Math 116 Final Exam April 19, 2018

Size: px

Start display at page:

Download "Math 116 Final Exam April 19, 2018"

Norah Manning
5 years ago
Views:

1 EXAM SOLUTIONS Math 6 Final Eam April 9, 8. Do not open this eam until ou are told to do so.. Do not write our name anwhere on this eam.. This eam has 5 pages including this cover. Do not separate the pages of this eam. If pages do become separated, write our UMID on ever page and point this out to our instructor when ou hand in the eam. 4. There are problems. Note that the problems are not of equal difficult, so ou ma want to skip over and return to a problem on which ou are stuck. 5. Note that the back of ever page of the eam is blank, and, if needed, ou ma use this space for scratchwork. Clearl identif an of this work that ou would like to have graded. 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. Show an appropriate amount of work (including appropriate eplanation) for each problem, so that graders can see not onl our answer but how ou obtained it. 8. The use of an networked device while working on this eam is not permitted. 9. You ma use an one calculator that does not have an internet or data connection ecept a TI-9 (or other calculator with a qwert kepad). However, ou must show work for an calculation which we have learned how to do in this course. You are also allowed two sides of a single 5 notecard.. For an graph or table that ou use to find an answer, be sure to sketch the graph or write out the entries of the table. In either case, include an eplanation of how ou used the graph or table to find the answer.. Include units in our answer where that is appropriate.. Problems ma ask for answers in eact form. Recall that = is a solution in eact form to the equation =, but = is not.. Turn off all cell phones, smartphones, and other electronic devices, and remove all headphones, earbuds, and smartwatches. Put all of these items awa. 4. You must use the methods learned in this course to solve all problems. Problem Points Score Problem Points Score Total

2 (7) + (7) = + (.5) = 5 Math 6 / Final (April 9, 8) page. [8 points] Once again, Giuseppe Li has a nightmare. This time he is roaming the plains in a covered wagon. His stepfather, Gerd Hömf, plotted the - and -coordinates of Giuseppe s dream wagon measured in astral miles as a function of dream das t. The positive -direction is north, and the positive -direction is east. (t) t (t) t a. [ points] At what time(s) t was Giuseppe s wagon moving directl north? Directl south? If there are no such times, write none. Directl north at t = 5 Directl south at t = b. [ points] How far, in astral miles, from where it began was Giuseppe s wagon after 8 dream das? astral miles c. [ points] What was Giuseppe s approimate speed, in astral miles per dream das, at time t = 7? speed at t = 7 is given b.5 astral miles per dream da. d. [ points] Approimate the total distance, in astral miles, that Giuseppe s wagon traveled during the first dream das. On the first da Giuseppe travels from (, ) to (, ) and on the second da from (, ) to (, ) along an approimatel straight path each da. Hence the total distance is approimatel ( ) + ( ) + ( ) + ( ) = astral miles.

3 Math 6 / Final (April 9, 8) page. [9 points] For a class project, Yennifer is studing the accumulation of dead leaves on the ground in a particular region on Nichols Arboretum. a. [4 points] She finds that the dead leaves accumulate at a constant rate of 6 grams per square centimeter per ear. At the same time, the leaves on the ground decompose at a continuous rate of 8 percent per ear. Write a differential equation for the total quantit Q of dead leaves, in grams per square centimeter, at time t, in ears. dq dt = 6.8Q b. [5 points] Yennifer finds that if she covers the ground in purified muck, then the total quantit P, in grams, of dead leaves per square centimeter satisfies the differential equation dp = (P ) cos(πt) dt In addition, when she first applies the muck (at t = ), the ground is covered with gram per square centimeter of leaves. Use separation of variables to find a formula for P (t). Show our work. Separating variables gives P dp = cos(πt) dt ln P = π sin(πt) + C P = C e π sin(πt) +. Using the initial condition P () = we find = C +, hence C =. P (t) = e π sin(πt)

4 Math 6 / Final (April 9, 8) page 4. [ points] Consider the function f() graphed below. 8 7 = f() a. [ points] Let F () = 4 f(t) dt. Find the -coordinates of all local etrema of F () and classif them as local maima or local minima. Write none if there are none. Local maima at = 8 b. [ points] Let G() = Local minima at = 6 f(t) dt. Compute G ( ). Using the second fundamental theorem of calculus we compute G () = f( ) f(). Therefore, G ( ) = f() f( ) = 6 G ( ) = 6 c. [ points] Which approimation method is guaranteed to underestimate MID TRAP LEFT RIGHT none of these 4 f() d? d. [ points] Which approimation method is guaranteed to overestimate 5 f() d? MID TRAP LEFT RIGHT none of these

