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Unformatted text preview: Math 31L.01  Test 3 READ INSTRUCTIONS BELOW BEFORE STARTING THE EXAM 0 Do not Open this test booklet until you are directed to do so. 0 You will have 105 minutes to complete the exam. If you ﬁnish early, go back and check your
work. 0 Throughout the exam, show your work so that your reasoning is clear. Otherwise no credit
will be given. Circle your answers. 0 Do not spend too much time on any one problem. Read them all through ﬁrst and attack
them in an order that allows you to make the most progress. I have adhered to the Duke Community Standard in completing this assignment.
Narne: Signed: 1. (6 pts) In each part below check the answer which most accurately characterizes the deﬁni
tion of the given expression. ' wifﬂaa
__ It’s a limiting valueof sums.
__ It‘s the family of antiderivatives of f.
__ It’s the limit of a diiference quotient.
__ It’s a diifereuce in values of an antiderivative of f.
__ It’s the family of derivatives of f.
___.. It’s the average value of f. o)ﬁﬂaa
_ It’s the average value of f over [a,b].
__ It’s a family of derivatives.
__ It’s F(b) — F(a), where F is an antiderivative of f.
_ It’s a family of antiderivatives.
__ It’s the limiting value of Left and Right Hand Riemann Sums.
__ It’s the slope of any antiderivative of f. 2. (6 points) The graph of a derivative, f’(:r:), is shown below. Given that f (2) = 13, ﬁnd the pre
cise value of f (6). If you use a major theorem, state the name of the theorem and write an
7 expression (or equation) to show how you’re using the theorem. '
—_—..__..____,________ 3. (4 points) Assume that (2f — 5) dzt: 50. Find the value of f; f (as) data Be sure to show
all of your work. ' 4. (10 points) Let F(m) = ffaﬁh — 5)~./t4 + 1 dt, for a: 2 —3. (a) Find F’(:1:). Name any major theorems that you use. (b) Does F have anyr local maxima or minima‘? If so, where and what type? (6) Graph the funtion (t — 5)\/t4 + 1, and use this graph to determine whether has a
global maximum and/or a. global minimum for x 2 m3. ' 5. (8 points) Consider the differential equation a! = t —— 23; with the initial condition y(0) = 2.
Using Euler’s method with a step size of At: .2, approximate y(.6). 6. (8 points) Solve the initial value problem g: 2 y‘2 secE(a:), given that y(0) = 0.01. Your
answer should express 3; in terms of 3:. 7. (10 pts) Suppose that the function T(t) = 70 + 1108—009“ gives the temperature of a cup of .
coffee t minutes after it was poured into a. ceramic mug. (a) What happens to the temperature of the coffee as t gets large? (b) Note that by the end of an hour (t 2 60), the coﬁee has cooled almost to room temper—
ature. Compute the average temperature of the coffee over that hour. (c) The average of T(0) and T(60) is about 125 degrees. Explain why we should have ex
pected your answer to (b) to be higher/lower (choose one) than 125. ' 1000 ) none of these *0» b
1000
m4 999
0 >Zf(a+k k
f (:6) b—a
) 50 ﬁx) I 6" 50 (as) which passes through the point (0, 3) and whose derivative
b — a 0 Zf(a+k k: the following ineqaulity be true?
1‘ (1:) = \/:E Circle ONE answer below: res) = . _2
138”. 8. (4 points) Deﬁne a function F
I 9. (4 points) Let [151, b] be an interval on the x axis. For which one of the functions below would 10. (6 points) Circle the differential equation below that matches the slope ﬁeld: Eifffffﬁrtﬁrf 7/ a N. \N \V \\v\\.¢ Ll!
slit??? 9, a 35%. r,
Iffffflajlil; « M \Axxxuxkrtt
llwlfsTIuEIJJraErfiif :/ a_ __w x». \NxNXHKK xv a x / a N a a / .. .5. Hz a_ a fur] (If...de ffqztreuif fail.le Tr eroaJﬁffi /,”//x’//// f,’////,’f// 11. (10 points) This problem is based on the density lab1 but this time our mosquitoes are located in a rectangular region of the :1: — y plane as shown below. (The axes are marked off in miles.)
The density of mosquitoes (in thousands of mosquitoes per square mile) at any point depends
only on the distance to the :1: axis and is given by m; i.e., if (22,9!) is any point in the
rectangle, then the mosquito density at that point is 4 thousand mosquitoes per square mile. How many mosquitoes are in the rectangle? (In your response, you should Show clearly _ 1+2;
the method you are using to solve this problem.) 12. (4 pts) Suppose that p(h) gives the climb rate of an airplane, expressed in feet per minute and where h, is the altitude expressed in feet. What is the meaning of folo'ooo ﬂ dh? What are the units attached to the value of this integral? Answer: I 13. (10 points) Students in a physics class at Duke built an experimental toy rocket. On the rocket
the students put a transmitter which would transmit its velocity every 2 seconds. After the
rocket was launched, it transmitted the following data for the ﬁrst 12 seconds: ﬂﬂﬂ . men (a) We assume that the rocket’s velocity increased throughout this time period. Fill in the
blanks below with the best statement you can make. Below the sentence show the justi—
ﬁcation for your answers. The rocket rose to a height of at least __ feet and at most feet. (b) How often would the transmitter have had to transmit the velocity, if the students would
like to have estimated the'height accurate to within 10 feet? 14. (10 points) Determine whether the following statements are true or false. You do not have to
justify your choice. (a) If f (9:) is a continuous function, then f(:r) d2: = —— f: f(:r:) 019:. (b) if f is a continuous function, then the midpoint sum is equal to the average of the
LHS and RHS. (c) It is always possible to construct an antiderivative for a continuous function.
((1) If f is a continuous even function, then If“ f (x) dcr: = 0. (e) You can only construct a slopeﬁeld for a separable differential equation. ...
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 Spring '08
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 Calculus, Derivative, Continuous function

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