Exam IIIA - Math 1743 Fall 2007 Exam III FORM A Name: Key...

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Unformatted text preview: Math 1743 Fall 2007 Exam III FORM A Name: Key Section #: i.d.#: Instructor: Read all questions carefully and answer them completely. Show appropriate work to receive maximum credit. In general, if you can label something with units= you should label it with units. When computing and writing numerical answers, use rounding guidelines discussed in class unless the problem specifies otherwise. Every model you write (and that includes rate of change models) should be complete. You will have 75 minutes for the exam and point values are as indicated. 1. Suppose that the population of a certain collection of Brazilian ants is given by A(t) = (t + 100)- In (t + 2) ants after t days. (15 points) a. Find the complete rate of change model for the population of the ants, A'(t). A(t):(1)-1n(t+2)+(t+100).; =1n(t+2)+ t+100 ants er da , aftert da s. t+2 t+2 p y y b. What was the ant population after 15 days? A(15) = 325.82; 325 or 326 ants c. Find and interpret A'(1 5) . A’(15)= 9.6; After fifteen days, the collection of ants was growing by 9.6 ants per day. On problems 2-4, you do n_oz‘ need to simplifil your answers. 2. Given g(x) = 123" + (3 — x)”, find g'(x). (8 points) g’(x) = (11112)123x -3 +12(3 —x)“ -—1 3. Given S(p) = 38.964p(0.858P), find S’(p). (8 points) S'(p) = 38.964(0.858P)+38.964p-1n 0.858(0.858P) 43 , find (11 points) 4. Gi = * VCn V(x) ex - (1n x)5 dx V(x) = 43e"x - (In x)_5 v'(x) = 43e‘X -—1 - (In x)‘5 + 43e‘x -—5 - (In x)_6 x 01' v(x) = 43(e" -(1n X)5)_1 X v'(x) = 43- —1(e" -(ln x)5)_2 [ex -(ln x)5 + e" .5(1n x)4 5. The weekly weight loss of a person on a weight-loss program is given by w(t) = 1.62te'°'38t pounds afier t weeks on the program. (16 points) In parts a & b, give both input and output, round coordinates to the hundredths place, and please label them with units. a. Find the absolute maximum weekly weight loss between weeks one and seven on the program. t = 2.63 weeks w = 1.57 pounds b. Find the absolute minimum weekly weight loss between weeks one and seven on the program. t = 7 weeks w = 0.79 pounds c. Based on your answers to the questions above, fill in the following: On the interval [1, 7], w'(t) is positive between t = l and t = 2.63 . On the interval [1, 7], w’(t) is negative between t = 2.63 and t = 7 6. A soda bottling company bottles 20,000 cases of lime soda each year. The cost to set up the machine for a production run is $1400, and the cost to store a case for one year is $18. (19 points) a. If x is the number of cases in production run, write an expression representing the number of production runs necessary to produce 20,000 cases. 20000 X b. Write an expression for the total set up cost. [20000](1400) X 0. Assume that the average number of cases stored throughout the year is half the number of cases in production run. Write an expression for the total storage cost for one year, using x as the number of cases in each production run. on (1. Use your responses to parts b & c to write a complete model for the combined set up and storage costs for one year. 200 . C(X) = [ 00)(1400)+[§](18) = M + 9x dollars, where x cases are In each x x production run. e. What size production run minimizes the total yearly cost? Show some work (i.e. calculus) to support your answer. Round to the nearest whole number. C(x) = 28000000x‘1 + 9x ; C'(x) = —28000000x_2 + 9 =0 x = 1763.83; 1763 or 1764 cases f. How many production runs will minimize total yearly production cost? Round to the nearest tenth, assuming they can make a partial run to end up with 20,000 cases. 11.3 runs 7. An international wildlife agency determined that the population of Black-capped vireo for given years is: (23 points) 1966 1971 1976 1981 1986 1991 1996 Black-cappedvireo 35.1 33.7 29.8 17.4 11.2 (thousand birds) a. Find a complete logistic model for this data. Align the data so that the year 1966 corresponds to an input of zero. ' 40.748 f x =— () 1+0.117e°'146" thousand birds, x years afier 1966 = 40.748(1+ 0.1176014“)—l b. Find the complete model for rate of change of the number of population of Black- capped vireo. f’(x) = 40.748-—1(1+ 0.1176,°-““"‘)‘2 611760-14“ -O.146 = —O.696e°'146"(1+ 0.117e‘"“‘6")_2 thousand birds per year, x years after 1966 c. Find the inflection point for the model you generated in part a. Round both coordinates to the hundredths place and label them with appropriate units. x = 14.67 years afier 1966; f = 20.37 thousand birds ...
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Exam IIIA - Math 1743 Fall 2007 Exam III FORM A Name: Key...

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