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1 GENTLY REMOVE THIS PAGE. Note that it WILL be collected with your test. MATH 2070 Test 3 Formula Sheet When you are asked to show work or the notation that leads to your answer, be sure to write the notation with specific parameter values and function names for that problem. No credit will be given for simply copying a formula from this sheet or for using a function location (Y1) instead of a function name. Correct mathematical notation: D( p) dp VS. Incorrect examples: Mean, Variance, and Standard Deviation of x µ = x f ( x) dx Uniform Density Function 1 when a x b f( x) = b a 0 elsewhere σ = ( x µ ) f ( x) dx b µ = a + a or µ = a+b 2 2 Normal Density Function & The Empirical Rule nd 2 VARS DISTR normalpdf( X, µσ, ) 68, 95, 99.7 Compensating For Change Formulas pmax D( p) dp OR ( Y1 ) dp p0 2 ( x ) ( ) σ = µ f x dx Given f( xy,, ) dy dx f = x and f y dy y x dx The Determinant Test fxx fxy Dab (, ) = = fxx fyy fxy f yx f f yx yy ( ab, ) ( ab, ) If Dab (, ) > 0and f ( ab, ) < 0, then f has a relative maximum at ( ab., ) xx If Dab (, ) > 0and f ( ab, ) > 0, then f has a relative minimum at ( ab., ) xx If Dab (, ) < 0, f has a saddle point at ( ab., ) If Dab (, ) = 0, the test does not given any information about ( ab., )

2 This page WILL be collected with your test, but any work or answers written here will NOT be graded.

3 Name: (PRINT) Section Instructor: DO NOT OPEN THIS EXAM UNTIL INSTRUCTED TO DO SO. Statement of Academic Integrity: Any communications with any person (other than the instructor or the designated proctor) during this test in any form, including but not limited to written, signed, verbal or digital, is understood to be a violation of academic integrity. All devices, including but not limited to, computers, cell phones, and smart watches must be turned off while the student is in the testing room. All watches must be removed from your wrist and placed in your bag or in your pocket. The only calculators to be used are TI-83, TI-84, TI-83+, TI-84+, or TI-84+ CE. No exceptions may be made. You may not borrow or share a calculator with another person taking this test. The test is closed notes and closed book. No outside resources may be on your desk at any time during the testing period. As a rule, no questions are answered during the test. It is very disruptive to other students. If you believe a question is incorrectly written, you may write a note to that effect, but do try to give the best answer you can. Your proctor will not interpret any question for you. By signing below, the student acknowledges that (s)he has read and understands these testing conditions. In addition, the student acknowledges that (s)he has neither given nor received any inappropriate or unauthorized information at any time before or during this test. This table is for instructor use only Page Question Points Points Possible Earned 1a b 3 1c 2 1d a 2 5b 2 5c 6 5d 3 5e 2 5f 3 Scantron 1 Free Response (includes scantron) 51 Multiple Choice 49 Total 100 Student Signature: Page 1 of 12

4 GENERAL DIRECTIONS: Show work where possible. Answers without supporting work (where work is appropriate) may receive little or no credit. Work in pencil only. Do not round intermediate calculations. Answers in context ALWAYS require units. Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make sense in the context of the problem, OR the directions specify another type rounding, OR the complete answer has less than 3 decimal places. When you are asked to show the mathematical notation that leads to your answer, be as specific as possbile and do not use calculator notation. i.e. Write f(3) f(2) instead of f( b) f( a) or Y1(3) Y1(2). When you are asked to write a model, include all components of a model: an equation, a description of the output including units, a description of the input including units, and the input interval when known. When asked to write a sentence of interpretation, answer the questions When?, What?, How? and How much? using ordinary, conversational language. Do not use math words, terms, jargon or unnecessary phrases. Always use a ruler when estimating values off of a graph. HINTS FOR TROUBLESHOOTING YOUR CALCULATOR: If you lose your L1, L2, 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 L2. Turn off/on scatter plots by using the arrows to highlight the item you want off/on and using the enter key to change the status. DIM MISMATCH error usually means that the lists in L1 and L2 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 extra calculator available, he/she will loan it to you for a few minutes. An unexpected 0 value when integrating an improper integral involving an exponential function may mean that you have increased the bounds beyond what the calculator can compute. Try smaller bounds. NO SIGN CHANGE error usually means you are attempting to solve an equation that has no solution. SINGULAR MATRIX error usually means you are attempting to invert a matrix that is not invertible. Check your matrix entries for errors. It is recommended that you enter all single-variable functions in your calculator in terms of x. The use of other variables may make calculations more difficult, and will prohibit the use of the graphing capabilities of your calculator. Page 2 of 12

