -11, -13} D) {13, 11, 9, 7, 5}

Find the open intervals where the function is concave upward or concave downward. Find any inflection points. 65)

A) Concave upward on ( B) Concave upward on ( -2) C) Concave upward on ( and (2, -2) D) Concave upward on ( -2)

, -1) and (2, ); concave downward on (-1, 2); inflection point at (0, -1) , -1) and (1, ); concave downward on (-1, 1); inflection points at (-1, -3) and (1, , -1) and (1, ); concave downward on (-1, 1); inflection points at (-3, -5), (0, -1), , -3) and (2, ); concave downward on (-3, 2); inflection points at (-1, -3) and (1,

10

Evaluate the definite integral. 0 (4 + x2 ) dx 66) -1 A) 4

67) 1

13 3

C) 0

D) -2

4 2 t +1 dt t

A)

5

68)

B)

92 5

B) 32

C)

72 5

D)

77 5

B) - 5

C)

1 3

D)

65 3

(x - 4)2 dx

0

A) 65

Find the open interval(s) where the function is changing as requested. 1 69) Increasing; f(x) = x2 + 1 A) (- , 1)

B) (- , 0)

C) (1, )

D) (0, )

C) 120x

D) 160x2 + 10

C) x = -3, x = 0, x = 3

D) x = -2, x = 0, x = 2

Find the indicated derivative of the function. 70) f(4)(x) of f(x) = 2x5 - 5x2 - 2x + 1 A) 240x

B) 160x + 10

Find the x-values where the function does not have a derivative. 71)

A) x = -3, x = 3

B) x = -2, x = 2

Find the slope of the line tangent to the graph of the function at the given value of x. 72) y = 4x3/2 - 5x1/2; x = 16 A)

197 8

Find the derivative of the given function. 73) y = (x2 + 4)3 A) 3x5 + 48x3 + 96x

B)

91 4

C)

B) 6x5 + 40x3 + 96x

101 4

C) 6x5 + 48x3 + 96x

11

D)

187 8

D) 6x5 + 24x3 + 48x

Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. 74) Use a graphing utility to approximate the instantaneous rate of change of f(x) = x1/x at x = 5.

A) -0.0423

B) -0.0673

Find dy/dx by implicit differentiation. 75) x3 + y3 = 5 A)

y2 x2

B) -

C) 0.0750

x2 y2

C) -

76) y5 ex + x = y3 x

y2 x2

A)

dy y3 - 1 = dx 5y4 ex - 3xy2

B)

dy y3 - 1 = dx 5y4 ex - 3xy2 + 1

C)

dy y3 - y5 ex - 1 = dx 5y4 ex - 3xy2 - 1

D)

dy y3 - y5 ex - 1 = dx 5y4 ex - 3xy2

D) -0.0336

D)

x2 y2

Find the equation of the tangent line to the curve when x has the given value. x3 ;x=4 77) f(x) = 2 A) y = 8x + 64

B) y = 24x - 64

C) y = 64x + 24

Use a graphing calculator to find f'(x) when x has the given value. 78) f(x) = x + 3; x = 6 1 1 A) B) 6 6 79) f(x) = x10/x; x = 9 A) 2.1689

B) -3.0684

C) -

5 6

C) -1.6981

D) y = 8x - 64

D)

5 6

D) -0.4545

Give an appropriate answer. 80) Let lim f(x) = 9 and lim g(x) = -5. Find lim [f(x) - g(x)]. x 10 x 10 x 10 A) 10

B) 9

C) 14

81) If g (3) = 4 and h (3) = -1, find f (3) for f(x) = 5g(x) + 3h(x) + 2. A) 23 B) 25 C) 19

D) 4

D) 17

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.

82) How is the graph of y = f(x) = x5 - 4x + 3 related to the graph of y = g(x) = (x + 4)5 - 4(x + 4) + 3? How is the slope of the graph of g(x) at x = a related to the slope of the graph of f(x) at x = a + 4? 83) Explain how the graph of y = (1/2)x - 2 can be obtained from the graph of y = 2 x.

