Math 105 / Final (December 17, 2013) page 5

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1 Math 105 / Final (December 17, 013) page 5 4. [13 points] Severus Snake is slithering along the banks of a river. At noon, a scientist starts to track Severus s distance awa from the edge of the river. After a few minutes, the scientist realizes Severus s distance awa from the edge of the river is a sinusoidal function. Let D(t) be Severus s distance, in centimeters, awa from the edge of the river t seconds after noon. a. [5 points] At noon eactl, the scientist notes that Severus is 97 centimeters awa from the edge of the river, which is the farthest awa he ever gets. Three seconds after that, Severus s distance is 65 centimeters awa from the river, the closest he gets. Graph = D(t) for 0 t 1. (Clearl label the aes and important points on our graph. Be ver careful with the shape and ke features of our graph.) b. [6 points] Find the period, amplitude, equation of the midline, and a formula for the sinusoidal function D(t). (Include units for the period and amplitude.) Period: Amplitude: Midline: Formula: D(t) = c. [ points] How far awa from the river is Severus 11 seconds after noon? Give our answer accurate to at least two decimal places.

2 Math 105 / Final (December 17, 013) page 6 5. [5 points] Find a formula for one polnomial p(z) that satisfies all of the following conditions: lim p(z) = and lim p(z) = z z The onl zeros of p(z) are z =, z = 1, and z = 3. The point (, 1) is on the graph of p(z). The degree of p(z) is at most 5. Show our work and reasoning carefull. You might find it helpful to first sketch a graph. There ma be more than one possible answer, but ou should give onl one answer. p(z) = 6. [5 points] Find all solutions to the equation ( 5 tan + π ) 13 = 1 for between 0 and 5. Show our work carefull and give our answer(s) in eact form. =

3 Math 105 / Final (December 17, 013) page 8 8. [11 points] On the beaches of Meico, there is a population of pick snails that wait for special shells to wash up onto the shore. These snails can onl live in these particular shells, as the snails have become accustomed to the comfort in these shells. Suppose the number of hundreds of special shells on the beaches of Meico t ears after the beginning of 013 is h(t) = (t + 7)(4t 7) and the population, in hundreds, of pick snails t ears after the beginning of 013 is p(t) = (t ) (8t + 30). Throughout this problem, remember to clearl show our work and reasoning. a. [3 points] Find the leading term and an zeros of h(t). If appropriate, write none in the answer blank provided. Answers: Leading Term: Zero(s): b. [3 points] The number of shells per snail is Q(t) = h(t) p(t). Find the equations of all vertical asmptotes ( V.A. ) and horizontal asmptotes ( H.A. ) of the graph of = Q(t). If appropriate, write none in the answer blank provided. Answers: V.A.: H.A.: There is a competitive population of crabs that live on the same beaches. Suppose that there are 100 of these crabs at the beginning of 013, and that the population grows at a continuous annual rate of 35%. Let c(t) be the population, in hundreds, of these crabs t ears after the beginning of 013. c. [ points] Find a formula for c(t). c(t) = d. [3 points] The crabs like the same special shells as the snails do. Write a formula for the ratio of the number of shells to the number of crabs t ears after the beginning of 013. In the long run, what happens to the ratio of the number of shells to the number of crabs? In other words, assuming the functions described in this problem continue to be accurate models, what happens to this ratio after man, man ears? You must clearl indicate our reasoning in order to receive an credit for this problem.

4 Math 105 / Final (December 17, 013) page 9 9. [5 points] Note that throughout this problem, ou are not required to show our work. A portion of the graph of a sinusoidal function h() is shown below. 4 = h() a. [ points] Which, if an, of the figures below shows part of the graph of = 1 h()? Note that the scale is smaller than in the original graph above. Be sure to pa attention to the scale indicated on the aes (Option A) (Option B) (Option C) Circle our one final answer below. (Onl the answer ou circle below will be graded.) Option A Option B Option C none of these b. [3 points] Which, if an, of the figures below shows part of the graph of = h( + )? Note that the scale is smaller than in the original graph above. Be sure to pa attention to the scale indicated on the aes (Option A) (Option B) (Option C) Circle our one final answer below. (Onl the answer ou circle below will be graded.) Option A Option B Option C none of these

