Printed Name: Section #: Instructor:

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1 Printed Name: Section #: Instructor: Please do not ask questions during tis exam. If you consider a question to be ambiguous, state your assumptions in te margin and do te best you can to provide te correct answer. Refer to tis page and te last page of te test for formulas, general directions, and calculator troublesooting tips. Any communication wit any person (oter tan te instructor or te designated proctor) during tis exam in any form, including written, signed, verbal or digital, is understood to be a violation of academic integrity. All devices, suc as computers, cell pones, cameras, watces and PDAs must be turned off and stowed away wile te student is in te testing room. Te only calculators to be used are TI-83, TI-83+, TI-84 or TI-84+. You may NOT borrow or sare a calculator wit anoter person taking tis test. Statement of Academic Integrity: I ave not and will not give or receive improper aid on tis test. In signing below, I acknowledge tat I ave read, understand, and agree to tese testing conditions. Student s Signature: (Tis test will not be accepted for grading unless it bears te signature of te student.) FR#1 FR # FR #3 FR #4 scantron Free Response Total Multiple Coice Total Total Possible Points Points Earned Useful Formulas: ( ) f x + f ( x) f ( x) = lim ; 0 f ( x) f ( a) f ( a) = lim x a x a rt r F( t) = P(1 + rt) F( t) = Pe F( t) = P 1+ n r r APY = ( e 1)100% or APY = % n n ( nt ) Page 1/1

2 MULTIPLE CHOICE: 64 points (3 points eac, unless oterwise noted) Use a # pencil and completely fill eac bubble on your scantron to answer eac multiple coice question. (For future reference, circle your answers on tis test paper.) Tere is no penalty for guessing on multiple coice. If you indicate more tan one answer, or you leave a blank, te question will be marked as incorrect. D C C B B D A B C A D A B C A D C B D C A A Use te following to answer te next tree questions: Te grap below sows te city of Clemson s population, in tousands of people, for various years. Draw appropriate lines on te grap to answer te questions below. 1. Between 1980 and 005, Clemson s population increased on average by tousand people per year. a b c. 0.0 d In 1975, Clemson s population was increasing by tousand people per year. a b c d Consider te slope of tangent lines to te above grap at te points x= 10, x= 0, x= 30, and x= 55. Order te points from least to greatest slope. a. 10, 0, 30, 55 b. 30, 55, 0, 10 c. 55, 30, 0, 10 d. 55, 0 30, 10 Page /1

3 Use te following table for te next two problems. x 1 f ( x) f (1) x 1 + x 1 f ( x) f (1) x Wat does te number in te saded cell represent? [ pts] a. Te instantaneous rate of cange at x= b. Te slope of te line passing troug points at x= 0.99 and x= 1. c. f ( x) f (1) lim. x 1 x 1 d. Te percentage rate of cange between x= 0.99 and x= Based on te above table, f (1) =. [ pts] a. 0 b..5 c. d. does not exist 6. For a smoot continuous function (wit no sarp points), to find te instantaneous rate of cange at a point T,. a. find te slope of te secant line at te point T b. find te average of te slopes of tangent lines nearby te point T c. find te limit of slopes of tangent lines at points nearby T, as tose points get closer and closer to T d. find te limit of slopes of secant lines at points nearby T, as tose points get closer and closer to T Page 3/1

4 Use te following grap for te next two problems. 7. Given tat f ( x ), graped to te rigt, is continuous at te inflection point at x = 3, wic one of te following is a reason wy te instantaneous rate of cange does not exist at x = 3? a. Tere is a vertical tangent line at x = 3. b. Tere is a orizontal tangent line at x = 3. c. Tere is a sarp corner at x = 3. d. lim f ( x) does not exist. x 3 8. Find te interval(s) on wic te slope grap of f ( x ) lies below te x-axis and is decreasing. a. (, 3),( 3,1) b. (, 3) c. ( 3,1) d. (1, ) 9. A cake taken out of a 35 o F oven will completely cool off to a room temperature of 68 o F after 37 minutes. T ( m ) degrees Fareneit gives te temperature of te cake, m minutes after it as been taken out of te oven, 0 m 37. Wen m= 15, determine weter T ( m) is positive or negative and give te units for T ( m). a. Positive; degrees Fareneit per minute b. Positive; degrees Fareneit c. Negative; degrees Fareneit per minute d. Negative; degrees Fareneit per our 10. According to Te Weater Cannel, istorically, te mean low temperature in Greenville for te mont of January is 30 o F wile te mean low temperature in Greenville for te mont of July is 68 o F. Complete te following sentence: Between January and July, te mean low temperature in Greenville increases by %. a b c d Page 4/1