12

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 84) If Q = 89 - e-0.6t what happens to Q and to Q' as t increases? A) Q increases and Q' decreases. B) Q increases and Q' increases. C) Q decreases and Q' increases. D) Q decreases and Q' decreases. 85) True or false? If average product is increasing then the marginal product must be increasing. A) True B) False Complete the table and use the result to find the indicated limit. x-4 , find lim f(x). 86) If f(x) = x-2 x 4 x f(x)

A)

B)

C)

D)

3.9

3.99

3.999

4.001

4.01

4.1

x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993

4.001 4.01 4.1 ; limit = 1.20 1.20007 1.20075 1.20745

x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993

4.001 4.01 4.1 ; limit = 1.20007 1.20075 1.20745

x 3.9 3.99 3.999 f(x) 5.07736 5.09775 5.09978

4.001 4.01 4.1 ; limit = 5.10 5.10022 5.10225 5.12236

x 3.9 3.99 3.999 f(x) 3.97484 3.99750 3.99975

4.001 4.01 4.1 ; limit = 4.0 4.00025 4.00250 4.02485

Find an equation in slope-intercept form (where possible) for the line. 87) Through (4, 0) and (0, -5) 5 4 4 A) y = x - 5 B) y = x - 5 C) y = - x - 5 4 5 5 88) Through (6, -9), perpendicular to -3x + 5y = -63 5 3 A) y = - x + 1 B) y = - x + 3 3 5

C) y = -

89) Through (4, 5), with undefined slope 5 A) x = 4 B) - x + 4y = 0 4

C) -

5 x 3

4 x + 5y = 0 5

D) y = -

D) y =

8 x 5

B) y =

8 96 x+ 5 5

C) y =

13

4 64 x+ 5 5

5 x-1 3

D) y = 5

Find the equation of the tangent line to the graph of the given function at the given value of x. 90) f(x) = (x2 + 28)4/5 ; x = 2 A) y =

5 x-5 4

D) y =

8 64 x+ 5 5

Find all values of x for the given function where the tangent line is horizontal. x 91) f(x) = (x2 + 7)3 A) 0

B) ±

7 5

C) 0, ±

35 5

D) ±

35 5

Solve the following.

92) At what points on the graph of f(x) = 2x3 - 3x2 - 78x is the slope of the tangent line -6? A) (4, -232), (1, -79) B) (-4, 136), (-232, -207) C) (0, 0), (-3, 153) D) (4, -232), (-3, 153)

Provide the proper response. 93) True or false? If the graph of a function f is concave up on its entire domain, then f' is decreasing. A) True B) False b

94) If r(t) is the rate of change of revenue, then

r(t) dt is

a i) the total revenue up to time b. ii) the total revenue from time a to time b. iii) the change in revenue at any time. A) Only i is correct. C) Only iii is correct.

B) Only ii is correct. D) None of the above is correct.

Suppose that the function with the given graph is not f(x), but f (x). Find the open intervals where f(x) is increasing or decreasing as indicated. 95) Decreasing

A) (

, -1), (1, )

B) (

, 0)

C) (-1, 1)

14

D) (0, )

Suppose that the function with the given graph is not f(x), but f (x). Find the open intervals where the function is concave upward or concave downward, and find the location of any inflection points. 96)

A) Concave upward on (-2, 2); concave downward on ( , -2) and (2, ); inflection points at -120 and 120 B) Concave upward on ( , 0); concave downward on (0, ); inflection point at 0 C) Concave upward on (-2, 2); concave downward on ( , -2) and (2, ); inflection points at -2 and 2 D) Concave upward on ( , -2) and (2, ); concave downward on (-2, 2); inflection points at -2 and 2 Suppose that the function with the given graph is not f(x), but f (x). Find the locations of all extrema, and tell whether each extremum is a relative maximum or minimum. 97)

A) Relative maxima at -2 and 2 C) Relative minimum at -2; relative maximum at 2

B) Relative minimum at -4 D) Relative maximum at -2; relative minimum at 2

Solve the equation. 1 98) 5 x = 625 A) 4

B)

1 125

C) -4

D)

B)

3 5

C) 3, -2

D) No solution

99) log5 x2 = log5 (1x + 6) A) 3

15

1 4

Find the domain of the function. 100) f(x) = log3 (16 - x2 ) A) -16 < x < 16

Give the domain of the function. 101) g(z) = 25 - z 2 A) (-5, 5)

B) x < -4 and x > 4

C) -4 < x < 4

D) -4

B) [0, )

C) (- , )

D) [-5, 5]

