[PDF] Math 115 Final Exam April 24 2017 - University of Michigan

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Math 115 | Final Exam | April 24, 2017

EXAM SOLUTIONS1.Do not open this exam until you are told to do so.

2.Do not write your name anywhere on this exam.

3. This exam has 11 pages including this co ver.There are 10 p roblems. Note that the problems are not of equal diculty, so you may want to skip over and return to a problem on which you are stuck. 4. Do not separate the pages of this exam. If they do b ecomeseparated, write y ourUMID (not name) on every page and point this out to your instructor when you hand in the exam. 5. Note that th ebac kof ev erypage of the exam is blank, and, if needed, y ouma yu sethis space for scratchwork. Clearly identify any of this work that you would like to have graded. 6. Please read the instructions for eac hindividu alproblem carefully .One of t heskills b eing tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 7. Sho wan appropriate amoun tof w ork( includingapprop riateexplanation) for eac hproblem, so that graders can see not only your answer but how you obtained it. 8. The use of an ynet workeddevice while w orkingon this exam is not permitted. 9. Y ouma yuse an yone calculator that do esnot ha vean in ternetor data connection except a TI-92 (or other calculator with a \qwerty" keypad). However, you must show work for any calculation which we have learned how to do in this course.

You are also allowed two sides of a single 3

00500notecard.

10. F oran ygraph or t ablethat y ouuse to nd an answ er,b esure to sk etchthe graph or wr ite out the entries of the table. In either case, include an explanation of how you used the graph or table to nd the answer. 11. Include un itsin y ouransw erwhere th atis appropriate. 12. Problems ma yask for answ ersin exact form. Recall thatx=p2 is a solution in exact form to the equationx2= 2, butx= 1:41421356237 is not.

13.Turn o all cell phones, smartphones, and other electronic devices, and remove all

headphones, earbuds, and smartwatches. Put all of these items away. 14. Y oum ustuse the metho dslearned in this cour seto solv eall problems. ProblemPointsScore 110
29
312
45

59ProblemPointsScore

611
79
816
99
1010

Total100

Math 115 / Final (April 24, 2017)page 21.[10 p oints]The gr aphof f(x) shown below consists of lines and semicircles.654321123456789

211234y=f(x)xy

For the following problems, you do not need to show work. If there is not enough information, write\nei". a.[2 p oints]F orwhic hv aluesof 6< x <9 is the functionf(x) discontinuous?Solution:x=4 andx= 2.

b.[2 p oints]F orwhic hv aluesof 0 < x <9 doesf(x) appear to not be dierentiable?Solution:x= 2, 6 and 7.

c.[2 p oints]Find lim h!0f(4 +h)f(4).Solution:limh!0f(4 +h)f(4) = 3. d.[2 p oints]Find lim x!1f2xx+ 1 .Solution:limx!1f2xx+ 1 = 4

e.[2 p oints]Let g(x) = ln(4 +f(x)). Findg0(6:5).Solution:g0(x) =f0(x)4 +f(x), theng0(6:5) =f0(6:5)4 +f(6:5)=25

Math 115 / Final (April 24, 2017)page 32.[9 p oints]The gr aphof f(x) shown below consists of lines and semicircles.654321123456789

211234y=f(x)xy

Use the graph above to calculate the answers to the following questions. Give your answers as exact values. You do not need to show work. If any of the answers can't be found with the information given, write\nei". a.[3 p oints]Find the a veragev alueof f(x) on [4;2].Solution: 16 Z 2

4f(x)dx=16

(12 (1:5)2+12 (4)(3)) =16 (98 + 6) =948 + 1 b.[2 p oints]Find the v alueof Z 9 4 jf(z)jdz.Solution: Z 9 4 jf(z)jdz=Z 6 4 f(z)dz+Z 9 6 f(z)dz=14 (2)2+12 (3 + 2)(2) = 5 + c.[2 p oints]Find the v alueof 4 < T9 such thatZ T 4 f(x)dx= 0.Solution:We need to nd a value ofTfor which Z T 4 f(x)dx=Z 6 4 f(x)dx+Z T 6 f(x)dx= 0:

From the graph

Z 6 4 f(x)dx=andZ T 6 f(x)dx=12 ((T6) + (T7))(2) = 2T13.