5 Math 105 / Final (December 17, 013) page [4 points] Suppose that the number of acorns in Squish squirrel s nest is proportional to the cube of the number of squirrels currentl living there. If there are 113 acorns in his nest when there are two squirrels living there, how man acorns will there be in Squish s nest when there are four squirrels living there? Remember to show our work carefull. 11. [7 points] Wolfgang the wolf is on a 10-foot long leash that is tied to a post that is 40 feet west of a fence. Wolfgang θ Post Q 40 feet Fence Fence P(0,0) Because he dislikes being on his leash, he stas 10 feet awa from the post at all times. a. [4 points] Suppose we think of the origin at the point P as shown in the diagram and that the unit of measurement is feet so that the coordinates of the post are (0, 0). Find Wolfgang s coordinates when he is at the angle θ shown in the diagram. (Your answer should be in terms of θ.) Wolfgang s coordinates are (, ). b. [3 points] Wolfgang starts walking counterclockwise from the point Q. The angle θ through which Wolfgang has walked is a function of the amount of time he has been walking. Let θ = z(t) be the angle (in radians) through which Wolfgang has walked after he has been walking for t minutes. Let A(t) be the distance Wolfgang has traveled along the circle in t minutes. Find a function f(t) such that A(t) = f(z(t)). f(t) =

6 Math 105 / Final (December 17, 013) page [10 points] Consider the functions f, g, and h defined as follows: f() = a + b g() = c d h() = w(1 + r) for nonzero constants a, b, c, d, r, and w with r > 1. For each of the questions below, circle all the correct answers from among the choices provided, or circle none of these if appropriate. a. [ points] The graph of which function(s) definitel has at least one horizontal intercept? f() g() h() none of these b. [ points] The graph of which function(s) definitel has at least one horizontal asmptote? f() g() h() none of these c. [ points] Which function(s) is(are) definitel invertible? f() g() h() none of these d. [ points] How man times could the graph of f() intersect the graph of h()? more than 4 e. [ points] Suppose the graph of h is concave up. Which of the following is(are) definitel true? w > 0 w < 0 r > 0 r < 0 none of these

7 Math 105 / Eam (November 1, 01) page 5 6. [14 points] The number of hours of dalight in Ann Arbor varies from a minimum of 9.1 hours of dalight on December 1 to a maimum of 15.3 hours of dalight on June 1 (and then back down to 9.1 hours on the following December 1). Let L = D(m) be the number of hours of dalight in Ann Arbor on a da that is m months after December 1, 010. Assume that D(m) is a sinusoidal function. a. [4 points] On the aes provided below, graph two periods of the function L = D(m) starting with m = 0. (Clearl label the aes and important points on our graph. Be ver careful with the shape and ke features of our graph.) b. [4 points] Find the period, amplitude, and midline of L = D(m). (Include units for the period and amplitude.) Period: Amplitude: Midline: c. [4 points] Find a formula for D(m). D(m) = d. [ points] Use our formula from part (c) to estimate the number of hours of dalight in Ann Arbor on April 1. (Show our work and round our answer to the nearest 0.1 hour.)

8 Math 105 / Final (December 14, 01) page 3 3. [9 points] Note that the problems on this page are not related to each other. (You do not have to show work. However work shown ma be used to award partial credit.) a. [3 points] A salesperson at a local department store earns a base salar of $750 per month plus a commission (bonus) of 8% of her total sales. Let M(d) be the emploee s total earnings, in dollars, in a month in which she sells d dollars worth of merchandise. Find a formula for M(d). M(d) = b. [3 points] Suppose that the half-life of caffeine in a student s bloodstream is 5 hours. If the student drinks a latte that contains 150 mg of caffeine at 8 am, find a formula for C(h), the amount of caffeine (in milligrams) from that latte that remains in the student s bloodstream h hours after 8 am. C(h) = c. [3 points] The monthl revenue of a local business varies seasonall from a low of $35,000 in Februar to a high of $75,000 in August (and back down to $35,000 the following Februar). Let R(t) be this compan s monthl revenue, in thousands of dollars, t months after Januar. (Note that t = 0 represents Januar, t = 1 represents Februar, etc.) Assuming that R(t) is a sinusoidal function, find a formula for R(t). R(t) =

