Department of Mathematics

MAC 2233Exam 2A

Spring 2017UF UNIVERSITY of

FLORIDA

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 number

3) Section number

C. Under "special codes" code in the test ID numbers 2, l.1.34567890.234567890

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

E. 1) This test consists of 17multiple choice questions, ranging from one point to five pointsin value, plus two sheets (four pages) of free response questions worth 30 points.The test is counted out of 80points, and there are 8 bonus points available.

2) The time allowed is 90minutes.

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 and tearoff sheets to your exam proctor. Beprepared to show your picture I.D. with a legible signature.

3) The answers will be posted in CANVAS within one day after the exam. Your scorewill be posted in CANVAS within one week of the exam.

2A MAC 2233 Exam 2

NOTE: Be sure to bubble the answers to questions 1-17 on your scantron.

Questions 1 - 8 are worth 5 points each.

1. Find the equation of the tangent line to f(x) = v'~ 2 at x = 4.

1 5a. y = -S"x +"2

1d. y = -"2x + 4

1b. y = 4x + 1

1 3e. y = -S"x -"2

2. A sewage treatment plant disposes of treated waste through a pipelinethat empties into a nearby stream. The spread of the treated waste

from the pipeline can be modeled by the formula C(x) = (4X2~O2)2

where C(x) is the concentration of the treated waste in parts per millionmeasured x meters from the opening of the pipeline.

Find the average rate of change in concentration as the distance fromthe pipeline increases from 0 to 2 meters, and the rate at which theconcentration is changing at a distance of exactly 2 meters from thepipeline.

Average rate of changefrom x = 0 to x = 2

Rate of changeat 2 meters

Concentration (in parts per million per meter) is

a. increasing by 1 ppm/meter

b. decreasing by 24 ppm/meter

c. decreasing by 48 ppm/meter

d. decreasing by 24 ppm/meter

e. decreasing by 48 ppm/meter

increasing by 2.4 ppm/meter

decreasing by 0.4 ppm/meter

decreasing by 1.6 ppm/meter

decreasing by 1.6 ppm/meter

decreasing by 0.4 ppm/meter

MAC 2233 Exam 2 3A

{

x2 -13. Let f(x) = ~2 - 4x + 3

x=l

Which of the following statements is/are true?

P. lim f(x) = O.x----+l

Q. f(x) has a removable discontinuity at x = 1.

R. f(x) can be made continuous at x = 1 by redefining f(l) = -1.

S. f(x) has a nonremovable discontinuity at x = 3.

a. P and S only b. Q and R only c. Q, Rand S

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

4. Find an expression for the slope of the tangent line to x2 - xy = 8 - y3at any point on the curve.

dy y - 2x b dy = 2x - y + 8 dy 2x+ya -= c. -_. dx 3y2 - X . dx x - 3y2 dx X+3y2

d dy = 8 - 2x dy 2x. dx 3y2 - 1

e.dx 1+ 3y2

5. Find the value of k so that f is continuous at x = 2 if

{

2X -4x<2

f(x) = Ix - 21kx + 1 x >2

a. k = 13

b k=--. 21

c k =-. 2 d. k =-1

e. f(x) cannot be made continuous at x = 2.

4A MAC 2233 Exam 2

6. A spherical balloon is being inflated so that its volume is increasing ata constant rate of 1271"in3 per second. At what rate is the radius ofthe balloon increasing with respect to time when that radius is exactly 24inches? Note: the volume of a sphere is given by the formula V = 371"r3.

a. 3 in/sec b. 571"in/sec 3 . /c. "2 III sec4

d. 371"in/sec3 . /e. "4 In sec

7. Find each x-value at which the graph of f(x) = x~3x + 4 has ahorizontal tangent line.Hint: write your derivative as a single fraction.

4a. x = - - and x = -1

36

d. x = -5 only

b. x = -1 only4 6

c x = - - and x = --. 3 5

e. There are no horizontal tangent lines.

8. The demand and cost functions for a new large screen TV are2

p(x) = 1600 - x and C(x) = 1200x - ~ + 24,000, where p is the unit

price when x items are sold. Suppose that sales are currently increasingby 3 units per week. At what rate is profit changing with respect totime when 100 units are sold? Hint: what is the profit function?

a. Profit is increasing by $300 per week.

b. Profit is decreasing by $250 per week.

c. Profit is increasing by $900 per week.

d. Profit is decreasing by $100 per week.

e. Profit is increasing by $750 per week.