Solving forTon 2T13 =, we getT=+ 132

d.[2 p oints]Find the v alueof Z 7

8f(x+ 2) + 1dx:Solution:

Z 7

8f(x+ 2) + 1dx=Z

5

6f(x) + 1dx=Z

5

6f(x)dx+ 1 = 1:5 + 1 = 2:5:

Math 115 / Final (April 24, 2017)page 43.[12 p oints]Virgil, Duncan, Jasp erand Zander are all w atchinga to ywind-up mouse mo ve

across the oor. Their person places the toy on the oor 2:3 meters away from Virgil, and it moves in a straight line directly away from Virgil at a strictly decreasing velocity. Below are some values ofv(t), the velocity of the toy mouse, in meters per second,tseconds after the person places it on the oor, where a positive velocity corresponds to the toy moving away from Virgil.t00.250.50.7511.251.51.752 v(t)3.192.391.861.431.110.860.540.420.11 a.[4 p oints]Estimate the v alueof Z 1:75

0:25v(t)dtusing a left-hand Riemann sum with t= 0:5.

Be sure to write down all the terms in your sum. Is your answer an over- or underestimate?Solution:Left hand sum=0:5(2:39 + 1:43 + 0:86) = 2:34.

This is (circle one):

an overestimatean underestimate not enough information b.[3 p oints]Ho woften should the v aluesof v(t) be measured in order to nd upper and lower estimates forZ 1:75

0:25v(t)dtthat are within 0:1 m of the actual value?Solution:We can estimate the size of tusing the formula

jv(1:75)v(0:25)jt=j0:422:39jt= 1:97t0:1:

This yields t0:10:970:0507 seconds.

c.[2 p oints]Find the v alueof Z 1:25

0:5v0(t)dt.Solution:Using the Fundamental Theorem of CalculusZ

1:25

0:5v0(t)dt=v(1:25)

v(0:5) = 0:861:86 =1. d.[3 p oints]Whic hof the follo wingrepresen tsho wm uchthe distance from the to ymouse to Virgil increases during the 2 ndsecond after it has been placed on the oor? Circle the onebest answer. i. 2 :3Z 2 1 v(t)dt ii. 2 :3Z 2 1 v0(t)dt iii. Z 2 1 v(t)dtZ 1 0 v(t)dtiv. Z 2 1 v(t)dtv. Z 2 1 v0(t)dt vi.v(2)v(1)

Math 115 / Final (April 24, 2017)page 54.[5 p oints]Consider the function Z(w) = arctan(kw)(w+1) wherekis a nonzero constant.

Use the limit denition of the derivative to write an explicit expression forZ0(2):Your answer should not involve the letterZ. Do not attempt to evaluate or simplify the limit.Please write your nal answer in the answer box provided below.Solution: Answer:Z0(2) =limh!0arctan(k(2 +h))((2 +h) + 1)(arctan(2k) + 1)h

5.[9 p oints]A cylindrical bar of radius Rand lengthL(both in meters) is put into an oven. As

the bar gains temperature, its radius decreases at a constant rate of 0:05 meters per hour and its length increases at a constant rate of 0:12 meters per hour. Fifteen minutes after the bar was put into the oven, its radius and length are 0:4 and 3 meters respectively. At what rate is

the volume of the bar changing at that point?Be sure to include units.Solution:The volume of the cylindrical bar isV=R2L. Dierentiating with respect tot,

we obtaindVdt

2RdRdt

L+R2dLdt

You are given that after 15 minutes:R= 0:4,L= 3,dRdt =:05 anddLdt = 0:12. Then dVdt =2(0:4)(0:05)(3) + (0:4)2(0:12)=0:3166:

Answer:The volume of the bar is (circle one):

increasing decreasingnot enough information at a rate of 0:3166 m3per hour.

