Math 115 Final Exam December 14, 2017

Transcription

1 On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname): Insrucor Name: Secion #:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam.. This exam has pages including his cover. There are 0 problems. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem on which you are suck.. Do no separae he pages of his exam. If hey do become separaed, wrie your UMID (no name) on every page and poin his ou o your insrucor when you hand in he exam. 5. Noe ha he back of every page of he exam is blank, and, if needed, you may use his space for scrachwork. Clearly idenify any of his work ha you would like o have graded. 6. Please read he insrucions for each individual problem carefully. One of he skills being esed on his exam is your abiliy o inerpre mahemaical quesions, so insrucors will no answer quesions abou exam problems during he exam. 7. Show an appropriae amoun of work (including appropriae explanaion) for each problem, so ha graders can see no only your answer bu how you obained i. 8. The use of any neworked device while working on his exam is no permied. 9. You may use any one calculaor ha does no have an inerne or daa connecion excep a TI-9 (or oher calculaor wih a qwery keypad). However, you mus show work for any calculaion which we have learned how o do in his course. You are also allowed wo sides of a single 5 noecard. 0. For any graph or able ha you use o find an answer, be sure o skech he graph or wrie ou he enries of he able. In eiher case, include an explanaion of how you used he graph or able o find he answer.. Include unis in your answer where ha is appropriae.. Problems may ask for answers in exac form. Recall ha x = is a soluion in exac form o he equaion x =, bu x =.567 is no.. Turn off all cell phones, smarphones, and oher elecronic devices, and remove all headphones, earbuds, and smarwaches. Pu all of hese iems away.. You mus use he mehods learned in his course o solve all problems. Problem Poins Score Problem Poins Score Toal 00

2 Mah 5 / Final (December, 07) page. [ poins] The graph of a porion of a funcion y = h(x) is shown below. Noe ha he graph is linear where i appears o be linear, including on he inervals [7, 8] and [0, ). y = h(x) 5 x a. [ poins] A which of he following poins p is h(x) no coninuous a x = p? Circle all such values. p = p = p = p = p = 5 none of hese b. [ poins] For which of he following values a is lim h(x) = h(a)? Circle all such values. x a + a = a = a = a = 5 a = 6 none of hese For pars c. e., find he exac value of each of he expressions. If he value does no exis, wrie DNE. If here is no enough informaion, wrie NI. c. [ poins] Calculae he average value of h(x) on he inerval [, ]. d. [ poins] Suppose g(x) = h(h(x)). Calculae g (.5). Show all your compuaions o receive full credi. e. [ poins] Calculae h (x) dx. g (.5) = h (x) dx =

3 Mah 5 / Final (December, 07) page. [0 poins] Jane has a company ha produces a proein powder for an energy shake. The cos, in dollars, of producing m pounds of proein powder is given by he funcion (m + ) m < 6 C(m) = m m 0. The revenue, in dollars, of selling m pounds of proein powder is given by R(m) = 5m. a. [ poin] Wha is he price, in dollars, a which Jane sells each pound of he proein powder? b. [ poin] Wha is he fixed cos, in dollars, of producing Jane s proein powder? c. [ poins] Find all values of 6 m 0 for which Jane s profi is posiive. d. [ poins] Find all he values of 0 m 0 where he marginal cos is equal o he marginal revenue for he proein powder. Show all your work o jusify your answer. e. [ poins] Wha is he maximum profi ha Jane can make if she sells a mos 0 pounds of proein powder? Use calculus o find and jusify your answer, and make sure o provide enough evidence o fully jusify your answer.

4 Mah 5 / Final (December, 07) page. [8 poins] A group of biologiss is sudying he populaion of rou in a lake. Le k() be he rae a which he populaion of rou changes, in housands of rou per monh, monhs afer he biologiss sared heir sudy, and le P () be he populaion of rou, in housands, monhs afer he sudy begins. The graph of y = k() is shown below for 0 6. y = k() 5 6 a. [ poins] Fill in he numbers I. - V. in he blanks below o lis he quaniies in order from leas o greaes. I. The number zero. II. P () P () III. k() d IV. V. 5 5 k() d k(5) d b. [ poins] Suppose P () = 8.6. Use he graph o find a formula for L(), he linear approximaion for P () near =. L() = c. [ poin] Use L() o approximae he populaion of rou, in housands,.75 monhs afer he sudy sars.

5 Mah 5 / Final (December, 07) page 5. [0 poins] Gabe he mouse is swimming alone in a very large puddle of waer. He keeps rack of his swimming ime by logging his velociy a various poins in ime. Gabe sars a a poin on he edge of he puddle and swims in a sraigh line wih increasing speed. A able of Gabe s velociy V (), in fee per second, seconds afer he begins swimming is given below V () a. [ poins] Give a pracical inerpreaion of he inegral problem. Be sure o include unis. 5.5 V () d in he conex of he b. [ poins] Esimae 5.5 Make sure o wrie down all erms in your sum. V () d by using a righ-hand Riemann sum wih equal subdivisions. c. [ poin] Is your esimae from above an overesimae or an underesimae of he exac value of 5.5 V () d? Circle your answer. overesimae underesimae no enough informaion d. [ poins] Suppose Gabe wans o use a Riemann sum o calculae how far he raveled beween = and = 5.5, accurae o wihin 0.5 fee. How many imes would he have o measure his velociy in his inerval in order o achieve his accuracy? Jusify your answer.