5 11. Wic one of te following sows a correctly drawn tangent line at bot points A and at point B (were te function is decreasing most rapidly)? a. b. c. d. 1. A car dealer models te number of cars sold at eac of er two locations after te start of a sale. 3 n ( x) = x cars were sold at te first location and a( x) = 1.1( x ) cars were sold at te second location during te t x week after te start of a sale, 1 x 8. Find T ( x ), te total number of cars sold in bot locations during te Ten find te rate of cange formula T ( x) =. t x week after te start of te sale. 1 x 3 a. 8.68x b x + 1.1x c. 8.68x x d. + 3 x x Page 5/1

6 1 x 13. For te function g( x) = +, g ( x) =. x a. 1 x b. x x 3 c. 1 x d. x + x Wic one of te following graps is te slope grap for te function sown in te grap below? a. b. c. d. Page 6/1

7 15. Wic one of te following graps is te slope grap for te function f ( x) sown in te grap below? a. b. c. d. 16. Suppose a function f ( x) is a smoot and continuous function (wit no sarp points). If te slope grap of f ( x ) as a zero (an x-intercept) at x=, te grap of f ( x) must ave a at x=. If te slope grap of f ( x) as a relative maximum at x= 5, te grap of f ( x) must ave at x= 5. a. zero; inflection point b. orizontal tangent; zero c. zero; orizontal tangent d. orizontal tangent; inflection point Page 7/1

8 Use te following to answer te next tree questions: T ( o ) minutes gives te amount of time it takes a mecanic to perform is Also, T (5) = Write a sentence of interpretation for T (5) = 3. t o oil cange of te day. a. Wen te mecanic performs is 5 t oil cange of te day, it takes im 3 minutes. b. After 5 minutes of canging oil, te number of oil canges te mecanic performs decreases by 3 oil canges per minute. c. Wen te mecanic performs is 5 t oil cange of te day, te time it takes im to perform an oil cange is decreasing by 3 minutes per oil cange. d. Between te 5 t and 6 t oil cange of te day, Wen te mecanic performs is 5 t oil cange, te time it takes te mecanic to perform an oil cange decreases on average by 3 minutes. 18. If it takes 16 minutes for te mecanic to perform is 5 t oil cange, find te percentage rate of cange for te 5 t oil cange. a b c d Wat are te units for te percentage rate of cange in te previous question? a. percent b. percent per minute c. minutes per oil cange d. percent per oil cange Use te following information to answer te following tree questions. Joe opens an investment account in a bank wit 3,150 dollars. Te bank manager offers im an investment (call it Plan A) tat pays an annual percentage rate (APR) of 3.8% compounded quarterly. 0. How long would Joe ave to wait for te initial investment of $3150 to double? a. 18 years 3 monts b. 18 years 4 monts c. 18 years 6 monts d. 19 years 1. If tere are no oter deposits or witdrawals, find te balance in te account if Joe decides to witdraw te money after 8 monts. a. $3,49.79 b. $3,440.6 c. $9,08.78 d. $3,495.7 Page 8/1

9 . If te bank manager offers im anoter investment tat pays an annual percentage rate of 3.75 compounded continuously, wat is te annual percentage yield (APY) of tis second investment (call it Plan B) and wic investment sould e coose? a. 3.81%; Plan A b. 3.81%; Plan B c. 3.75%; Plan A d. 3.75%; Plan B FREE RESPONSE: 36 points Sow work were possible. Read te directions at te back of te test on rounding, inclusion of units, and writing sentences and models. 1. Te following table sows te percentage of people, over 14 years old, awake in te US at various times after 9:00 PM. Hours after 9 pm Percent awake A. Use te table to find te average rate of cange in te percentage of people awake between 11:00 pm and 7:00 am. Sow work, rounding te answer to tree decimal places. Write a sentence of interpretation = Between 11:00 pm and 7:00 am, te percentage of people in te US, over 14 year old, wo are awake increases on average by.396 percentage points per our. Part A) 5 pts 1 pt wen 1 pt wat 1 pt increases on avg by 1 pt ow muc 1 pt units B. Te data in te table can be modeled as follows: A( x) =.731x x percent gives te percentage of people, over 14 years old, awake in te US, x ours after 9:00 PM, were 0 x 10. Ceck point: A (1) = Find te instantaneous rate of cange in te percentage of people awake at midnigt. Round te answer to tree decimal places. Write a sentence of interpretation. A (3) = At midnigt, te percentage of people in te US, over 14 year old, wo are awake is decreasing by percentage points per our. Part B) 5 pts 1 pt wen, 1 pt wat, 1 pt was decreasing by 1 pt ow muc 1 pt units units ( /10 pts ) Page 9/1

