UFFLoRIf>ADepartment of Mathematics

MAC2311Final Exam A

Fall 2015

A. Sign your bubble sheet on the back at the bottom in ink.

B. In pencil, write and encode in the spaces indicated:

1) Name (last name, first initial, middle initial)

2) UF ID number3) Section number

C. Under "special codes" code in the test ID numbers 4, 1.123.567890.234567890

D. At the top right of your answer sheet, for "Test Form Code", encode A.• BCD E

E. 1) This test consists of 26 multiple choice questions, ranging from two points to fourpoints in value. The test is counted out of 80 points, and there are 12 bonuspoints available.

2) The time allowed is 120minutes.3) You may write on the test.4) Raise your hand if you need more scratch paper or if you have a problem with your

test. DO NOT LEAVE YOUR SEAT UNLESS YOU ARE FINISHED WITHTHE TEST.

F. KEEP YOUR BUBBLE SHEET COVERED AT ALL TIMES.

G. When you are finished:

1) Before turning in your test check carefully for transcribing errors. Any mis­takes you leave in are there to stay.

2) You must turn in your scantron to your discussion leader or exam proctor. Beprepared to show your picture I.D. with a legible signature.

3) The answers will be posted online within one day after the exam.

/

NOTE: Be sure to bubble the answers to questions 1-26 on your scantron.

Questions 1 - 20 are worth 4 points each.

1. Determine the value of the constant A so that f(x) is continuous at x = 3:

{

x - 3

f(x)= 2-Jx+1A-x

x<3

x>3

a. A =-2 b. A =-1 c. A=O d. A= 1 e. A = 2

sin4(2x)2. Evaluate the limit: lim 3 4

x-+o X

3 b. ~2

2c.

3d 16. 3 e. does not exista.

16

3. Which of the statements below is true for f(x) = {IXI x < 1 ?lnx x > 1

P. f(x) is continuous but not differentiable at x = O.

Q. f(x} is continuous on (-00, (0).

R. f (x) is differentiable at x = 1.

a. P only b. R only c. P and R only

d. P, Q and R e. None

2A

2X2 - 3x - 54. Let f(x) = I I . If p = lim f(x) and q = lim f(x),x x - 1 x--+1+ x--+-oo

find p and q.

a. p = +00 and q = 2

b. p = -00 and q = -2

c. p = -3 and q = -00

d. p = -00 and q = 2

e. p = +00 and q = -2

2ex5. Consider the function f(x) = . Which of the following statements

eX - 1is/ are true?

P. The domain of f is all reals.

Q. f(x) has two horizontal asymptotes y = 0 and y = 2.

R. f(x) has no vertical asymptotes.

a. Q only b. R only c. P and Q only

d. Q and R only e. None

6. Evaluate: lim arctan [In(x2~l.x--+l 1- x

7ra. 4 4

d. -1b.O c.

3A

57re. 4

7. The local extrema of f (x )x-values?

x2el-2x occur at which of the following

a. local minimum at x = 0 and local maximum at x = 1

b. local minimum at x = 1 and local maximum at x = 0. 1

c. local minimum at x = "2 and local maxima at x = 0 and x = 1

d. local minimum at x = 0 and local maximum at x = -21

e. local minima at x = - 2 and x = 0 and local maximum at x = "2

8. If the radius r and height h of a circular cylinder both change at a rate of2 cm/sec, how fast is the volume V of the cylinder increasing at the instantwhen r. 10 cm and h = 20 cm?

dVa. dt = 407rcm3/sec

dVb. dt = 807rcm3/sec

dVc. dt = 1007rcm3/sec

dVd. dt = 8007rcm3/sec

dVe. dt = 10007rcm3/sec

h

9. If f(x) = (1+X)4/x, use logarithmic differentiation to find the slope of thetangent line to f(x) at x = 1.

a. 32 - 64ln2 b. 2 -ln4 c. -321n2

d. 16 - 8ln2 e. 64 -161n2

4A

10. The volume V in liters and pressure P in atmospheres of a certain gas24

satisfy the equation P = V. If a measurement yields V = 4 liters with

a maximum possible error of ±O.3 liters, use differentials to estimate themaximum error in calculating the pressure P of the gas.

a. ±O.55 atmospheres

d. ±1 atmosphere

b. ±4.5 atmospheres

e. ±O.l atmospheres

c. ±0.45 atmospheres

111. If f(x) = 2"X2 + lnx, which of the followingstatements is/are true?

Be sure to consider the domain of f (x) .