Find the derivative of the function. 102) y = ln (3 + x2 ) A)

1 2x + 3

103) y = log

3 3 A) ln 3

x 4

B)

2 x

C)

2x 2 x +3

D)

6 x

B)

3 ln3 3x + 1

C)

3 2(ln 3)(3x + 1)

D)

3 ln 3 (3x + 1)

3x + 1

Find the equation of the tangent line at the given value of x on the curve. 104) xy + x = 2, x = 1 1 1 1 3 A) y = x + B) y = - x + C) y = 2x - 1 2 2 2 2

D) y = -2x + 3

The table lists the values of the functions f and g and their derivatives at several points. Use the table to find the indicated derivative. x 1 2 3 4 f(x) 4 3 1 2 105) f'(x) -5 -3 4 1 g(x) 3 4 1 2 g'(x) -3 6 3 4 Find Dx(g[f(x)]) at x = 2.

A) 3

B) -3

C) -9

D) 24

Use the properties of limits to help decide whether each limit exits. If a limit exists, find its value. 2 106) Let f(x) = x + 1 if x < 0 . Find lim f(x). -3 if x 0 x -3 A) Does not exist

B) 10

C) 1

Find the instantaneous rate of change for the function at the given value. 107) g(t) = 5t2 + t at t = -4 A) -41

B) -14

C) -39

Find the largest open intervals where the function is concave upward. 108) f(x) = x2 + 2x + 1 A) (- , -1)

B) (-1, )

C) (- , )

16

D) -3

D) 6

D) None

Use the product rule to find the derivative. 109) f(x) = (x2 - 2x + 2)(4x3 - x2 + 4)

A) f'(x) = 4x4 - 32x3 + 30x2 + 4x - 8 C) f'(x) = 20x4 - 36x3 + 30x2 + 4x - 8

B) f'(x) = 4x4 - 36x3 + 30x2 + 4x - 8 D) f'(x) = 20x4 - 32x3 + 30x2 + 4x - 8

Find the slope of the line passing through the given pair of points. 110) (-1, -4) and (-7, 9) 6 13 A) B) 13 6

C) -

5 8

D) -

13 6

Find all values x = a where the function is discontinuous. x-3 111) f(x) = ln x+7 A) a = -3, 7

B) a = 3, -7

C) a = -7

Use natural logarithms to evaluate the logarithm to the nearest thousandth. 112) log5 0.203 A) -0.991

B) -1.009

C) -0.693

Find the equation of the tangent line at the given point on the curve. 113) y x + 1 = 4; (3, 2) 1 11 1 5 1 1 A) y = - x + B) y = x + C) y = x + 4 4 4 4 2 2

D) Nowhere

D) 24.631

D) 2y = -

1 7 x+ 2 2

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. x2 - 25 114) lim 2 x 5 x - 8x + 15 A) 0

115)

lim x -

B) Does not exist

C) 5

B) 0

C)

D)

5 2

x 5x - 15

A)

Write the logarithmic equation in exponential form. 116) log2 8 = 3 A) 2 8 = 3

B) 8 3 = 2

1 5

C) 3 2 = 8

D) -

1 5

D) 2 3 = 8

Graph the function on the indicated domain, and use the capabilities of your calculator to find the location and value of the indicated absolute extremum. 117) f(x) = (x - 3)(x + 3); [0, ) Minimum A) -8.75 at x = 0.5 B) -9 at x = 0 C) -8.51 at x = 0.7 D) -8.91 at x = 0.3

17

Find all values of x (if any) where the tangent line to the graph of the function is horizontal. 118) y = x3 + 5x2 - 333x + 37 A) -

37 ,9 3

B) 9

C) -

37 37 , ,9 3 3

D)

37 , -9 3

Use the graph to evaluate the function f(x) at the indicated value of x. 119) Find f(1).

A) 0 C) 1

B) 2 D) None of these are correct.

Find the value of the constant k that makes the function continuous. 2 if x 9 120) h(x) = x x + k if x > 9 A) k = 9 B) k = 72 C) k = 90 Find

D) k = -9

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

121) f(x) = 5x - 10 A) 5

B) 10

C) -5h

D) 2

C) Does not exist

D) 3.5

Use a graphing utility to find the limit, if it exists. 49x2 + 5x + 6 122) lim 2x x A)

B) -3.5

Using a graphing calculator, find the values of x for which f (x) = 0, to three decimal places. 123) f(x) = 3 - x2 x2 - 3 A) 0 C) 0, -1.033, 1.033

B) 0, -1.538, 1.538 D) There are no real values of x for which f (x) = 0.

