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Unformatted text preview: MAT 286 Final Exam May 7, 2002 Signature: Instructions: Write the answers and show the main steps of your" work on this test sheet.
There are 12 questions on 14 pages (including this cover). Be sure you have all 14 pages (7
sheets) and that you do all 11 problems! The Final Exam is scored on a basis of 100 points
and will count 20% of your ﬁnal grade. PUT YOUR NAME ON THE TOP OF EACH SHEET  NOW! This Exam has 4 parts corresponding to the four sections of the course. You should spend
no more than 25 minutes on each part to be sure that you get to the easier problems in each
part. Where indicated, you must show your work to get full credit! DO NOT WRITE ON THE REST OF THIS COVER SHEET! Part I:
Part II:
Part III: Part IV: Total: Have a nice and safe Summer! JVIAT 286 Final Exam May 7, 2002 Part I Problem 1 (9 points) For each of the following questions, circle the correct answer. 12
a. Which of the following Reimann sums approximates / e2 do: using the left endpoints 10
of 10 subintervals?
1 g(810+§+610+§+ _ , . . __+eIO+§); é<elo+610+%+ , I . _ H+€10+§); 1 10+L 10+l 10+lQ IO—(e 10+e 10+ . . . . ..+e 10); $160310 +810+fa + . . _ . _.+610+I95); none of the above. df 7 x3 1 _r
b. If; = ~22 +3 — 3e and f(0) = e, then 1 1 1 r7 1 f($)=—:7‘7+ECII4—"2€_r, f<f13>=7+132+561+€
7 5 f($) = $7 + :3 + 8’1 + 8; none of the above. c. The graph of the function f(x) is given below; circle the label of the graph of f: f (t) dt
from the next page: MAT 286 Final Exam May 7, 2002 A. MAT 286 Final Exam iVIay 7, 2002 4 Problem 2 {10 point) a. (4 point) Make a rough sketch of the region between y = g and 5113—132 5 y: b. (2 point) Compute the r coordinates of the points of intersection. Show and label the
intersection points on th above graph. c. (2 point) Write down the integral that gives the area of this region. d. (2 point) Compute the area of this region. MAT 286 Final Exam May 7. 2002 5 Problem 3 (6 points) At time t=0, a sand pile contains 100 kilograms of sand. It is observed
that the rate of increase of the sand pile is given by: kilograms per day, where time is measured in days. a. (4 point) Compute the increase in the number of kilograms in the pile in the ﬁrst I
days [INCLUDE APPROPRIATE UNITS!]. b. (2 point} As time goes to inﬁnity, how many kilograms of sand will be in the pile? MAT 286 Final Exam lVIay 7. 2002 6 Part II Problem 4 (12 points) For each of the following questions, circle the correct answer. a. The daily sales of CDs in a record store are given, in dollars, by 3000t 365 — t '2, t
p(t) = ——3(T——)— + cos Where time, t, is measured in days. The average daily sales over one year (365 days) is:
$192,500 $18,250 $1,182,500 $182,500 $318,259. b. Suppose that the random variable X has an exponential distribution with a mean of 2.
Which of the following is most likely to occur: 0.25 S X S 0.50 1 S X S 1.5.
. . . _ 2x + 1 c. The mean of the probability denSity function f (I) = 6 on 0 S 3: S 2 is: 13 11 1 x/ﬁ — 1 2 9 3 2 . . . . . 2x + 1 d. The median of the probability densxty function f = 6 on 0 S :1: S 2 is: 13 11 1 v 13 — 1 2 9 3 2 . Problem 5 (3 points) The function f(3:) = c  a:  (6 — :c) is a probability density function
on the interval (0.3). Find the value of c. Show work. MAT 286 Final Exam May 7, 2002 7 Problem 6 (5 points) The length of yard sticks manufactured at a certrain plant can best
described by a normal density function. Suppose that the mean length of the yard sticks is
36. 05 inches and the standard deviation is 0.1inches. a. (3 points) A yard stick is randomly selected. What is the probability that the length of
’ the yard stick is between 35.95 and 36.05 inches? b. {2 points) A yard stick is randomly selected. What is the probability that the length of
the yard stick is greater than 36.05 inches? Problem 7 (5 points) Consider a continuous stream of income at the constant rate of $1,500
per year deposited directly into a savings account accruing interest at an annual rate 0f4.25%
compounded continuously. Find the future value. 35 years from now, of this savings account.
Be sure to write down any integrals you calculate and formulas that you use. MAT 286 Final Exam Afay 7, 2002 8 Part III Problem 8 (5 points) On the axes below, sketch the solution to the diﬁerential equation
,2 (y—4)(y+'2) 2 that passes through the origin. The window is: 7.5 S :c S 7.5, ~25 S y
y S 4.5. Problem 9 {4 points) Find the solution in the form ”y =” of the diﬁerential equation
312 + 3 29 I y that satiﬁes the initial condition y(1) = 3. Problem 10 (8 points) In part (a), circle the correct answer.
a. Which one of the slope ﬁelds on page 9 is the slope ﬁeld of y’ = $2 — if“)? A B C D b. On slope ﬁeld D (not necessarily the solution to the previous problem) on page 9, sketch
the solution through (2,0). NIAT 286 Final Exam May 7, 2002 All slope ﬁelds are drawn in the window
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C MAT 286 Final Exam NIay 7, 2002 10 Problem 11 (8 points)
a. A population grows at a rate proportional to the number of people present at time t, with constant of proportionality 0.95. (i) Write a differential equation for the number of people in the population, P, as a, function of time, t, in years. (ii) Compute the general solution to this diﬁerential equation. d3 . . . . . .
b. The velocity :1? of a projectile at time t (in minutes) is onefourth times the square root of the distance 3 ( in yards ) already traveled by the projectile. (i) Give the diﬂerential equation for the distance 3 the projectile has traveled at time t. (ii) Compute the general solution to this differential equation. MAT 286 Final Exam NIay 7, 2002 11 Part IV Problem 12 (12 points) The following differential equations represents the interaction be
tween two species: dz ' dy
2E — —3:c+2$y and a? — y+5:r:y. The sizes of the two populations (in thousands) are given by a: and y, respectively. a. (4 points) Describe how the two species interact. Speciﬁcally, how would each species
do in the absence of the other species? Are they helpful or harmful to each other? Explain/give reasons for answers. b. (4 points) When a: = 2 and y = 1, is
at increasing decreasing (circle one ); y increasing decreasing (circle one). c. (2 points ) Give the differential equation relating the two populations: (1. (4 points) Find the equilibrium point {other than (0,0)) for this system: as: y: MAT 286 Final Exam May 7, 2002 12 Problem 13 (13 points) Consider the SIR model of a flu epidemic given in the text with
the following modiﬁcation ' ’ d3 d1 E = —0.0015I, a; = 0.00131 — 1.451 and 5+ I + R = 4,000. a. (5 points) I = I(t) represents the number of school boys who are infected at time t
(measured in days) and S = S'(t) represents the number of school boys who, at time t,
are susceptable to getting the ﬂu but are not yet sick. What does R = R(t) represent? (ii) Explain the meaning of the equation (15 ——=—. I.
dt 00033 b. {4 points) The diﬁerential equation relating I and S MAT 286 Final Exam Afay 7, 2002 c. (4 points) Let T denote the threshold value for this model. Compute T. (ii) What is the practical interpe'rtation of the threshold of this model?s 13 MAT 286 F inal Exam Afay 7, 2002 USE THIS PAGE FOR SCRATCH 11 ...
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