Printed Name: Section #: Instructor:

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1 Printed Name: Section #: Instructor: Please do not ask questions during this eam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you can to provide the correct answer. Refer to this page and the net for formulas, general directions, and calculator troubleshooting tips. Any communication with any person (other than the instructor or the designated proctor) during this eam in any form, including written, signed, verbal or digital, is understood to be a violation of academic integrity. All devices, such as computers, cell phones, cameras and PDAs must be turned off while the student is in the testing room. The only calculators to be used are TI-83, TI-83+, TI-84 or TI-84+. You may NOT borrow or share a calculator with another person taking this test. Statement of Academic Integrity: I have not and will not give or receive improper aid on this test. In signing below, I acknowledge that I have read, understand, and agree to these testing conditions. Student s Signature: (This test will not be accepted for grading unless it bears the signature of the student.) FR#1 FR # FR #3 FR #4 FR #5 FR #6 scantron Free Response Total Multiple Choice Total Total Possible Points Points Earned Useful Formulas: ( ) f + h f ( ) f ( ) = lim ; h 0 h f ( ) f ( a) f ( a) = lim a a Page 1/14

2 General Directions: Show work where possible. Answers without supporting work (where work is appropriate) may receive little credit. Do not round intermediate calculations. Answers in contet ALWAYS require units. Assume end of the year data unless stated otherwise. Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make sense in the contet of the problem OR the directions specify another type rounding OR the complete answer has fewer than 3 decimal places. When asked to write a model, include all components of a model: an equation, a description of the input including units, a description of the output including units, and the input interval when known. When asked to write a sentence of practical interpretation, answer the questions: when?, what?, and how much? using ordinary, conversational language. DO NOT use math words, terms, or unnecessary phrases. Always use a ruler when estimating values off of a graph. HINTS FOR TROUBLESHOOTING YOUR CALCULATOR: If you lose your L1, L, etc., you may reinsert them using STAT 5 (set-up editor) enter. The SCATTER PLOT will not show unless Plot 1 has been turned on and there is data in L1 and L. ZOOM 0 may not work for graphing if Plot 1 is turned on. DIM MISMATCH error usually means that the lists in L1 and L are not of equal length. DATA TYPE error usually means that you already have something in Y1 and you need to clear it before you can paste a new equation. INVALID DIM error usually means that your plot(s) are on, but that you have no data in the lists. Refer to the second hint above. If your batteries die, raise your hand and hold up your calculator. If your instructor has an etra calculator available, he/she will loan it to you for a few minutes. SYNTAX ERROR: Try GO TO. This will happen if you use a subtraction minus sign when you should use a negative sign. If you need to CLEAR MEMORY, use nd +, 7:Reset, 1:All Ram, :Reset Page /14

3 MULTIPLE CHOICE: 60 points (3 points each, unless otherwise noted) D A B A C D C D A C A B D B B D A C B B C D Use a # pencil and completely fill each bubble on your scantron to answer each multiple choice question. (For future reference, circle your answers on this test paper.) There is no penalty for guessing on multiple choice. If you indicate more than one answer, or you leave a blank, the question will be marked as incorrect. Consider the table in answering the following two questions. 3 f ( ) f (3) f ( ) f (3) What is a correct interpretation of the number 5.34 in the shaded cell? a. The difference in the values of f ( ) between = 3 and = b. The instantaneous rate of change of the function f ( ) at the point = c. The slope of the tangent line to the function f ( ) at the point = 3. d. The slope of the secant line to the function f ( ) between = 3 and = Suppose that f ( ) feet gives a ball s height above the ground, seconds after being propelled into the air, Which one of the following statements follows from the information in the table? a. 3 seconds into flight, the ball s height is increasing by 5.5 feet per second. b. 3 seconds into flight, the ball s height is 5.5 feet above the ground. c. Between.9 seconds and 3.1 seconds after being propelled into the air, the ball s height increased on average by 5.5 feet per second. d. The height of the ball was decreasing 3 seconds into flight. 3. Use the following information to compare the steepness of the graph of the function f at points A, B, C, and D. Specify the correct order in terms of least to greatest steepness. f ( A) = 0, f ( B) =.94, f ( C) = 3.04, and f ( D) = a. D, B, A, C b. A, B, C, D c. C, A, B, D d. D, C, B, A Page 3/14

