MATH 2070 Test 1 (Sections )

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1 MATH 070 Test 1 (Sections ) Spring 018 Multiple Choice: Use a # 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 1 multiple choice questions. Each question is worth points for a total of 9 points. For future reference, circle your answers on this test paper as you will not receive your Scantron back with your test. The rate of change in the percentage of smokers in the United States from 1990 to 010 is given by p( t) 0.009t 0.t 0.6 percent per year, t years since Check: p(5) 0.11 Use this information to answer the net si questions. 1. Which of the following statements is TRUE? a. In 1990, 6.% of people in the U.S. smoked. b. In 199, the percentage of smokers in the U.S. reached a relative maimum. c. In 001, the percentage of smokers in the U.S. was not changing. d. In 001, the percentage of smokers in the U.S. reached a relative minimum.. What is the total area trapped between the graph of p( t ) and the t-ais between t = 0 and t = 0? a b c d. 9.. Which of the following correctly completes the interpretation of p( t) dt 0.5? 0 From 1990 to 199, a., the percentage of smokers in the U.S. increased by 0.5 percentage points. b., the change in the percentage of smokers in the U.S. increased by 0.5 percentage points. c., the percentage of smokers in the U.S. was increasing by 0.5 percentage points per year. d., the rate-of-change in the percentage of smokers in the U.S. was increasing by 0.5 percentage points per year.

2 MATH 070 Test 1 (Sections ) Spring 018 contet continued. Which of the following describes the graph of the accumulation function P( ) p( t) dt 0 on the interval ? a. decreasing and concave up b. decreasing and concave down c. increasing and concave up d. increasing and concave down 5. On what interval, would the graph of the accumulation function P( ) p( t) dt be 0 decreasing at a faster rate? a b only c d. 0 only 6. In 1991, 5.7% of people in the United States were smokers. What percentage of people in the United States were smokers in 199? a % b % c % d %

3 MATH 070 Test 1 (Sections ) Spring Find the value of C in the specific antiderivative function F() when F (1). f ( ) and a. C b. C.5 c. C d. C.5 8. Given F is an antiderivative of f, d f ( t) dt d =. 0 a. F ( ) F (0) b. f (0) c. f ( ) d Given F is an antiderivative of f, f ( t ) dt =. 1 a. F () F ( 1) b. f () f ( 1) c. f () d e d a. e C b. 1 e C 1 c. e C d. e C 5

4 MATH 070 Test 1 (Sections ) Spring A car was traveling at 9 feet/ second. The driver noted her speed at time t = 0. She maintained this speed until t = 6 seconds. The figure shows her velocity in terms of time. At t = seconds she noticed a stop sign at the side of the road. v(t) feet per second At t = 6 seconds she reacted and applied the brakes. The car came to a rest at t = 9 seconds. How far did the car travel between the time she noticed the stop sign and when the car came to a rest? a. 9.5 feet b feet c feet d. 5.5 feet 1. Consider the graph of f ( ) shown to the right. Which of the following is the graph of f ( ) d? a. b. c. d. 6

5 MATH 070 Test 1 (Sections ) Spring 018 Consider the graph of c( ) shown above. The area between c( ) and the -ais in each region is denoted with A = area. 1. Which of the following ordered pairs is on the graph of the accumulation function C( ) c( ) d? 0 a. (, 8) b. (, -8) c. (, ) d. (8, 9) Check your Scantron now to make sure it will successfully run. Refer to the last page of the test for specifics. If it does, you will earn one point. (1 pt) 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. 7

6 MATH 070 Test 1 (Sections ) Spring 018 RE-READ the directions on the second page of the test regarding rounding, units, etc. 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. This portion is 60%. 1. Find each of the following. Use proper notation throughout your work. You do not need to simplify your coefficients. ( pts, pts, pts) 1 a. e e e d e ln C 1 pt each of the first three terms, ½ pt C, ½ pt notation -½ pt for missing absolute value sign on the third term 1 d 5 d 5 C ln( ) 1 pt each of the first three terms, ½ pt C, ½ pt notation b c. 5 d d C pt each of the first three terms (½ pt partial per term can be earned for rewriting correctly), ½ pt C, ½ pt notation; - pts for f ( ) g( ) d f ( ) d g( ) d OR f ( ) d g( ) d -½ pt per term for an incorrect rewrite as long as the product is being simplified, but then follow; -½ pt for simplification error. Possible notation errors in parts a c: misuse of equal signs, missing d, leaving the integral after the antiderivative is found, etc. 8

7 MATH 070 Test 1 (Sections ) Spring 018. Algebraically evaluate the following integral to obtain the eact answer. (7 pts) Show all steps necessary to obtain the answer algebraically using proper notation throughout your work. Simplify fractions. i.e. 17 should be simplified to 15 which should be simplified to 5. Do not approimate values such as 1, ln(), or e. Keep all values eact. Combine like terms and simplify when possible. 5 e d 5 e 5() 5() e () e () 5 0 e e 5 e OR 1.5 e If the antiderivative is never attempted or if only the general antiderivative was found with no other notation, no credit was awarded (-7 pts) pts anti-derivative (1 pt each term; okay if +C is included) 1 pt for the evaluation notation OR a clearly labeled general antiderivative. pts for F(b) F(a) - for F(a) F(b) - for F(b) + F(a) 1 pt for distributing the negative (point can not be received if F(b) + F(a) in previous step or if their incorrect antiderivative only has one term; if incorrect antiderivative was found in the first step, this point is only awarded if the distribution of the negative is shown eplicitly) 1 pt for the correct, simplified, final answer (correct answers that do not follow from the work shown receive no credit); deduct ½ pt from this step if their incorrect antiderivative only has one term as they have worked a simpler problem. -½ pt for unsimplified answers or decimal approimations or incorrect simplifications such as combining the e terms. -½ pt for each notational error up to -1 pt total (equal signs between steps, missing bounds, missing parentheses or brackets, sign errors, leaving the d or integral symbol in the antiderivative, etc.) 9