MAC 2233 Exam 2 5A

Problems 9 - 11 are worth 3 points each.

9. At what value of x is the tangent line to f(x) = e3x perpendicular tothe line x + 6y = 127

ln6a. x=-

3b. x = 6

2c x =-. 3

d x = ln2. 3 e. x = ln3

10. If f(x) = (x: 2)4' then f'(x) =----

x-3a. (x + 2)3

2 - 3xd. (x + 2)5

b 1. 4(x + 2)3

3x - 4e. (x + 2)8

5x+2c. (x + 2)5

11. The position of an object moving in a straight line is given by the2function s(t) = 3t3 - 4t2 + 9t where s(t) is measured in centimeters and

t is in seconds. Find each time (in seconds) during its journey at whichthe velocity of the object is exactly 3 cm/sec.

a. t = 1only3

b t = - and t = 4. 2 c. t = 2 and t = 4

3d. t = "2 only e. t = 1 and t = 3

12. (2 points) Find the slope of the tangent line to f(x) = 3x2+2x+l at x = o.

a. 6ln3 b. 0 c. 3 d. 3ln3 e. 6

Be sure to answer the bonus questions on the next page.

6A MAC 2233 Exam 2

Bonus! Answer the following for the function f(x) graphed below.Be sure to consider tangent lines.

, I

I····· \ j ····1·········1·:···

verticaltangent

13. (2 pts.) On which of the following intervals is the derivative f'(x) < O?

a. (-1,1) only b. (0,1) U (4,00) c. (-1,1) U (2,00) d. (-00, -2)

14. (1 pt.) f'(O) = _ a. -1 b.13

c. 0 d. -3

(1 point each) Indicate whether the following are true or false.

a. True b. False

15. The graph of the derivative of f(x) has a vertical asymptote at x = -2.

16. f(x) is differentiable but not continuous at x = 1.

a. True b. False

17. f'(x) = 0 at both x = -1 and x = 2.

a. True b. False

MAC 2233 Test 2A Part II Spring 2017

Sect # Name _

UFID Signature _

SHOW ALL WORK TO RECEIVE FULL CREDIT.

1. Let f(x) = JX+1.

(a) Use the limit definition of derivative only to find f'(x).

(b) Verify your limit by finding f'(x) using a rule of differentiation.

(c) Write the equation of the tangent line to f(x) at x = o.

y = ---------

8A MAC 2233 Exam 2A

2. The cost of producing x units of a new product is given by the cost functionO(x) = 600+ 8x - 0.02X2. Include units in your answers to the following:

(a) Find the cost of producing 100 items.

(b) Find the marginal cost when x = 100.

(c) Use parts (a) and (b) to approximate the cost of producing 101 items.

0(101) ~ _

(d) Suppose that the manufacturer of the product has determined that the numberof items produced after t days is given by x = 50+ 2t2• Use the Chain Rule tofind the rate at which cost is changing with respect to time after 5 days.Hint: What is the value of x when t = 57 Look at part (b).

dOdt

Name --------------------------- Section --------------------Problem 2, continued.Given C(x) = 600+8x - 0.02X2 where C(x) is the cost of producing x items.

(e) Find the average cost function C(x) = C(x).x

C(x) = _

(f) Find the marginal average cost when x = 100.

Average cost is by '

. {X4 + 3x - 1 x < 03. Consider the function f(x) = - .

3 - x2 x> 0

Fill in the blanks. Give a reason for your answers.

(a) True or False: f(x) is continuous at x = O.

(b) Write the letter of each interval on which f(x) must have a zero according to theIntermediate Value Theorem. Be sure to consider part (a).

B. [-2, -1] C. [-1,1] D. none of these

IDA MAC 2233 Exam 2A

1x+3

4. Consider the function f(x) = 2 - x

x<O

vx - 2 x> 2

(a) Sketch the graph of f(x). (b) Find the limits if possible.If not possible, write" DNE" .

lim f(x) = _x---*-oo

lim f(x) = _x---*-3-

lim f(x) = _x---*o-

lim f(x) = _x---*o+

lim f(x) = _x---*o

lim f(x) = _x---*2

(c) List all discontinuities of f(x) and state whether they are jump, infinite or remov­able.

(d) At each discontinuity from (c), state whether or not f(x) can be defined to makef continuous there. If so, find that value of f(x).