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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 24, 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)+38.964p1n 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 weightloss 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 Blackcapped vireo
for given years is:
(23 points) 1966 1971 1976 1981 1986 1991 1996
Blackcappedvireo 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 61176014“ 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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 Fall '08
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