Math 105 Second Midterm

Transcription

1 Math 05 Second Midterm November, 06 UMID: Solutions Section: Instructor:. Do not open this eam until ou are told to do so.. This eam has 9 pages (not including this cover page) and there are 9 problems in total. 3. Turn off and put awa all cell phones, pagers, headphones, smartwatches and an other unauthorized electronic devices.. Note that the problems are not of equal difficult, so ou ma want to skip over a problem if ou get stuck and return to it later. 5. Do not separate the pages of this eam. If the do become separated, write our UMID on ever page and notif our instructor when ou hand in the eam. 6. Please read the instructions for each individual problem carefull. One of the skills being tested on this eam is our abilit to interpret mathematical questions, so instructors will not answer questions about eam problems during the eam. 7. You must show an appropriate amount of work (including appropriate eplanation) for each problem unless indicated otherwise: the graders will be looking not onl for our answer, but also how ou obtained it. Include units in our answer where appropriate. 8. You ma use a TI-8, TI-89, TI-Nspire or other approved calculator. However, ou must show work for an calculation which we have learned how to do in this course. You are not allowed a notecard for this eam. 9. If ou use a graph that wasn t provided to ou in answering a problem, be sure to include an eplanation and sketch of the graph. If our solution to a problem makes reference to a table, it should be clear which entries of the table ou re using. 0. You must solve the problems on this eam using the methods ou have learned in this course. Problem Points Score Total 00 I recognize that there are severe penalties for academic dishonest and I affirm that I have done nothing to compromise the integrit of this eam, nor do I intend to help anone else do so. Initials:

2 Math 05 / Eam (November, 06) Page of 9. [0 points] You do not need to show an work for this problem, but ou should write our answers in the spaces provided. a. [ points] Let j() be an odd function with domain (, ) and j( ) =. Evaluate j(0) and j(). j(0) = 0, and j() = b. [ points] Let f() = log. Write down an epression for a function g() which is a transformation of f() that has a vertical asmptote at =. g() = log( + ) c. [3 points] Let f() = log and h() = log(0.5). B how much, and in which direction, must the graph of = f() be shifted verticall to obtain the graph of = h()? Your answer must be eact. The graph of = f() must be shifted verticall down b log(0.5) d. [ points] Let k() = b sin() 0 (for some constant b) be a periodic function with amplitude. List all possible values of b, and find the equation of the midline of = k(). The midline is = 0, and b could be or e. [3 points] Consider the graph of = tan( + ). Write down the equations of one horizontal asmptote and one vertical asmptote of this graph, or write none if there are no asmptotes of a particular tpe. Your answer must be eact. A vertical asmptote is = 0.5π, and a horizontal asmptote is none f. [6 points] Let R() = L(7 3) +. List the transformations ou need to appl to the graph of = L(), in order, to obtain the graph of = R(). Fill each space with either a number or one of the phrases below, as appropriate. shift it horizontall to the right shift it horizontall to the left shift it verticall upwards shift it verticall downwards compress it horizontall stretch it horizontall compress it verticall stretch it verticall To get the graph of = R(), we start with the graph of = L(). First, we compress it horizontall b /7, and then we shift it horizontall to the right b 3/7, and then we stretch it verticall b, and then we shift it verticall upwards b

3 Math 05 / Eam (November, 06) Page of 9. [ points] A drone starts at the origin O, and flies in a straight line to a point P with coordinates (a, b). From there, it travels counterclockwise around a circle of radius 8 centered at the point C = (0, 5), until it reaches the point Q. This is illustrated in the diagram below, though the diagram is not drawn to scale. Q D β C ϕ K θ P O Note that θ, β and ϕ are the positive measures of the angles P CK, DCQ and QCK (respectivel) given in radians. You do not need to show an work for this problem, but ou should write our answers in the spaces provided. a. [ points] Find the length of the line segment OP in terms of a and b alone. b. [ points] Find a formula for ϕ in terms of β alone. The length of OP is a + b ϕ = π β c. [3 points] Find the length of the (bolded) circular arc P Q in terms of θ and β alone. The length of the circular arc P Q is d. [ points] Write a formula for b in terms of θ alone. 8(θ + (π β)) b = 5 8 sin θ