8 MATH 070 Test 1 (Sections ) Spring 018. The rate of change in the amount of gasoline consumed in the United States between 010 and 016 is given by g( ) billion gallons per year, years since 000. Check: g(1).96 a. Complete the following sentences with the correct word/values. (6 pts) In 01, the amount of gasoline consumed in the United States was increasing increasing/decreasing by billion gallons per year. From 010 to 01, the amount of round to three places gasoline consumed in the United Stated _decreased_ by _.56 billion gallons. increased/decreased round to three places 1 g( ) d.56 and g(1) pt for each word; pts for each number; -½ pt for sign errors or incorrect suffi b. Find the general antiderivative function, G( ). Coefficients do not have to be simplified. ½ pt each term, all or nothing ( pts) G( ) g( ) d 5.68 C or C c. In 011, 11.7 billion gallons of gasoline were consumed in the United States. Find the specific antiderivative function and complete the specific antiderivative model below. Show your work in the space provided and then complete the model below. (6 pts) G(11) 11.7 C OR 0.081(11) 0.508(11) 5.68(11) C 11.7 C G( ) billion gallons output units gives the function gasoline consumption in the U.S. output description years after 000, pts for correct C pts for work [showing G(11) = 11.7 or values plugged in] -½ pt for g(11) = 11.7 if G was actually used to find C -½ pt for notation errors such as using Y 1 instead of G (see the directions on page of the test) pts for the correct C value (1 pt additional partial credit can be earned here for additional algebraic work shown; the second point should only be awarded for the correct C value i.e. don t follow work for incorrect answers in part a.) ½ pt for rewriting the general antiderivative as a specific antiderivative (only award if C-value is other than 11.7) ½ pt for output units (all or nothing) 1 pt for output description (all or nothing) d. In 006, the White House stated that they wanted to cut gasoline usage to 18 billion gallons by 016. Was this goal achieved? ( pts) 1 pt for correct conclusion based on relevant work (no credit Circle One: Yes or No without relevant work) pts relevant work: G(16) = value OR Work to Justify Your Answer: 1 pt for the integral from 1 to 16 and 1 pt for change -½ pt for notation errors (Y 1 instead of G or g, missing d, etc.) G(16) pts for finding G(6) instead of G(16) or - pts for finding the G(16) G(6) or the integral of g() from 6 to as this does not answer the given question g( ) d

9 MATH 070 Test 1 (Sections ) Spring 018. The misery inde is defined as the unemployment rate plus the inflation rate. The rate of change of the misery inde from 01 to 017 can be modeled by the function m( ) percent per year where is the number of years after 01. Check: m() 1.08 a. When did the misery inde reach a relative minimum between 01 and 017? ( pts).77 years after 01 round to three decimal places b. Sketch two midpoint rectangles of equal width to estimate the change in the misery inde between 01 and pt per rectangle Shade in your rectangles. ( pts) ½ pt per rectangle c. What are the units of measure on: ( pts) 1 pt each (values given along with correct units were ignored) i. the height of the rectangles: percent per year ii. the width of the rectangles: years iii. the area of the rectangles: percent or percentage points d. Estimate the change in the misery inde between 01 and 017 using the two rectangles you sketched above. Show your work, round to two decimal places and include units with your answer. Follow work from b. ( pts) m(1) m() % or.68 percentage points 1 pt each height, ½ pt each area, ½ pt final answer, ½ pt units Function notation does not have to be shown, individual height values can be shown instead. - pts if integral is calculated; Do not deduct for notational errors such as using incorrect variable names (f instead of m). e. Write the integral notation that would be evaluated to find the sum of the signed areas between m() and the -ais between = 0 and =. You do not need to find this amount. ( pts) m( ) d OR 0.77 m( ) d m( ) d Follow from part a if needed f. Write the integral notation that would be evaluated to find the total area between m() and the -ais between = 0 and =. You do not need to find this amount. ( pts).77 m( ) d m( ) d OR m( ) d m( ) d Follow from part a if possible.77 0 e & f: -1 to -1.5 pts for one or two empty integrals with correct bounds; -1/ if top limit of integration was instead of One time -½ pt for notation errors in part e and/or f (missing d, wrong function name, Y 1 instead of m, etc.) f: -1.5 pts for the incorrect -intercept value used correctly in the integrals -1.5 pts if the two integrals are shown with the correct bounds but the incorrect operation (missing absolute values or addition instead of subtraction); -1 pt for m instead of m() if that notation was used 11

10 MATH 070 Test 1 (Sections ) Spring m(t) t Consider the graph of m( t) shown above. Use this graph to determine the characteristics of the associated accumulation function M ( ) m( t) dt, whose graph is not given. a a. Circle the correct answers. There may be more than one circle per line. ( pts) M ( ) is decreasing most rapidly at equals: a b c d e f g h M ( ) has a relative maimum at equals: a b c d e f g h M ( ) has an inflection point at equals: a b c d e f g h ½ pt per circle; -½ pt per etra circle up to the point total per line b. Determine an interval on which M ( ) has the following characteristics. For eample, if your answer is between a and b write a < < b on the lines. ( pts) 1 pt each interval, all or nothing M ( ) is increasing and concave up: _f < < g M ( ) is decreasing at a slower rate: _d < < _ f (also accept e) 1