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Unformatted text preview: FORM B
Math 1743 Fall 2007 Exam II 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 and in your text. You will have 75 minutes for the exam and point values are as indicated. 1. The table shows the amount of garbage (in thousands of tons) accumulating at the
Norman Dum, from 1995 throuh 2000. 11 oints a. Find the change in the amount of garbage between 1995 and 1999. 1434.2 — 85.7 = 1348.5 thousand tons
b. Find the average rate of change in the amount of garbage between 1996 and 1998. 1195.6 — 168.9 = 513.35 thousand tons per year
1998 — 1996 c. Find the percentage change in the amount of garbage between 1999 and 2000. 1522.1 — 1434.2
1434.2 100%=6.1% 2. Two competing banks are offering excellent interest rates on their CDs. The ﬁrst bank
is offering a CD with 5.8% interest compounded monthly, and the second bank is
offering a CD with 5.5% interest compounded continuously. Determine which bank’s
CD is the better investment by answering the following (round your answers to the
hundredths place as percentages): (8 points) a. Find the effective rate (APY) for the ﬁrst bank’s CD 12t
A=P[1+9‘—1025—8] =P(1.0596‘); APY= 5.96% b. Find the effective rate (APY) for the second bank’s CD A = Pe°055t = P(1.0565‘); APY = 5.65% 4. Use the numerical method to ﬁnd the slope of the line tangent to the graph of
f (x) : 6x1/4x — 3 at x = 7. Organize your work in a table (or pair of tables). Although most of the work is done on the calculator, be sure it is clear what values you are
plugging in, and what you are plugging them into. Use the way the numerical method
was presented in class as a guide for your solution here. (11 points) f (7) — f (other endpt)
7 — other endpt 1999999999 _m 6.999 9.9999
46.62678 46.78272 46.79827 46.79983 secant slope = 999 999.999 7.999 7.9991
46.97238 46.81728 46.80173 46.80017 tangent slope = 46.8 5. a. State the formal (limit) deﬁnition of the derivative of a function. (5 points) f,(x):1im f(x + h) — f(x) h—90 h b. If c(x) = —1.15x2 + 3.94x + 34.44 gives European auto sales (million cars), x years since 1990, use the algebraic method (fourstep method) to show that
c’(x) = 2.3x + 3.94 (13 points) c(x + h) '2 —1.15(x + h)2 + 3.94(x + h) + 34.44
= —1.15X2 — 2.3Xh —1.15h2 + 3.94x +3.94h +34.44 , . —1.15x2 —2.3xh —1.15h2 +3.94x +3.94h +34.44—(—1.15x2 +3.94x +3444)
c(x)=ggng————h——_ ~ 1.15x2 —2.3xh —1.15h2 +3.94x +3.94h+34.44+1.15x2 —3.94x —34.44 = lim
h—>0 h
. — 2.3 — . 2 . . — . ~ . . . =11... M 2m 11W 2 11m(_ 23,. 4.15.. +3.94)
h—>O h h—>0 h h—>0 = 2.3x + 3.94 c. Write an interpretation of c'(3)= 2.96 (5 points) In 1993, European auto sales were decreasing by 2.96 million cars per year. 6. Match the given graph on the leﬂ with its slope graph on the right by writing the appropriate letter in the spaces provided. (12 points)
(a)
b
(b)
d .
(0)
e .
(d) i : i: (6.) ¥% 7. Use rules and formulas to ﬁnd the derivatives of the following ﬁmctions. You do not
need to simplify your answer. (10 points) a. y = 9.06(4.28") y' = 9.06  In 4.28  (428") b. t =15e‘—t24+§=15et—t24+6t_1
g t g'(t) = 15et — 24t23 + 6  —1t‘2 8. In an experiment designed to study memory, a subject was given a list of words to
memorize and asked to recognize these words in a reading sample given after various
elapsed times (see data below). (17 points) a.
Wordsrecognized 73.0 61.7 58.3 55.7 50.3 46.7 40.3
(percentage) a. Find a complete logarithmic model for the data (do not align the input). f (x) = 81.976 — 6.648 In x percentage of words recognized after x minutes b. Use rules/formulas to ﬁnd the derivative of the model in part a, writing it as a complete
rate of change model. f ’(x) = 0 — 6.648  1 percentage of words recognized per minute after x minutes X c. Evaluate your rate of change model at an input of 46 and write a complete
interpretation of your answer. f ’(46) = —0. 14; After 46 minutes, the percentage of words recognized is decreasing by 0.14 percentage points per minute ...
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 Fall '08
 DavidsonRossier
 Derivative, European auto sales

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