4 Math 05 / Eam (November, 06) Page 3 of 9 3. [ points] At a wildlife sanctuar in central Africa, conservationists are carefull monitoring the population of various species of animals. For the following parts, write our answers in the spaces provided. Your answers for this problem can either be eact, or accurate to three decimal places. a. [3 points] On Januar, 008, the population of lions in the sanctuar was estimated to be 850, and was decreasing eponentiall at a continuous rate of 5% each ear. Find a formula for the population L(t) of lions in the sanctuar t ears after Januar, 008. You do not need to show an work for this part. L(t) = 850e 0.5t b. [5 points] On the other hand, the number of elephants in the sanctuar increased b 60% ever 7 ears. Let E(t) be the number of elephants in the sanctuar t ears after Januar, 008. What is the (annual) continuous growth rate of the function E? You should carefull show our work for this part. Solution: The function E(t) is eponential, so we have E(t) = ae kt for some constants a and k. We know that E(7) =.6a, so:.6a = ae 7k.6 = e 7k 7k = ln.6 k = ln.6 7 The continuous growth rate of E is ln(.6) 7 per ear. For the following parts, ou do not need to show an work, but ou can receive partial credit for work shown if our final answer is incorrect. c. [3 points] Let B(m) = 60(3) 0.5m be the number of buffalo in the sanctuar m months after Jul 5, 06. What is the (monthl) continuous growth rate of the function B? The continuous growth rate of B is ln ( 3 0.5) per month. d. [3 points] Let H() be the total value of donations received b the sanctuar s governing organization (in thousands of dollars) ears after Jul 5, 06. The function H is eponential, with continuous growth rate e What is the annual percentage growth rate of the function H? The annual percentage growth rate of H is e (e0.77 )

5 Math 05 / Eam (November, 06) Page of 9. [ points] The comets A and B each orbit the sun in an ellipse, as illustrated in the diagram below. Let A(t) be the distance (in millions of miles) between comet A and the sun and B(t) the distance (in millions of miles) between comet B and the sun t ears after June, 03. B A It takes 7 ears for comet A to complete a full orbit (i.e. return to its initial position), and 0 ears for comet B to do the same. The functions A(t) and B(t) have been graphed for the time it takes the comets to travel through a complete orbit and return to their starting positions. A(t) B(t) t t Note that the graphs are not drawn to the same scale. You do not need to show an work for this problem. a. [ points] What is the period of the function A(t)? Write our answer in the space provided. The period of A(t) is 3.5 ears b. [ points] What is the period of the function B(t)? Write our answer in the space provided. The period of B(t) is 0 ears c. [3 points] The closest that comet A gets to the sun is.75 million miles, and the function A(t) has midline =.5. What is the furthest that comet A gets from the sun? Write our answer in the space provided, and include units. The furthest that comet A gets is 6.75 million miles. d. [ points] Comet C (not shown above) also orbits the sun in an ellipse. Let C(t) be the distance (in millions of miles) from comet C to the sun t ears after June, 03. The function C(t) is periodic with period. Between t = 0 and t =, comet C is the closest to the sun at time t = 3. Which of the following must be true? Circle our answer(s) from the options listed; if none of the options are correct, circle none of these. C(7) = C(3) C(8) = C(3) C(3) > C() C() C() C(t) is the largest at t = none of these

6 Math 05 / Eam (November, 06) Page 5 of 9 5. [9 points] At low temperatures, pure tin can deteriorate and become brittle. Martín was recentl awarded a tin medal, which he received in the mail; unfortunatel, it had alread begun to degrade b the time he had received it. Let P (t) be the fraction of the medal that has degraded t das after Martín first opened the package containing the medal, which is given b the formula: where k > 0 is a constant. P (t) = + ( kt ) a. [ points] What fraction of the medal had alread degraded at the time Martín first opened the package? Circle our answer from the options below. If none of the options are correct, circle none of these. 5 9 none of these b. [ points] Let U(t) be the fraction of the medal that has not et degraded t das after Martín first opened the package. Find a formula that epresses U(t) as a (combination of) transformation(s) of P (t). You do not need to show an work for this part, but ou should write our answer in the space provided. U(t) = P (t) c. [5 points] Eactl das after first opening the package, Martín finds that the fraction of the metal that has degraded is now Set up an equation representing this fact, and use this to find k. Your answer must be found algebraicall, and must be eact. You should carefull show our work for this part, and write our answer in the space provided. Solution: We have P () = 0.95, which gives us: 0.95 = + ( k ) + ( k ) = 0.95 ( k ) = 0.95 k = ( ) 0.95 ( ( k ln = ln 0.95 k = ln ln ( )) ( 0.95 )) k = ( ( )) ln ln 0.95