5 Multiple Choice: Use a #2 pencil and completely fill in each bubble on your scantron to indicate the answer to each question. Each question has one correct answer. If you indicate more than one answer, or leave a blank, the question will be marked as incorrect. In this section there are 17 multiple choice questions worth a total of 49 points. Each question is worth 3 points, unless otherwise noted. For future reference, circle your answers on this test paper as you will not receive your Scantron back with your test. The weekly number of visitors at an amusement park is Vtp (, ) thousand visitors when the temperature is t and the admission price is p dollars per ticket. Some of the contours of Vtp (, ) are shown in the graph below. Use this graph and description to answer the next 5 questions. 1. How many and what types of critical points exist on the graph? a. 2 relative maximums and 2 saddle points b. 1 relative maximum, 1 relative minimum, and 0 saddle points c. 1 relative maximum, 1 relative minimum, and 2 saddle points d. 2 relative maximums and 2 relative minimums 2. At what temperature and ticket price is the number of visitors greatest? a. 94 ; $71 b. 80 ; $51 c ; $80 d. 79 ; $49 Page 3 of 12

6 Context continues from previous page. 3. From point A, which event results in the greatest change in the number of visitors? a. a 5 decrease in temperature b. a 5 increase in temperature c. a $5 decrease in ticket price d. a $5 increase in ticket price Point B, marked on the graph, is a critical point of Vtp. (, ) Answer the next 2 questions regarding this point. 4. Which of the following MUST be true at point B? V I. V < 50 II. V t = 0 III. = 0 p a. I, only b. I and II, only c. II and III, only d. I, II, and III 5. Let D represent the determinant of the second partials matrix for Vtpat (, ) point B. Which of the following is true? a. D < 0 b. D = 0 c. D 45 d. Answers a, b, and c are all false. Page 4 of 12

7 p Gt (, p) = tons of garbage is the amount of garbage produced by an amusement park on a 2 t day when the high temperature is t and the admission price is p dollars per ticket. Check: G (2,3) = Use this information to answer the next three questions. 6. On a day when the high temperature is 85 and the admission price is $60. tons of garbage are produced. a b c d If 7.8 tons of garbage is produced on a day when the high temperature is 70, then the admission price is dollars per ticket. a b c d G 8. Complete the interpretation of t ( 80,40) On a day when the admission price is $40 and the high temperature is 80, a. the amount of garbage produced is decreasing by 0.12 tons per. b. the temperature is decreasing by 0.12 per ton of garbage. c. the admission price is decreasing by 0.12 dollars per. d. the amount of garbage produced is decreasing by 0.12 tons per dollar of admission price. 9. Given the function C(2, y ). 2 2 Cxy (, ) = 0.5x 6x+ 0.2y 10y+ 240 find the cross-sectional function a. 2 C(2, y) = 0.2y 10y+ 240 b. 2 C(2, y) = 0.2y 10y c. 2 C(2, y) = 0.2y 10y+ 230 d. C(2, y) = 0.4y 10 Page 5 of 12