4 4. Identify the correctly drawn tangent line to the graph of the function f at the inflection point. a. Line A b. Line B c. Line C d. Line D Use the following to answer the net two questions. Let g( t) be a function that gives the price of a gallon of gasoline in US dollars, t months after the start of Two ways to represent the slope of the tangent line to the graph of g( t) at t= 4 are and. a. g(5) g(4) 5 4 ; g (4) b. dg ; g (4) c. g (4) ; d = 4 dg dt t= 4 d. dt ; g (4) dg = t 4 6. The output units for the derivative of g( t ) are. a. US dollars b. gallons per month c. years per US dollar d. US dollars per month Page 4/14

5 Use the following graph to answer the net three questions. The graph below shows the tide levels in Charleston, SC, on a September day from noon to 6pm. 7. Use the graph to find how quickly the tide levels are changing at 1:30 pm. a feet per hour b.. feet per hour c. 1.4 feet per hour d. 3.6 feet per hour 8. At 3:30 pm, tide levels are feet and tide levels are decreasing by feet per hour. Calculate the percentage rate of change in the tide levels at 3:30 pm to fill in the blank in the following sentence. At 3:30 pm, tide levels are decreasing by percent per hour. a b c d What calculation must be made in order to fill in the blank in the following sentence? Between noon and 3 pm, tide levels increase on average by feet per hour. a. Find the slope of the secant line between t= 0 and t= 3. b. Find the slope of the tangent lines at t= 0 and t= 3 and find the average of the two slopes. c. Find the limit of slopes of secant lines between t= 3 and nearby points. d. Find the slope of the tangent line at t= 3. Page 5/14

6 10. H ( t) gives the number of home runs that a certain professional baseball player hits in a season, in year t of his 1 year professional career. According to the following graph of H ( t ), which one of the following statements is FALSE? a. In year 3, the rate of change in the number of home runs in a season was positive. b. In year 5, the player reached the highest home run total in his career. c. The number of home runs in a season was decreasing most rapidly in year 6 of the player s career. d. Between year 1 and year 5 of the player s career, the number of home runs in a season increased on average by 1 hit per year. 11. The table shows the population of a particular city for various years. Year Population, in million people Complete the following sentence: Between 00 and 008, the population of the city increased by percent. a b c..833 d Page 6/14

7 1. c( t) billion bottles gives the number of bottles sold of a popular brand of cola, t years after 000, 0 t 9. Which one of the following is a correct sentence of interpretation for c(5) c() = 45? 5 a. Between 00 and 005, the number of bottles of a popular brand of cola sold was 45 billion bottles. b. Between 00 and 005, the number of bottles of a popular brand of cola sold increased on average by 45 billion bottles per year. c. In 005, the number of bottles of a popular brand of cola sold was increasing by 45 billion bottles per year. d. Between 00 and 005, the number of bottles of a popular brand of cola sold increased by 45%. 13. Which one of the following graphs is the slope graph for the function f ( ) shown in the graph below? a. b. c. d. Page 7/14

8 14. Which one of the following graphs is the slope graph for the function f ( ) shown in the graph below? a. b. c. d. 15. [ points] Find f ( ) = for f ( ) = 4( π ). a. (ln 4)( π ) b. 4(ln π )( π ) c. (ln 4 π )(4 π ) d. 4π [ points] The derivative of r( ) = e is r '( ) =. a. e b. e c. e d. e Page 8/14

9 17. Which one of the following graphs is the slope graph for the function f ( ) shown in the graph below? a. b. c. d. 18. [ points] Find the derivative of y = 5. dy a. 5 d = b. dy d = 0 c. dy d = 1 d. dy 5 d = Page 9/14

10 [ points] For the function p( ) = +, p ( ) =. 4 a b c d [ points] f ( ) = for the function f ( ) =. a b. c. 7 + ( ln ) d [ points] Find the derivative of the function ( ) 3 f = e 5(ln ) + 3. a. c. 1 f ( ) = 3e 5 e + (ln )3 b. 5 f ( ) = 3 e + (ln 3)3 d. 5 f ( ) = 3e + 3 ln 5 f ( ) = 3(ln ) e + (ln 3)3. Suppose that f () = 4 on a continuous function f. Which one of the following statements is false? a. The slope of the tangent line at = is 4 b. The limit of the slopes of the secant lines between = and nearby points, as nearby points get closer to =, is 4. c. The slope of the graph of f at = is 4. d. The slope of the secant line between = and any nearby point is 4. Page 10/14