P. f is increasing on (0,00).

Q. f has inflection points at x = -1 and x = 1.

R. The shape of the graph of y = f(x) is J on (1,00).

a. P only b. P and R only c. P and Q only

d. Q and R only e. P, Q, and R

12. Find the equation of the tangent line to the curve x2/3 + y2/3 = 2 at thepoint (-1, 1).

a. y = -x + 2 b. y = x - 2

4 1e. y = --x --

3 3

c. y =-x

d. y = x + 2

5A

13. Find the area of the largest rectangle that can be inscribed in a righttriangle with legs of length 3 in and 4 in if two sides of the rectangle lie alongthe legs. Hint: write the equation of the line with intercepts x = 3 and y = 4.

7 . 2a. - In2

d 4. 2. - In3

3 . 2e. - In2

x2

14. Determine the intervals where f(x) =1 t(t - 1) dt is increasing.

a. (-00, -1) U (1,00)

d. (-00,0) U (1,00)

b. (-00, -1) U (0,1)

e. (-I,O)U(I,oo)

c. (-1,0) U (0,1)

15. The slope of a curve y = j(x) at any point is given by sec2 x.- sinx. If

the origin is a point on the curve, find j (;).

1d y3--. 2

1 1b. y3+ '2

1e. y3+ '2

y3a 3--. 2

6A

16. A particle starting from rest moves in a straight line so that its acceler­ation at time t is given by a(t) = 6t - 3 in/sec''. Find the total distancetraveled by the object in the first two seconds (t = 0 to t = 2).

a. 2 inches b. ~. inches c. 6 inches d. 3 inches7.

e. "2 Inches

17. Find the maximum and minimum values of f(x) = V4x - x2 on [0,3].Use these values and the Comparison Property to find lower and upper

bounds for the definite integral 13f(x) dx.

a. 0 <13f (x) dx < 6

c. 3,,13 <13f(x) dx < 6

e. 0 <13f(x) dx < 3V3

b. 0 < l'f(x) dx < 2

d. 0 <13 f(x)dx <V3

18. Find the area of the region bounded by f(x) = 4 on [1, e3].xVI + ln z

a. 2 b. 4 - V2 c. 8 d. 16 e. 1

19. It is projected that t years from now the rate of increase of the populationof a certain country, measured in million people per year, can be modeledby the function P'(t) = e°.4t, where P(t) is the population of the country inmillions. According to this model, what will be the net increase in population(in millions of people) in the next 10 years (t = 0 to t = 10)?

a. 2 4c. -e5

d. 4e4 - 4 e.

7A

J 4JX-420. Evaluate the integral: 3/2 dx2x - 3x + 1

a. 2ln 12x3/2 - 3x + 11+C

c. ~ In 12x3/2 - 3x + 11+ C

4e. "3 In 12x3/2 - 3x + 11+C

-4b. 3(2x3/2 _ 3x + 1)2+ C

-3d. 4(2x3/2 _ 3x + 1)2+ C

Bonus problems 21 - 26 are worth 2 points each.

e2+2h e221. Use the definition of derivative to evaluate the limit: lim h-

h-+O

a.O e. does not exist

22. Use the definition of definite integral to evaluate the limit:

n (.). • 7r1, 7rhmLsln --n-sco n n

i=O

a. -2 b. -1 c. 0 d. 1 e. 2

23. Suppose that g( x) is the inverse of a differentiable function f (x) suchthat f(-1) = 8 and 1'(-1) = 12. Find the value of g'(8).

2a. g'(8) = 3 b. g'(8) = ~ c. g'(8) = -1 d. g'(8) = 112

e. Not enough information to evaluate

8A

24. Find all critical numbers of f(x) = X1/4 - x9/4.Hint: be sure to consider the domain of f(x).

1a. x = 3 only

1d. x = -3 only

1b. x = 0, 3 1 1

c. x = -3' 0, 3

e. x = ° only

25. Using substitution, the integral l' sin2 (~) cos (~) dxis equivalent to _

d. 2 r1 u2 duJo .

26. Suppose that f (x) = x3 sin( x). Which of the following statementsis/ are true?

P. f (x) is an odd function.

Q. {f(X)dX=O.

R. {f(X)dX=2 1'f(X)dX.

a. R only b. P and Q only c. P and R only d. None are true.

Have a great holiday!

9A