Math 115 / Final (April 24, 2017)page 66.[11 p oints]Supp oseh(x) is a function andH(x) is an antiderivative ofh(x) such thatH(x) is

dened and continuous on the entire interval3x4. Portions of the graphs ofh(x) and

H(x) are shown below.3211234

21123y=h(x)xy

3211234

21123y=H(x)xya.[4 p oints]Use the p ortions

of the graphs shown to ll in the exactvalues of

H(x) in the table below.x32124

H(x)1:00:52:01:03:0b.[7 p oints]Use the axe sab oveto sk etchthe missing p ortionsof the graphs of b othhand

Hover the interval3x4.

Be sure that you pay close attention to each of the following: the values ofH(x) you found in part (a) above whereHis/is not dierentiablewhereHandhare increasing, decreas- ing, or constant the concavity of the graph ofy=H(x) Math 115 / Final (April 24, 2017)page 77.[9 p oints]Consider the family of functions f(x) =ax2ebx whereaandbare positive constants. Note that f

0(x) =ax(2bx)ebx:

a.[4 p oints]Find the exact v aluesof aandbso thatf(x) has a critical point at (4;e2).Solution:Since (4;e2) is a critical point off(x), thenf0(4) = 0 or 0 = 4a(24b)e4b.

From this equation we get that 24b= 0 (sincea;e4b>0). Thenb= 0:5. We also know that the point (4;e2) is in the graph ofy=f(x), thene2= 16ae4b. Plugging the value ofb, we gete2= 16ae2. This yields 1 = 16a, soa=116 b.[5 p oints]Using y ourv aluesof aandbfrom the previous part, nd and classify the local extrema off(x). Use calculus to nd and justify your answers, and be sure to show enough

evidence that you have found them all. For each answer blank, writenoneif appropriate.Solution:With the values found abovef0(x) =116

x(20:5x)e0:5x. The critical points are found by solving f

0(x) =116

x(20:5x)e0:5x= 0 In this case, we havex= 0 or 20:5x= 0 (since:e0:5x>0). Hence the critical points arex= 0 andx= 4. To classify them we use the rst derivative test: f0(1) =116 (1)(20:5(1))e0:5(1)=0:257 f0(1) =116 (20:5)e0:5= 0:0568 f0(5) =116 (5)(20:5(5))e0:5(5)=0:0128 OR f0(1) = ()(+)(+) = f0(1) = (+)(+)(+) = + f0(5) = (+)()(+) =

Answer:Local max(es) atx= 4 Local min(s) atx= 0

Math 115 / Final (April 24, 2017)page 88.[16 p oints]An apple farmer start sharv estingap pleson her orc hard.They start collecting

apples at 6 am. Leta(t) be the total amount of apples (in hundreds of pounds) that have been harvestthours after 6 am. Some of the values of the invertible functiona(t), its derivative a

0(t) and an antiderivative functionb(t) are shown below.

t34.567.5910.512 a(t)1.5234.566.59 t36912 a

0(t)0.41.20.51.8

t36912 b(t)612.525.543

a.[2 p oints]Use the tables to estimate the v alueof a00(9). Show your work.Solution:Possible approximations:

a

00(9)1:80:51290:433,a00(9)0:51:296 :233 ora00(9)0:433:2332

= 0:1

b.[3 p oints]Find the v alueof ( a1)0(6). What are its units in the context of this problem?Solution:(a1)0(6) =1a

0(a1(6))=1a

0(9)=10:5= 2 hours per hundreds of pounds of

apples. c.[3 p oints]Use the fact that a0(10) = 3:2 to complete the sentence below, including units, to give a practical interpretation in the context of this problem that can be understood by someone who knows no calculus.