9 Math 105 / Final (December 14, 01) page 8 8. [1 points] Suppose Cato is riding a stationar eercise biccle. His foot moves a pedal in a circle. Let h(t) be the height (in cm) of the pedal above the ground at time t (in seconds). A formula for h(t) is given b h(t) = 0 sin (πt) a. [3 points] On the aes provided below, graph two periods of the function P = h(t) starting with t = 0. (Clearl label the aes and important points on our graph. Be ver careful with the shape and ke features of our graph.) b. [ points] Find the period and amplitude of P = h(t). (Include units.) Period: Amplitude: c. [4 points] Find all the times t for 0 t when the pedal is eactl 45 cm above the ground. (Find at least one answer algebraicall. Show our work carefull and check that our answers make sense.) Answer(s): d. [3 points] Find the length of the arc through which the pedal travels between t = 0 and the time the pedal first reaches a height of eactl 45 cm. (Show our work and reasoning. It ma help to sketch a picture.) Answer(s):

10 Math 105 / Eam (March 0, 01) page [10 points] Ivanka is a student at a nearb college. Let C(h) be the total tuition, in thousands of dollars, the college charges her if she takes h credit hours, and let a be the average number of credit hours students take at the college. For each of the following, pick the one epression from the list of Answer Choices that best represents the described quantit. Clearl write the capital letter of our choice on the answer blank provided. Answer Choices A. C(3) B. C 1 (3) C. C(a) D. C 1 (a) E. C(a + 3) F. C(a) + 3 G. C(a 3) H. C(a) 3 I. 3C(a) J. C(3a) K. 3C 1 (a) L. C 1 (a)/3 M. C(a)/3 N. C(a/3) O. C 1 (a/3) P. C 1 (3a) Q. 3C(C(a)) R. C(3C 1 (a)) S. C 1 (3C(a)) T. C 1 (C(3a)) a. [ points] Ivanka s tuition (in thousands of dollars) if she takes a total of 3 credit hours b. [ points] Ivanka s total tuition (in thousands of dollars) if she takes 3 credit hours more than average c. [ points] Ivanka s tuition (in thousands of dollars) if she takes one third the average number of credit hours d. [ points] The amount (in thousands of dollars) that Ivanka pas for tuition if she takes the average number of credit hours but has a scholarship that covers three thousand dollars of her tuition e. [ points] The number of credit hours Ivanka takes if her total tuition is three times as much as the tuition for taking the average number of credit hours

11 Math 105 / Final (April 19, 01) page 6 5. [6 points] Let g be the function defined b g() = 10( 1)( ) ( + 1)( + 1). Find all zeros, -intercepts, and horizontal and vertical asmptotes of the graph of = g(). If appropriate, write none in the answer blank provided. (Show our work and write our answers in eact form.) zero(s): -intercept(s): horizontal asmptote(s): vertical asmptote(s): 6. [5 points] Find all solutions to the equation 5 sin(t) = π for 0 t π. (Show our work clearl and give our final answer(s) in eact form.) Answer(s):

12 Math 105 / Final (December 15, 011) page 1. [9 points] The graph of a function h() is shown on the right. Below are the graphs of several transformations of h(). For each of these graphs, write the letter of the one function from the list on the right of the page whose graph is shown. (Clearl write the capital letter of our choice on the answer blank provided.) No work or eplanation is required. = h() ( 5, ) ( 3, ) (1, ) (4, 4) (5, 3) a. [3 points] (, 3) (, 1) (, 1) (5, 5) (6, ) Answer Choices A. h( + 1) + 1 B. h( 1) + 1 C. h( + 1) 1 D. h( 1) 1 E. h( ) + 1 F. h( ) 1 G. h() + 1 b. [3 points] (, 8) H. h() 1 I. h( + 1) (5, 4) J. h( 1) K. h( ) L. h( ) M. h() ( 5, 6) c. [3 points] ( 1, ) (3, ) N. h( ) O. h() P. 1 h() Q. 1 h( ) R. 1 h() 1 ( 0.5, ) (1.5, ) ( 1.5, ) (3, ) (3.5, 3) 1 S. h( 1) T. h(( 1)) U. h( 1) V. h(( 1)) W. h( 1 1) X. h( 1 ( + 1)) Y. h( 1 ( 1)) Z. None of these