Evaluate the logarithm without using a calculator. 3 1 124) log2 4 A) -

2 3

B) -

3 2

C)

18

2 3

D)

3 2

Use a graphing calculator to find the location of all relative extrema (to three decimal places). 125) f(x) = x4 - 4x3 - 53x2 - 86x + 79 A) Relative maximum at x = 1.033; relative minima at x = -3.269 and x = 7.174 B) Relative maximum at x = 0.933; relative minima at x = -3.157 and x = 7.132 C) Relative maximum at x = -0.944; relative minima at x = -3.192 and x = 7.136 D) Relative maximum at x = 0.899; relative minima at x = -3.224 and x = 7.161

Let f(x) = 8x2 - 5x and g(x) = 7x + 9. Find the composite. 126) g[f(k)] A) 56k2 + 35k + 9

B) 392k2 + 973k + 603

C) 56k2 - 35k + 9

Graph the equation. 127) 4x + 5y = 9

A)

B)

19

D) 392k2 - 973k + 603

C)

D)

Approximate the area under the graph of f(x) and above the x-axis using n rectangles. 128) f(x) = 16 - x2 from x = -4 to x = 4; n = 2; use midpoints A) 32

B) 48

C) 8

Graph the function. 129) y = -2e-x/2 + 2

A)

B)

20

D) 96

C)

D)

Rewrite the expression as a sum, difference, or product of simpler logarithms. 5p 130) log4 4k A)

log45 + log4 p

B) log4 5 + log4 p - 1 - log4 k

1 + log4 k

C) log4 5p - log4 4k

D)

log45log4 p log4 k

Use the differentiation feature on a graphing calculator to find the indicated derivative. 131) f(x) = 0.73x3 - 1.25x2 + 3.69x - 0.8; f (3) A) 19.650

B) 15.100

C) 15.900

D) 0.900

Use a graphing utility to find the discontinuities of the given rational function. x2 + 4x + 4 132) f(x) = x3 + x2 + 7x - 26 A) -2 C) 3

B) 2 D) The function is continuous for all values of x.

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. 3 ; [1, 3] 133) f(x) = x3 A)

1 2

B)

1 3

Evaluate the function for the given value. 2x + 2 if x 5 2 ; f 134) f(x) = x - 5 m 11 if x = 5 2 2 2 if m , 11 if m = A) m 5 5 C)

(4m + 2) if m (2m- 5)

2 2 , 11 if m = 5 5

C)

4 3

B)

(4 + 2m) if m (2 - 5m)

D) 2 if m

21

D) 3

2 2 , 11 if m = 5 5

2 2 , 11 if m = 5 5

The function gives the distances (in feet) traveled in time t (in seconds) by a particle. Find the velocity and acceleration at the given time. 1 ,t=4 135) s = t+ 4

A) v =

2 1 ft/s, a = ft/s2 512 64

C) v = -

B) v =

1 2 ft/s, a = ft/s2 64 512

1 2 ft/s, a = ft/s2 64 512

D) v = -

2 1 ft/s, a = ft/s2 512 64

Use the quotient rule to find the derivative. x2 + 8x + 3 136) y = x A)

dy 2x + 8 = dx x

B)

dy 3x2 + 8x - 3 = dx x

C)

dy 2x + 8 = dx 2x3/2

D)

dy 3x2 + 8x - 3 = dx 2x3/2

Estimate the slope of the tangent line to the curve at the given point. 137)

A) -4

B)

1 4

C) -

Find the slope of the line. 138) A line parallel to -4y - 3x = -8 3 3 A) B) 4 4 139) A line perpendicular to -7x - 4y = 67 7 A) -7 B) 4

22

1 4

D) -

1 2

C)

8 3

D) -

4 3

C)

4 7

D) -

4 7

140)

A) 0

B) 2

C) -2

D) undefined

Write the function as the composition of two functions f and g such that y = f[g(x)]). 9 +6 141) y = x2 A) f(x) = x + 6, g(x) = C) f(x) =