10 3. f ( t) = t 0.086t t + 7 tousand tons gives te yearly consumption of fertilizer in te US, t years since 1965, 0 t 46. Ceckpoint: f () = A. Calculate amount of cange in te yearly US consumption of fertilizer between 1975 and Sow work, round te answer to tree decimal places, and include units wit te answer. f (30) f (10) = = 4.98 tousand tons Part A) pts Part B) 3 pts B. Find f ( t) = t 0.17t Part C) pts: (Do NOT round any of te coefficients in te answer.) 1 pt units 1/ pt rate of cange 1/ pt output description C. Complete te completely defined rate-of-cange model for f ( t) by filling in te blanks: f ( t) = equation is in part B tousand tons per year (units) gives te RATE OF CHANGE in te yearly consumption of fertilizer in te US, t years since 1965, 0 t 46. D. How quickly is fertilizer consumption in te US canging in 1970? Round te answer to tree decimal places. Include units wit te answer. f (5) = tousand tons per year Part D) pts ( / 9 pts ) Test continues on next page Page 10/1

11 4 3. f ( x) = + 5 degrees Fareneit is te temperature on a late-summer evening in soutcentral Micigan, x ours after sunset, 0 x 6. Ceckpoint: f () = (0.6x+ 0.0) e A. Fill in te following table to numerically estimate te instantaneous rate of cange of f ( x ) at x= 3. Entries in te table must be rounded correctly to THREE decimal places for full credit. (Answers will be marked wrong if not rounded to exactly tree decimal places.) x 3 f ( x) f (3) x 3 + x 3 f ( x) f (3) x Part A) 4 pts entries in table, correct to tree decimal places Part B) 1 pt slope, correct to tree decimal places. Part C) pts B. According to te table, df =.87 (correct to tree decimal places). dx = x 3 C. Fill in te blank in te following sentence, wit numerical answers correct to tree decimal places. According to te table, te temperature 3 ours after sunset on a late-summer evening in sout-central Micigan was decreasing by.87 degrees Fareneit per our. ( / 7 pts ) Test continues on next page Page 11/1

12 4. Use te limit definition of te derivative to find te derivative of f ( x) = 3x 1.x + 5. For full credit, continue from te general limit definition (provided below), clearly sowing all necessary algebraic steps (cancellations, expansions, etc.) and including proper use of notation and equal sign f ( x + ) f ( x) f ( x) = lim 0 [3( x + ) 1.( x + ) + 5] [3x 1.x + 5] = = lim 0 x x x x x = lim 0 x x x x x = lim 0 6x = lim 0 (6x ) = lim 0 = lim(6x ) 0 = 6x 1. Tus, f ( x) = 6x 1. [3( + + ) 1.( + ) + 5] [ ] pts: Find slope of secant using given function: [f(x+) f(x)]/ 1 pt: Square (x+) correctly pts: Distribute 3, -1., and te -1 (minus sign) correctly 1 pt: Combine like terms and sow te result. 1 pt: Sow te limit of a completely simplified expression 1 pt: Evaluate limit of simplified expression to find derivative. Deductions: -1 pt if limit notation is missing; -½ pt if te limit notation is written incorrectly trougout proof -½ pt if equal signs not in correct places and used trougout ( / 9 pts) 1 point for correctly filling out and bubbling te scantron wit a # pencil, a correct XID, a correct test version AND te front of te test is completed wit your signature on te academic integrity statement. END OF TEST Page 1/1

13 General Directions: Sow work were possible. Answers witout supporting work (were work is appropriate) may receive little credit. Do not round intermediate calculations. Answers in context ALWAYS require units. Assume end of te year data unless stated oterwise. Round your answers to 3 decimal places UNLESS te answer needs to be rounded differently to make sense in te context of te problem OR te directions specify anoter type rounding OR te complete answer as fewer tan 3 decimal places. Wen asked to write a model, include all components of a model: an equation, a description of te input including units, a description of te output including units, and te input interval wen known. Wen asked to write a sentence of practical interpretation, answer te questions: wen?, wat?, and ow muc? using ordinary, conversational language. DO NOT use mat words, terms, or unnecessary prases. Always use a ruler wen estimating values off of a grap. HINTS FOR TROUBLESHOOTING YOUR CALCULATOR: If you lose your L1, L, etc., you may reinsert tem using STAT 5 (set-up editor) enter. Te SCATTER PLOT will not sow unless Plot 1 as been turned on and tere is data in L1 and L. ZOOM 0 may not work for graping if Plot 1 is turned on. DIM MISMATCH error usually means tat te lists in L1 and L are not of equal lengt. DATA TYPE error usually means tat you already ave someting in Y1 and you need to clear it before you can paste a new equation. INVALID DIM error usually means tat your plot(s) are on, but tat you ave no data in te lists. Refer to te second int above. If your batteries die, raise your and and old up your calculator. If your instructor as an extra calculator available, e/se will loan it to you for a few minutes. SYNTAX ERROR: Try GO TO. Tis will appen if you use a subtraction minus sign wen you sould use a negative sign. If you need to CLEAR MEMORY, use nd +, 7:Reset, 1:All Ram, :Reset Page 13/1