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Unformatted text preview: Math 1743 Fall 2007 Exam III FORMB Name: Section #: i.d.#: Instructor: Read all questions careﬁJlly 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 + 200) ln(t + 5) ants aﬁer 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 +5)+(t+2oo)._1_: 1n(t+5)+ t+200 ants er da , aﬁertda s.
t+5 t+5 p y y b. What was the ant population after 10 days? A(10)= 568.69; 568 or 569 ants c. Find and interpret A’(10). A'(10) = 16.7; After ten days, the collection of ants was growing by 16.7 ants per day. On problems 24, you do n_ot need to simplifv your answers.
2. Given h(x) : 312x + (12 — x)3, ﬁnd h'(x). (8 points) h’(x) = (1n 3)312" 12 +3(12— x)2 —1 3. Given DQ)) =13.407p(1.036P), ﬁnd D’(p). (8 points) D'(p)=18.407(1)(1.036P)+18.407p1n1.036(1.036p) 4. Given w(x) = x—59—7, ﬁnd (11 points)
e (ln x) dx w(x) = 59e‘x (1n x)—7 w'(x) = 59e'“  —1'(1n X)_7 + 59e—x "7 ' (1“ X)_8 X 01' w(x) = 59(ex (1n x)7)_1 ' _ _ x. x74 x. 7 x. X6;
w(x)—59 1(e (In [e (lnx) +e 7(1n ) x) 5. The weekly weight loss of a person on a weightloss program is given by
w(t) = 1.87te‘0'23t pounds after 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 two and six on the
program. t = 4.35 weeks w = 2.99 pounds b. Find the absolute minimum weekly weight loss between weeks two and six on the
program. t = 2 weeks w = 2.36 pounds c. Based on your answers to the questions above, ﬁll in the following: On the interval [2, 6], w'(t) is positive between t = 2 and t = 4.35 . On the interval [2, 6], w’(t) is negative between t = 4.35 and t = 6 . 6. A soda bottling company bottles 10,000 cases of lemon soda each year. The cost to
set up the machine for a production run is $1200, and the cost to store a case for one year
is $15. (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 10,000 cases. 10000
x b. Write an expression for the total set up cost. [10000](1200) 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. [305) (1. Use your responses to parts b & c to write a complete model for the combined set up
and storage costs for one year. C(X) = [10000](1200)+[§2—](1 5) = w + 7.5x 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) =12000000x‘1 + 7.5x ; C'(x) = —12000000x_2 + 7.5 =0 x = 1264.91; 1264 or 1265 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 10,000 cases. 7.9 runs 7. An international wildlife agency determined that the population of Neosho madtom for
given years is:
(23 points) 1963 1968 1973 1978 1983 1988 1993
Neosho madtom 105.7 100.2 85.9 59.4 33.8 24.7 18.9
(thousand ﬁsh) * a. Find a complete logistic model for this data. Align the data so that the year 1963
corresponds to an input of zero. 121.021 1‘ x =—————
U 1+O.114e°'”4" thousand ﬁsh, x years after 1963 =121.021(1+ 011461144")—1 b. Find the complete model for rate of change of the number of population of Neosho
madtom. f'(x)=121.021—1(1+ 0.114e°“‘4" )‘2  0.114e0‘44x  0.144 = —1.987e°‘144x (1 + 0.114e0'144")_2 thousand ﬁsh per year, x years after 1963 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 = 15.09 years after 1963; f = 60.51 thousand ﬁsh ...
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 Fall '08
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