9 x2

B) f(x) =

9 , g(x) = 6 x2

1 9 , g(x) = + 6 x x

D) f(x) = x, g(x) =

9 +6 x

Find all the critical numbers of the function. -5x 142) f(x) = x+3 A) 0, -3

B) 3

C) -3

Find the average rate of change for the function over the given interval. 143) y = 2x between x = 2 and x = 8 3 A) 2 B) C) 7 10

D) -15, 0

D)

1 3

Approximate the expression in the form a x without using e. Round to the nearest thousandth when necessary. 144) e6x

A) 1.792x

B) 403.429x

C) 16.31x

23

D) 130.387x

Decide whether the limit exists. If it exists, find its value. 145) lim f(x) x 1+

A) 3

1 2

B) 3

Evaluate the function as indicated. 146) Find f(-1.2) when f(x) = 5x - 9.0. A) 3 B) -15

C) Does not exist

D) 4

C) -6.9

D) -3

Assume x and y are functions of t. Evaluate dy/dt. 147) xy2 = 4; dx/dt = -5, x = 4, y = 1 A)

8 5

B) -

5 8

C) -

8 5

D)

5 8

C) -

1 10

D) -

Find the requested value of the second derivative of the function. ln x ; Find f (1). 148) f(x) = 5x A)

ln 2 10

B) 15

3 5

Find any inflection points given the equation. 149) f(x) = ex - 5e-x - 6x A) Inflection point at

1 ln 5, - 3 ln 5 2

B) Inflection point at (2, -4)

C) Inflection point at (0, -4)

D) Inflection point at (ln 5, 4 - 6 ln 5)

Find an equation for the line tangent to given curve at the given value of x. x3 ; x=4 150) y = 4 A) y = 12x - 32

B) y = 4x - 32

C) y = 32x + 12

24

D) y = 4x + 32

Find the exact value of the integral using a formula from geometry. 7 (7 - x) dx 151) 0 A) 24.5 B) 12.25 Write the expression using base e rather than base 10. 152) 10x8 A) 10ex8

B) e(ln 10)x8

C) 49

D) 98

C) e10x8

D) x8 e10

Write a cost function for the problem. Assume that the relationship is linear. 153) Marginal cost, $40; 80 items cost $3400 to produce A) C(x) = 3x + 3400 B) C(x) = 40x + 200 C) C(x) = 3x + 200

D) C(x) = 40x + 3400

Use the properties of logarithms to find the value of the expression. A 154) Let logb A = 3.890 and logb B = 0.231. Find logb . B A) 0.899

B) 4.121

C) 3.890

D) 3.659

Find the location and value of all relative extrema for the function. 155)

A) Relative minimum of -1 at -3 ; Relative maximum of 2 at -1 ; Relative minimum of 0 at 2. B) Relative minimum of 0 at -2 ; Relative maximum of -1 at 2 ; Relative minimum of 2 at 1. C) Relative minimum of -3 at -1 ; Relative maximum of -1 at 2 ; Relative minimum of 2 at 1. D) Relative minimum of -1 at -3 ; Relative maximum of 2 at -1 ; Relative minimum of 1 at 2. Classify the function as even, odd, or neither. 156) f(x) = -4x3 + 5x A) Even

B) Odd

C) Neither

Decide if the given value of x is a critical number for f, and if so, decide whether the point is a relative minimum, relative maximum, or neither. 157) f(x) = -x2 - 16x - 64; x = 8

A) Critical number, relative minimum at (8, -144) C) Not a critical number

B) Critical number but not an extreme point D) Critical number, relative maximum at (8, -144)

Solve the equation. Round decimal answers to the nearest thousandth. 158) 40.17x = 60.52x A) -0.167

B) -1.31

C) 1.523

25

D) 0.000

Use the derivative to find the vertex of the parabola. 159) y = -2x2 + 12x + 4 A) (-3, 22)

B) (3, 22)

C) (3, -22)

D) (-3, -22)

C) x = -2, x = 0

D) x = 0, x = 2

Find all points where the function is discontinuous. 160)

A) x = -2, x = 0, x = 2

B) x = 2

Write an equation of the tangent line to the graph of y = f(x) at the point on the graph where x has the indicated value. -6x2 - 6 ,x=0 161) f(x) = 4x - 3 A) y = -