5 Math 6 / Final (April 9, 8) page 5 4. [ points] a. [6 points] Determine whether the following series converges absolutel, converges conditionall, or diverges, and give a complete argument justifing our answer. ( ) n sin ( ) n= n Converges absolutel Converges conditionall Diverges Justification: This series converges b the alternating series test, which applies, since sin( n ) is a positive decreasing sequence that converges to zero. It does not converge absolutel since for n We know the series so must n= sin ( n n= n ) = n= n sin ( n). diverges b p-test with p =. Then b the comparison test, ( ) n sin ( n). Alternativel, we can use the Limit Comparison Test. Since we know that sin(/n) sin(/n) sin() lim = lim = lim = <, n /n /n n and sin ( n) must either both converge or both diverge. Since n= n= n is the harmonic series, which we know diverges, n= n= sin ( n) must diverge as well.

6 Math 6 / Final (April 9, 8) page 6 b. [5 points] Compute the value of the following improper integral. Show all our work using correct notation. Evaluation of integrals must be done without a calculator. e ( + e ) d First we change to limit notation, then use u-substitution with u = + e. e ( + e d = lim ) b b b = lim b u du = lim b b u e ( + e ) d = lim b b = Alternativel, first compute the antiderivative using u-substitution. e ( + e ) d = u du = u = + e. Thus, e b e ( + e d = lim ) b ( + e d = lim ) b + e b + =.

7 Math 6 / Final (April 9, 8) page 7 5. [ points] Compute the eact value of each of the following. You do not need to show work. 5( ) n a. [ points] Find the radius of convergence R of n. n= B the geometric series test this series converges if < which implies that the radius of convergence is R =. b. [ points] n= e n R = e n = n= e e e n c. [ points] lim p(t) dt, where p(t) is a probabilit densit function. n n lim p(t) dt = n d. [ points] Find the function f() satisfing e d = e f() d Taking the derivative of both sides we find that e = e + e f(). Therefore f() = e. e. [ points] n= ( 4) n (n)! f() = e After rewriting the series as n= ( 4) n (n)! = n= ( ) n n, (n)! we recognize this as cos(). n= ( 4) n (n)! = cos()

8 Math 6 / Final (April 9, 8) page 8 6. [ points] Leight Vloss has instructed his Star Children to run laps on the Trail of Atonement. The Trail of Atonement is best described as the polar curve r = + cos(.5θ) where r is measured in kilometers. An aerial view of the trail is illustrated below. r = + cos(.5θ) a. [4 points] Leight stands on a pedestal in the center of the trail (at the origin). What is the furthest distance, in km, a Star Child gets from Leight on the Trail of Atonement? List all angles θ in [, 4π) where this distance r is achieved. Since the maimum value of cos(.5θ) is it follows that r is at most. To find the angles where this maimum r is achieved, we set + cos(.5θ) = So the values in [, 4π) are θ =, 4π, 8π. cos(.5θ) = θ =, π, 4π,... Alternativel, since we re looking for angles where r achieves its maimum, we can set the derivative of r = + cos(.5θ) equal to. Since r (θ) =.5 sin(.5θ) we have that θ =, 4π, 8π. Greatest distance: km θ = θ =, 4π, 8π

9 Math 6 / Final (April 9, 8) page 9 b. [ points] Write an integral in terms of θ which represents the total length, in km, of the Trail of Atonement. Since the Trail of Atonement is described b r = f(θ) = + cos(.5θ) with θ < 4π we use the polar arc length formula to write 4π 4π f(θ) + f (θ) dθ = ( + cos(.5θ)) + (.5 sin(.5θ)) dθ. Alternativel, if we use = ( + cos(.5θ) sin(θ) and = ( + cos(.5θ)) sin(θ), and then the parametric formula for arc length, we get where and d dθ 4π (d ) + dθ ( ) d dθ dθ =.5 sin(.5θ) sin(θ) + ( + cos(.5θ)) cos(θ) d dθ =.5 sin(.5θ) cos(θ) ( + cos(.5θ)) sin(θ) c. [4 points] The shaded innermost region of the trail is called the Sacred Heart. Write an epression involving one or more integrals which represents the area, in km, of the Sacred Heart. The corners of the Sacred heart occur when r = as we can see from the leftmost corner. The first two solutions of = + cos(.5θ) are θ = π, π. These angles describe / of the Sacred Heart, so the total area is π f(θ) dθ = π π π ( + cos(.5θ)) dθ.