11 FREE RESPONSE: 40 points Show work where possible. Read the directions at the front of the test on rounding, inclusion of units, and writing sentences and models. t 1. u( t ) = 7.11(1. ) million people gives the number of people using a ridesharing company, t years after 009, 0 t 6. Checkpoint: u () = A. Calculate and write a sentence of interpretation for the average rate of change in the number of people using the ridesharing company between 01 and 015. A) 4 pts Between 01 and 015, the number of people using the ridesharing 1 pt when company increased on average by million people per year. ½ pt what 1 pt increased on average by 1 pt how much ½ pt units B. Calculate and write a sentence of interpretation for the instantaneous rate of change in the number of people using the ridesharing company in 014. In 014, the number of people using the ridesharing company was increasing by 3.81 million people per year. B) 4 pts 1 pt when ½ pt what 1 pt was increasing by 1 pt how much ½ pt units ( / 8pts ). 3 T ( m) = 0.083m 0.333m +.083m degrees Fahrenheit gives the water temperature in an upstate SC lake, in month m of last year, 1 m 7. Checkpoint: T () = A. Write a completely defined rate-of-change model for T ( m) by filling in the blanks: T ( m) = 0.49m 0.666m degrees Fahrenheit per month (equation) (units) gives the rate of change in the water temperature in an upstate SC lake, in month m of last year, 1 m 7. A) 5 pts: pts equation 1 pt units 1 pt rate of change ; 1 pt output description B. How quickly is the water temperature of the lake changing in May of last year? Include units with the answer. T (5) = degrees Fahrenheit per month B) pts: 1 pt derivative, correct to three decimal places 1 pt units ( / 7 pts ) Page 11/14

12 N( ) =.5ln thousand people gives the total number of employees in a growing company, years after 005, Checkpoint: N () = A. Fill in the following table to numerically estimate the instantaneous rate of change of N ( ) at = 6. Entries in the table must be rounded correctly to THREE decimal places for full credit. (Answers will be marked wrong if not rounded to eactly three decimal places.) 6 N( ) N(6) N( ) N(6) B. Fill in the blank in the following sentence, with the answer correct to three decimal places. According to the table, in 011, the total number of employees in the company was increasing by thousand people per year. ( / 5pts ) A) 4 pts, entries in table, correct to three decimal places B) 1 pt, correct to three decimal places. 4. Find the derivative for each of the following functions. Simplify eponents and coefficients, and use eact numbers. For full credit, include proper use of derivative notation and equal signs. 5 1 A. f ( ) = = f ( ) = + 10 ½ pt correct notation 1 pt for derivative of first term 1 pt for derivative of second term B. f ( ) = 3( ) 4ln( ) 4 f ( ) = 3(ln )( ) ½ pt correct notation 1 pt for derivative of first term 1 pt for derivative of second ( / 5pts ) Page 1/14

13 5. Use the graph of f ( ) shown below to answer the following questions. Note that there is an inflection point at input value of D. 1 pt per blank Answer by choosing points at input values A, B, C, D, E, and F on the graph of f ( ). A. Name ONE point in which the derivative of f ( ) does not eist. B. Name ONE point in which the rate of change of f ( ) is equal to zero. C. Name ONE point in which the slope graph of f ( ) has a relative maimum. D. Name ONE point in which the slope graph of f ( ) is zero. = = A or C or B or = D = B or E E F E. On what interval(s) is the slope graph of f ( ) negative? Circle one: I. never II. (, A),( B, C),( E, F) III. ( A, C),( D, F ) IV. ( A, B),( C, E),( F, ) ( / 5pts ) TEST CONTINUES ON NEXT PAGE Page 13/14

14 6. Use the limit definition of the derivative to find the derivative of f ( ) = For full credit, continue from the general limit definition (provided below), clearly showing all necessary algebraic steps (cancellations, epansions, etc.) and including proper use of notation and equal signs. f ( ) = lim h 0 f ( + h) f ( ) h = lim h 0 h 0 = 5 9 Thus, f ( ) = 5 9 [.5( + h) 9( + h) + 4] [ ] [.5( + h + h ) 9( + h) + 4] [ ] = lim h 0 h + h + h h + + = lim h 0 h 5h +.5h 9 h h(5 +.5h 9) = lim = lim h 0 h h 0 h = lim(5 +.5h 9) h pts: Find slope of secant using given function: [f(+h) f()] / h 1 pt: Square (+h) correctly pts: Distribute.5, -9, and the -1 (minus sign) correctly 1 pt: Combine like terms and show the result. 1 pt: Show the limit of a completely simplified epression 1 pt: Evaluate limit of simplified epression to find derivative. Deductions: -½ pt if limit notation is not written correctly throughout proof -½ pt if equal signs not in correct places and used throughout proof ( / 9 pts) 1 point for correctly filling out and bubbling the scantron with a # pencil, a correct XID, a correct test version AND the front of the test is completed with your signature on the academic integrity statement. END OF TEST Page 14/14