8 x+2 3

Find f[g(x)] and g[f(x)]. 162) f(x) = 5x + 9; g(x) = 4x - 7 A) f[g(x)] = 20x - 29 g[f(x)] = 20x + 26

B) y =

8 x-2 3

C) y = -

B) f[g(x)] = 20x - 26 g[f(x)] = 20x + 29

8 x-2 3

C) f[g(x)] = 20x + 26 g[f(x)] = 20x - 29

D) y =

8 x+2 3

D) f[g(x)] = 20x + 29 g[f(x)] = 20x - 26

Find the location of the indicated absolute extremum for the function. 163) Minimum

A) x = -5

B) x = 5

C) x = -3

D) x = 3

Suppose the position of an object moving in a straight line is given by the specified function. Find the instantaneous velocity at time t. 164) s(t) = t2 + 5t + 4, t = 5

A) 14

B) 35

Write the exponential equation in logarithmic form. 165) 4 3 = 64 A) log4 64 = 3

B) log64 4 = 3 26

C) 54

D) 15

C) log4 3 = 64

D) log3 64 = 4

Evaluate the function. 166) f(x) = 3x2 + 5x + 1; Find f(a). A) 8a

C) 3a 2 + 5a + 1

B) 8a + 1

Give an appropriate response. 167) Find the limit of f(x) as x approaches 1 from the left. 2 if x < 1 f(x) = x + 3 if 1 x 3 6 if x > 3 A) 2 C) 6

D) 3a 2 + 5a

B) 4 D) The limit does not exist.

Find the area of the shaded region.

y = x3 - 4x

168)

A)

41 4

B)

33 4

C)

9 4

D)

17 4

Find f'(x) at the given value of x. 64 ; Find f (4). 169) f(x) = x A) -16

B) 32

C) - 4

D) 16

Find the equation of the secant line through the points where x has the given values. 7 170) f(x) = ; x = 5, x = 6 x A) y = -

7 x2

B) y =

7 77 x30 30

C) y = -

7 x 30

D) y = -

Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain. 171) f(x) = x3 - 3x2 ; [0, 4] Minimum A) No absolute minimum C) 16 at x = 4

B) 0 at x = 0 D) -4 at x = 2

27

7 77 x+ 30 30

Answer Key Testname: 1810 FINAL EXAM PRACTICE 1) B 2) A 3) B 4) B 5) B 6) B 7) C 8) D 9) B 10) A 11) A 12) D 13) B 14) B 15) C 16) A 17) C 18) A 19) B 20) C 21) A 22) C 23) B 24) B 25) D 26) A 27) C 28) D 29) C 30) D 31) A 32) C 33) D 34) D 35) D 36) C 37) A 38) C 39) D 40) A 41) B 42) D 43) C 44) A 45) A 46) B 47) C 48) A 49) A 50) C 28

Answer Key Testname: 1810 FINAL EXAM PRACTICE 51) A 52) D 53) B 54) B 55) A 56) A 57) A 58) D 59) D 60) B 61) D 62) D 63) D 64) C 65) B 66) B 67) C 68) D 69) B 70) A 71) B 72) D 73) C 74) D 75) B 76) D 77) B 78) B 79) C 80) C 81) D 82) The graph of g is translated 4 units to the left. The slopes are the same. 83) The graph is reflected over the y-axis and then shifted 2 units down. 84) A 85) B 86) D 87) A 88) A 89) A 90) D 91) D 92) D 93) B 94) B 95) A 96) C 97) D 98) C 99) C 100) C 29

Answer Key Testname: 1810 FINAL EXAM PRACTICE 101) 102) 103) 104) 105) 106) 107) 108) 109) 110) 111) 112) 113) 114) 115) 116) 117) 118) 119) 120) 121) 122) 123) 124) 125) 126) 127) 128) 129) 130) 131) 132) 133) 134) 135) 136) 137) 138) 139) 140) 141) 142) 143) 144) 145) 146) 147) 148) 149) 150)

D C C D C B C C C D B A A C C D B A C B A B B A C C A D D B C B C B C D C A C A A C D B B B D D A A 30

Answer Key Testname: 1810 FINAL EXAM PRACTICE 151) 152) 153) 154) 155) 156) 157) 158) 159) 160) 161) 162) 163) 164) 165) 166) 167) 168) 169) 170) 171)

A B B D D B C D B A D B D D A C A A C D D

31