10 Math 6 / Final (April 9, 8) page 7. [6 points] The function r(t), defined for all real numbers t, gives the position of a particle moving along the unit circle, r(t) = (cos(t t ), sin(t t )). a. [ points] Find all values of t where the particle stops moving. The particle stops moving when its speed is zero. The speed is given b ( sin(t t )( t )) + (cos(t t )( t )) = t. Therefore the speed is zero at t = ±. t = ± b. [ points] For which values of t is the particle moving counterclockwise? The parametric function r(t) moves counterclockwise precisel when f(t) = t t is increasing, which is the same as f (t) >. Since f (t) = t, this happens for t in (, ). < t <

11 Math 6 / Final (April 9, 8) page 8. [8 points] Let f() = e. a. [4 points] Find the first four nonzero terms of the Talor series for f() centered at =. b. [ points] Find the value of f (8) () Since f() is an odd function, there are no odd powers of in the Talor series epansion of f() centered at =. Since the coefficient of 8 is f (8) () 8!, it follows that f (8) () =. f (8) () = c. [ points] Compute the limit e lim 5. Using the Talor polnomial found in the first part we have e lim 5 = lim = lim = 5. Note that this answer is eact and not an approimation, since all later terms in the Talor series have n for n > 7, and so will go to even when divided b. e lim 5 = 5

12 Math 6 / Final (April 9, 8) page 9. [7 points] Brontel Muskell claims his phone s screen was broken b a penn falling from the top of the New Toledo television tower and now he wants mone from the insurance compan. Giuseppe Li, the actuar in charge of the case, does not believe Brontel s claim. The differential equation modelling the velocit v of a falling object subject to air resistance is v = g rv () where v is downward velocit, in m/s, g is acceleration due to gravit, and r is a positive constant depending on the shape and size of the falling object. a. [ points] The positive equilibrium solution v = v T to () is called the terminal velocit. Find v T in terms of g and r. Is v T stable or unstable? Solving = g rv for v we find v = g r. This is a stable equilibrium. v T = g r v T is (circle one) stable unstable b. [4 points] Giuseppe speaks with the insurance compan s lead scientist, Tamm Toppel, who conducts some eperiments and concludes that a penn needs to be moving at least m/s to break the phone screen. Use the compan s estimates that r = 4 for a falling penn and g = to compute the value of v T. Then use this value and our answer about the stabilit of v T to help Dr. Toppel write a response to Brontel s claim. Terminal velocit: m/s This finding (circle one) supports contradicts Brontel s claim because... (briefl give reasoning below) Since v T = meters per second is a stable equilibrium one would epect a falling penn from a great height to be moving at ver nearl this speed. Since this is slower than the speed required for a penn to break Brontel s phone screen, it seems he s ling.

13 Math 6 / Final (April 9, 8) page. [8 points] Recentl Debra McQueath was thinking about all the great things she used to make at Print.juice b revolving regions around the -ais. Those were the good das, weren t the? a. [4 points] There was that one time she designed the Juice Titan TM formed b rotating the region in the first quadrant bounded b = π and = 4 + cos(4) around the -ais. The densit δ() of the plastic was a function of the distance from the center of the juicer, although Debra cannot quite remember what it was. Help Debra write an integral that represents the total mass of the Juice Titan TM. Your integral ma include the densit function δ(). (in) 5 4 = 4 + cos(4) (in) 4 Use the shell method. b. [4 points] π πδ()(4 + cos(4)) d On Debra s last da at Print.juice her team made her a commemorative hat containing a hollow chamber filled with juice b rotating the region bounded b =, = and = around the - ais. The juice-filled hat still sits on her kitchen table; she sometimes wonders how much juice is in the hat. Write an integral that represents the total volume of juice in the hat. Note: juice fills the solid formed b rotating the shaded region..5 (ft).5 (ft) Use the washer method. π ( ( ) (( )/) ) d

14 Math 6 / Final (April 9, 8) page 4. [6 points] The polnomial P () = (+e) +5(+e) is the third-degree Talor polnomial approimating the function g() for near e. Find the following values. Write ni if there is not enough information. g ( e) = g( e) = g ( e) = P (4) ( e) = g() = ni P () = e + 5e. [6 points] Match the differential equations to their corresponding slope fields. i. = + D iv. = ( ) E ii. = B v. = ( ) F iii. = A vi. = + C (A) (B) (C) (D) (E) (F)

15 Math 6 / Final (April 9, 8) page 5 Known Talor series (all around = ): sin() = n= ( ) n n+ (n + )! =! + + ( )n n+ + for all values of (n + )! ( ) n n cos() = (n)! n= =! + + ( )n n + for all values of (n)! e = n= n n! = + +! + + n + for all values of n! ( ) n+ n ln( + ) = n n= = + + ( )n+ n + for < n ( + ) p = + p + p(p )! + p(p )(p )! + for < < = n = n + for < < n=

Math 116 Final Exam April 19, 2018

Math 116 Final Exam April 19, 2018 On m honor, as a student, I have neither given nor received unauthorized aid on this academic work. Initials: Do not write in this area Math 6 Final Eam April 9, 208 Your Initials Onl: Your U-M ID # (not