The amount of apples harvested between 4 pm and 4:30 pm ...Solution:increases by approximately 160 pounds of apples.

d.[3 p oints]Find the tangen tline appro ximationS(t) ofb(t) neart= 3.Solution:S(t) =b(3) +b0(3)(t3) = 6 + 1:5(t3).

e.[2 p oints]Use y ouransw erin dto approximate the value ofb(2).Solution:b(2)S(2) = 61:5 = 4:5. f.[1 p oint]Is y ouransw erin ean overestimate or an underestimate? Circle your answer.Solution: overestimate underestimatenot enough info

g.[2 p oints]Let m(t) be an antiderivative ofa(t) satisfyingm(9) =1. Findm(3).Solution:We know that two antiderivativesb(t) andm(t) ofa(t) satisfym(t) =b(t)+C.

Then usingt= 9 we get thatC=m(9)b(9) =125:5 =26:5. Hencem(3) = b(3)26:5 = 626:5 =20:5.

Math 115 / Final (April 24, 2017)page 99.[9 p oints]A Math 115 c oordinatoris trying to create functions with certain p ropertiesin order

to test students' understanding of various calculus concepts. a.[5 p oints]He w antsa function f(x) of the form f(x) =( ax2+ax+bexforx <0 a+ 2cos(x) forx0 whereaandbare constants. Find all value(s) ofaandbfor whichf(x) be dierentiable atx= 0. Show enough work

to justify your answer.Solution:In order forf(x) to be dierentiable atx= 0,f(x) has to be continuous.

Then limx!0ax2+ax+bex=b= limx!0+a+ 2cos(x) =a+ 2 =f(0):

Thenb=a+ 2. Iff(x) is dierentiable then

lim h!0f(0 +h)f(0)h = 2a(0) +a+be0=a+b: lim h!0+f(0 +h)f(0)h =2sin(0) = 0: Hencea+b= 0. Using both equation we obtaina=1 andb= 1. b.[4 p oints]The co ordinatoralso w antsa function g(x) =cxex, wherecis a constant, so thatg(x) has at least one critical point. What condition(s) oncwill make this true?

Find thex-values of all critical points in this case. Your answer may be in terms ofc.Solution:The functiong(x) has critical points at values ofxthat satisfy

g

0(x) =cex= 0:

Then the only potential critical point isx= ln(c). This critical point exists only ifc >0.

Math 115/ Final(April 24,2017) page10 10. [10p oints]TheHappyHiv esBee Farmsellshoney .Thegraph belo wshowsmarginal reven ue

MR(dashed) andmarginal costMC(solid), indollars per pound,wherehis then umberof poundsof honey. 102030405060708090100110120130140150160170180123456789101112y=MR y=MC h(pounds) y($/pound) a . [7p oints]Usethegraphto estimatethe answers tothe following questions.Y oudo not need tosho wwork.If ananswercan't be foundwith theinformationgiven, write\nei". i) Forwhatv alue(s)of hin thein terval[0;180] isthe costfunction Cminimized?

Answer:h= 0.

ii) Forwhatv alue(s)of hin thein terval[0;180] isMCminimized?

Answer:h= 100.

iii) Forwhatv alue(s)of hin thein terval[0;180] isprot maximized?

Answer:h= 130.

iv) Whatar ethexedc ostsof thef arm?

Answer:NEI

v) Forwhatv aluesof hin thein terval[0;180] isthe protfunction concav eup?

Answer:(0;100)S(160;180)

b . [3p oints]Thefarmcurrently sells20 pounds ofhoneybutis thinkingof increasingto 80 poundsof honey. Solution:The totalc hangeinprotfrom selling20 to80 pounds ofhoney isgiv enb yZ0 20

MR(q)MC(q)dq=12

(2 +4)(20) =60.

Will thisincrease ordecrease prot?(Circle one.)

increasedecrease

By approximatelyhowmuc hwilltheprot change?

60 dollars.

quotesdbs_dbs19.pdfusesText_25

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