6 Mah 5 / Final (December, 07) page 6 5. [ poins] A porion of he graphs of wo funcions y = s() and y = S() are shown below. Suppose ha S() is he coninuous aniderivaive of s() passing hrough he poin (0, ). Noe ha he graphs are linear anywhere hey appear o be linear, and ha on he inervals (, ) and (, 5), he graph of s() is a quarer circle. y = s() 5 y = S() 5 a. [ poins] Use he porions of he graphs o fill in he exac values of S() in he able below. S() b. [8 poins] On he axes above, skech he missing porions of boh s and S over he inerval < < 5. Make sure o pay aenion o: he values of S() from he able above where S is and is no differeniable where S and s are increasing/decreasing/consan he concaviy of he graph y = S().

7 Mah 5 / Final (December, 07) page 7 6. [ poins] Waer is being poured ino a large vase wih a circular base. Le V () be he volume of waer in he vase, in cubic inches, minues afer he waer sared being poured ino he vase. Le H be he deph of he waer in he vase, in inches, and le R be he radius of he surface of he waer, in inches. A formula for V in erms of R and H is given by V = πh(r + 8). a. [6 poins] Suppose ha he waer is being poured ino he vase a rae of 00 cubic inches per minue. When he deph of he waer is 5 inches, he radius of he surface of he waer is inches and he radius is increasing a a rae of. inches per minue. Find he rae a which he deph of he waer in he vase is increasing a ha ime. Show all your work carefully. R H = deph of waer b. [ poins] Esimae he insananeous rae of change of H when = if H Show your work and include unis. The problem coninues on he nex page

8 Mah 5 / Final (December, 07) page 8 c. [ poins] Recall ha R gives he radius of he surface of he waer, in inches, minues afer he waer sared being poured ino he vase. Suppose ha R is given by R = m() and m () = 0.7. Use hese facs o complee he following senence: Afer he waer has been poured ino he vase for hree minues, over he nex en seconds, he radius of he surface of he waer [7 poins] Le A and B be posiive consans and f(x) = A(x B), for x >. Noe ha x f (x) = A(x x + B) (x ) and f (x) = A(x 8x + B). (x ) 5 Find all values of A and B so ha f(x) has an inflecion poin a (8, ). Use calculus o jusify ha he poin (8, ) is an inflecion poin. If here are no such values, wrie none. A = B =

9 Mah 5 / Final (December, 07) page 9 8. [ poins] A ank conains 0 gallons of waer. Beginning a am, waer is pumped in and ou of he ank. Le A() be he rae, in gallons per minue, a which he waer is added ino he ank minues afer am. Similarly, le R() be he rae, in gallons per minue, a which he waer is removed from he ank minues afer am. The graphs of he funcions A() (solid line) and R() (dashed line) for 0 0 are shown below. y 0 A() 8 6 R() a. [ poins] For which values of is he oal amoun of waer in he ank decreasing? Esimae your answer. b. [ poin] A wha ime 0 0 does he ank have he leas amoun of waer? In pars c. and d., give a mahemaical expression ha may involve A(), R(), heir derivaives, and/or definie inegrals. c. [ poins] Find an expression for he oal amoun of waer, in gallons, ha was removed from he ank beween :0 am and :05 am. d. [ poins] Find an expression for he amoun of waer, in gallons, in he ank a :0 am. Problem coninues on he nex page

10 Mah 5 / Final (December, 07) page 0 For your convenience, he graphs of A() and R() for 0 0 are reprined below. y 0 A() 8 6 R() e. [ poins] Suppose ha here are 0 gallons of waer in he ank a :0 am. Which of he following graphs could be he graph of A() and R() for 0 0 in his case? Circle he one bes answer y R() A() y A() R() y A() R()

11 Mah 5 / Final (December, 07) page 9. [9 poins] For he following problems, choose he correc answer. If none of he choices are correc, circle none of hese. a. [ poins] Which of he following is an aniderivaive of he funcion /x + cos(x) for x > 0? Circle all correc answers. i. x sin(x) ii. ln(5x) + sin(x) iii. ln(x) + sin(x) 0 ( ) iv. ln x cos(x) v. x + sin(x) vi. none of hese b. [ poins] Suppose f(x) is a differeniable, inverible funcion defined on (, ) wih f (x) > 0 for all x. Suppose ha f() = 5 and f () =. Which of he following saemens mus be rue? Circle all correc answers. i. f (f (x)) = (f ) (x) iii. (f ) (x) = f (x) v. f () = 5 ii. f (x) is inverible iv. (f ) (5) = vi. none of hese c. [ poins] If p() is an even funcion ha is differeniable on (, ), which of he following mus be rue? Circle all correc answers. i. p() d = p() d iv. 8 6 p( + ) d = 5 p() d ii. p() d = 0. v. 5 5 p () d = 0 iii. Any aniderivaive of p() is an even funcion vi. none of hese d. [ poins] Suppose he limi definiion of he derivaive gives g c( +h) + a( + h) ( c a) ( ) = lim, h 0 h where a and c are nonzero consans. Which of he following could be he formula for g(x)? Circle he one bes answer. i. g(x) = cx + ax ii. g(x) = a(x ) + c x iii. g(x) = c a iv. g(x) = c(x+h) + ah v. g(x) = cx + ax vi. none of hese

12 Mah 5 / Final (December, 07) page 0. [8 poins] Consider he family of funcions g(x) = e x kx, where k is a posiive consan. a. [ poins] Show ha he poin ( ln(k), k k ln(k)) is he only criical poin of g(x) for all posiive k. Show all your work o receive full credi. b. [ poins] Show ha g(x) has a global minimum on (, ) a x = ln(k). Use calculus o jusify your answer. c. [ poins] Find all values of 0.5 k ha maximize he y-value of he global minimum of g(x) on (, ). Use calculus o jusify your answer. Wrie none if no such value exiss. k =