8 Mtr (, ) dollars estimates the monthly payment required to repay a $35,000 automobile loan over a term of t years when the interest rate is r %. Some values of Mtr (, ) are shown in the table below. r % t years Use this information to answer the next 3 questions. 10. Find an exponential cross sectional model (function only) for M(6, r ). a. r M(6, r ) = 0.002(1.014 ) b. r M(6, r ) = (0.999 ) c. r M(6, r ) = (1.029 ) d. r M(6, r ) = (0.817 ) 11. To create a cross-sectional model to estimate the monthly payment required to repay a $35,000 automobile loan over a term of 7 years when the interest rate is 4%, would a row or a column of the original data be needed? (2 pts) a. Row b. Column 12. What is an appropriate name for the cross-sectional model described in the previous question? (2 pts) a. M (7, 4) b. Mt (,4) c. Mt (,7) d. M(7, r ) In a six-month clinical trial, the amount of weight lost by females on a particular diet plan was found to be normally distributed with a mean of 20 pounds and a standard deviation of 5 pounds. Use this information to answer the next 2 questions. 13. According to the Empirical Rule, what percentage of women lost more than 10 pounds? a. 95% b. 97.5% c % d. 99.7% 14. Use the normal pdf function on your calculator to find the probability that a woman on the diet plan will lose between 20 and 36 pounds. a b c d. 0.5 Page 6 of 12

9 In a 4-year study, the number of years that a patient survives after an experimental medical procedure is a random variable, x, with probability density function 3 2 x 0 x 4 f( x) = 64. Check: f (3) = otherwise Use this information to answer the next 2 questions. 15. Find the probability that a randomly chosen patient survives at least 2 years after the procedure. a b c. 0.5 d What is the mean survival time for patients after receiving this procedure? a years b. 1 year c. 2 years d. 3 years 17. Carriage tours depart a carriage house every 15 minutes. So, when a tourist arrives at the carriage house, the amount of time, x, he/she will have to wait on the next tour to begin is uniformly distributed between 0 and 15 minutes. Find the standard deviation of the wait times. a minutes b minutes c. 7.5 minutes d minutes Check your Scantron now to make sure it will successfully run. If it does, you will earn one point. When you are not working on the multiple choice portion of the test, turn your Scantron over so that it cannot be read by others in the room. Page 7 of 12

10 Free Response: RE-READ the directions at the beginning of the test. Then read each question carefully. Provide only one clearly indicated answer to each question. If your answer is illegible, it will be graded as incorrect. Show all work. The free response portion, including the scantron point, is 51% of your test grade. For each question, set up the specific mathematical notation that is being evaluated to obtain your answer. No credit will be awarded for simply copying generic formulas from the formula sheet. Little or no credit will be awarded for answers without the corresponding notation. 1. Tailgate Essentials sells two types of tailgate tents, one basic tent and one with a team logo. The profit from the sale of these tents is given by Pbt (,) dollars when b basic tents and t team logo tents are sold. The following values are known: P (75,130) = 28,500 and P (75,130) = 44 and P (75,130) = 88 b a. Approximately how much would the profit change if the number of team logo tents sold decreased to 123 while the number of basic tents sold remained at 75? (3 pts) t b. Find the value of db dt (75,130). (3 pts) db c. Choose the appropriate units for. Circle your answer choice. (2 pts) dt (75,130) i. dollars per basic tent ii. dollars per team logo tent iii. basic tents per team logo tent iv. team logo tents per basic tent d. If Tailgate Essentials sells 123 team logo tents instead of 130, yet wants to maintain a profit of $28,500, how many basic tents must they sell? (5 pts) Page 8 of 12

11 5 3 c ac b 4a 2. Find the indicated derivatives of f( abc,, ) = 6ba + + a 2 ( 3 ) e b c+ 3cln( b). (10 pts) f a = f b = 3. Consider the function f( xy, ) 200 3x y 2 2 =. Check: (4,5) 127 f = Find the slope of the line tangent to the f( xy=, ) 127 contour curve at the point (4,5). Give an exact answer. Show the specific mathematical notation for your calculation as supporting work. (4 pts) 4. State the two criteria for a function, hx ( ), to be a valid probability density function. (2 pts) Criteria 1: Criteria 2: Is the function below a valid probability density function? Show specific calculations, pictures, etc. to support your answer. (3 pts) x 0 x 4 hx ( ) = Check: h (2) = otherwise Answer: (yes/no) Supporting work: Page 9 of 12