7 Math 05 / Eam (November, 06) Page 6 of 9 6. [6 points] A sculpture of a star of height S (measured in meters) is mounted on a base of height B (measured in meters). Reina is standing at a distance of 0 meters awa from the base of the statue, at the point R. L S R 6 0 m M P B The measures of the angles LRM and LRP are and 6 respectivel. Note that the diagram above is not drawn to scale. For this problem, ou should show our work, and write our answers in the spaces provided. a. [3 points] Write an epression for the height S of the star. Your answer ma involve the constant B. Solution: Since LP R is a right angle, we have: and solving for S gives us: tan(6 ) = S + B 0 S = 0 tan(6 ) B S = 0 tan(6 ) B b. [3 points] Find the height B of the base of the statue. Your answer for this part cannot involve S. Solution: Since MP R is a right angle and the measure of angle MRP is 6 = 0, we have: tan(0 ) = B 0 and solving for B gives us: B = 0 tan(0 ) B = 0 tan(0 )

8 Math 05 / Eam (November, 06) Page 7 of 9 7. [ points] In each of the following parts, ou are given an equation in which ou must solve for. Your answers must be eact and should be obtained algebraicall. You should show all our work, step-b-step, and write our final answers in the spaces provided. a. [3 points] ln( ) = Solution: We eponentiate both sides and solve for, which gives us: = e 3 7 = e 5 7 = ( e 5 ) 3 = 7 3 (e 5) b. [ points] e 7 = 5e 0 = 7 3 (e 5) Solution: We take the natural logarithm of both sides and use properties of the logarithm to simplif, which gives us: ln ( e 7) = ln ( 5e 0) 7 = (ln 5) = ln 5 = 3 ln 5 = ln 5 3 c. [ points] (log(a)) = 0, where a > 0 is a constant. Your answer for this part ma involve a. Solution: We ll first isolate the (log(a)) 3 on one side, then take a cube root and eponentiate to solve for : (log(a)) 3 = log(a) = 3 a = 0 3 = a 0 3 = a 0 3

9 Math 05 / Eam (November, 06) Page 8 of 9 8. [0 points] Twissell is attempting to determine the depth of a lake at various points b lowering a sensor to the bottom of the lake and measuring the intensit of the sun s light at that point. According to his calculations, if the intensit of the light (in lumens) that reaches the sensor is I, then the sensor must be at a depth of F (I) (measured in meters), which is given b the formula: F (I) = ( ) I log c where c > 0 is a constant. Note that the intensit of light I at all points underwater is positive and smaller than c, so F (I) is positive. In the following parts, ou must show all our work, step-b-step, and find our answers algebraicall to receive full credit. Your final answers must be eact, and should be written in the spaces provided. a. [ points] What is the intensit of the light that reaches the sensor when it is meters underwater? Your answer for this part ma include the constant c, and should include units. Solution: Two meters below the surface, the intensit I should satisf: = ( I log c ( ) I log = 8 c I c = 0 8 I = c 0 8 The intensit meters below the surface is b. [6 points] Twissell submerges the sensor at two different points in the lake. ) c 0 8 At the first point, the depth is d meters and the sensor measures the intensit of the sun s light to be 6K lumens. At the second point, the depth is D meters and the sensor measures the intensit of the sun s light to be K lumens. How much deeper is the lake at the second point compared to the first point? Your final answer should be simplified so that it does not include the constants K or c, but should include units. Solution: We know that: ) ) D d = ( K log c ( log = ( K c + ( 6K log c ) ( )) 6K log c which we can simplif using properties of the logarithm to get: = ( ( )) K log c c 6K = ( ) log 6 ) D d = log ( 6

10 Math 05 / Eam (November, 06) Page 9 of 9 9. [8 points] Consider the graph of = f() given below. - - For each of the graphs below, if the graph is a (combination of) transformation(s) of the graph of = f(), write an epression in the space provided that gives this (combination of) transformation(s). If the given graph is not a combination of vertical and horizontal shifts, stretches, compressions and reflections of the graph of = f(), write not a transformation. You do not need to show an work for this problem This is the graph of = f( ) This is the graph of = not a transformation This is the graph of = f( + ) This is the graph of = f( + )