12 5. Suppose the proportion of Mazda car owners who will buy a new Mazda can be predicted by 2 2 M ( c, f ) = c + 4.2c 40.8 f 20.3cf f where c is the proportion of Chrysler owners who remain loyal to Chrysler and f is the proportion of Ford owners who remain loyal to Ford, for 0.6 c 0.75 and 0.6 f Check: M (0.6, 0.6) = The first partial derivatives of M(, c f ) are as follows: M = c 20.3 f M = c f c f a. Find the second partial derivatives of M(, c f ). (2 pts) M cc = M cf = M fc = M ff = b. Write a system of equations that can be used to find the critical point of M(, c f ). (2 pts) c. Solve the system of equations you wrote in the previous question. Show supporting work (algebraic process or specific matrix calculations). Round results to 3 decimal places. Units are not necessary. (6 pts) c= f = M = Page 10 of 12

13 Continued from previous page. d. Find the determinant of the 2 nd partials matrix at the critical point and briefly explain how it indicates that the critical point is not a saddle point. (3 pts) D = e. What other value must be considered in order to complete the classification of the critical point? Show this value and its corresponding notation. (2 pts) f. Complete the interpretation of the critical point. (3 pts) The (maximum or minimum?) percentage of Mazda owners who will by a new Mazda is %. This will occur when % of Chrysler owners remain loyal to Chrysler and % of Ford owners remain loyal to Ford. Check to make sure you showed specific, mathematical notation throughout the free response section. Check your scantron again to make sure it is filled out completely. Then submit your test, scantron, and formula sheet to your instructor or proctor. Page 11 of 12

14 FILL OUT YOUR SCANTRON At the top of the scantron fill out: Name Your last name and first name Course: Math 2070 Section: see table below Instructor: see table below Test No. (see top, right page header) Test Version: (see top, right page header) Write AND Bubble in your CUID. Bubble in a 0 underneath the C. Bubble in the Test Version: (see top, right page header) The Scantron will not successfully run if: The XID is not filled in correctly: Check that you have the correct number and that you have bubbled in the number below where you write it in. The initial C is bubbled in as a zero. The Test Version is bubbled in incorrectly. Check to make sure that the Test Version you bubbled in (below the XID) agrees with the Test Version in the header of this page. Your responses are not bubbled in boldly enough. If your responses are only lightly shaded, the machine will not pick up your answers. Check to make sure your marks are heavy enough. You also will not earn the point if: Your name, your instructor s name and your section number are missing. Check to make sure all of these are clearly and legibly indicated. A list of section numbers, class times and instructors are included below for your convenience. Sec Time Instructor Sec Day/Time Instructor 1 8:00am-9:15am TTh Jennifer Hanna 9 3:30pm-4:45pm TTh Molly Honecker 2 9:30am-10:45am TTh Jennifer Hanna 10 2:00pm-3:15pm TTh Ebenezer Eduafo 3 11:00am-12:15pm TTh Joey Liu 11 9:30am-10:45am TTh Aaron Moose 4 11:00am-12:15pm TTh Jennifer Newton 12 11:00am-12:15pm TTh Anna Bachstein 5 12:30pm-1:45pm TTh Jennifer Hanna 13 12:30pm-1:45pm TTh Jennifer Newton 6 2:00pm-3:15pm TTh Jennifer Hanna 14 10:10am-11:00am MWF Jennifer Van Dyken 7 3:30pm-4:45pm TTh Kayla Javier 15 8:00am-8:50am MWF Tony Nguyen 8 8:00am-9:15am TTh Travis Alvarez Page 12 of 12