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MATH 116 — SECOND MIDTERM EXAM
Fall 2004
NAME: ID NUMBER:
INSTRUCTOR: SECTION NO:
- Do not open this exam until you are told to begin.
- This exam has 9 pages including this cover. There are 8 questions.
- Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you turn in the exam.
- Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam.
- Show an appropriate amount of work for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate.
- You may use your calculator. You are also allowed 2 sides of a 3 by 5 notecard.
- If you use graphs or tables to obtain an answer, be certain to provide an explanation and sketch of the graph to make clear how you arrived at your solution.
- Please turn off all cell phones and devices whose sounds might disturb your classmates. Please remove all headphones.
PROBLEM POINTS SCORE
TOTAL 100
- (16 points) For each of the three questions, fill in the blank(s) using the appropriate suggested answer(s). No explanation is required.
(a) The polynomial P 2 (x) = 1 + 3(x − a) − 2(x − a)^2 is the second degree Taylor polynomial ap- proximating the function f for x near a. The graph of f is given in the figure. Which of the points A, B, C, or D on the x-axis has a as its x-coordinate?
A B
C
D
x
f (x)
ANSWER :.
(b) Three of the tests for deciding the convergence or divergence of an infinite series are:
A. integral test, B. comparison test, C. ratio test.
Using each of these letters A, B, C exactly once, fill in the blank by each of the following infinite series with the label of the most appropriate test to use in deciding whether the series converges or diverges.
∑^ ∞
n=
(n!)^2 (2n)!
∑^ ∞
n=
n sin^2 n 1 + n^5 /^2
∑^ ∞
n=
n ln n
(c) The graph of the distribution p(t) is shown on the figure, where a > 0 is a constant. Fill in the blank with “greater than”, “equal to”, or “smaller than” to make the sentence below the graph correct.
− 2 a 0 a
1 /(2a)
p(t)
t
The median of the distribution p(t) is its mean.
- (10 points) A firm that manufactures and bottles apple juice has a machine that automatically fills bottles with 15 ounces (oz) of apple juice. There is some variation, however, in the amount of liquid dispensed in each bottle. Over a long period of time, the average amount dispensed into the bottles was 15 ounces, but the underlying measurements showed the distribution of the
ounces, x, of juice in the bottles was given by p(x) =
2 π
e−^
(^12) (x−15) 2 .
12 13 14 15 16 17 18
amount x of liquid (oz.)
p(x)
(a) What fraction of the bottles contained between 14 and 16 oz of juice? Explain.
(b) Give a graphical interpretation of your answer to part (a) on the figure.
(c) Find, as accurately as you can, the fraction of the bottles that contained at least 17 oz of juice inside them. Explain.
- (10 points)
(a) Find the radius of convergence R of the following power series. Show your work.
∑^ ∞
n=
(n + n^3 2 n) n^2 3 n^
(x − 1)n^.
ANSWER : R =.
(b) What is the interval of convergence of the series?
< x <.
- (10 points)
(a) Find the second order Taylor polynomial of the function f (x) =
4 + x for x near 0. You must show the calculations that lead to your answer.
(b) What is the Taylor series about x = 0 of the function sin x? No explanation required.
(c) Using your answers to parts (a) and (b) and without computing any derivatives, find the
second order Taylor polynomial that approximates g(x) =
4 + sin(2x) for x near 0. Show your work.
- (12 points) We have learned how to use slicing to calculate areas and volumes. This problem explores a different kind of slicing through a simple example. A right-isoceles triangle with sides of length 2 is covered by squares as illustrated and explained in the figure below.
step 1: one square of side length 1
step 2: two squares of side length 1/
step 3: four squares of side length 1/
step 4: eight squares of side length 1/
... etc ...
@ @ @ @ @ @ @ @ @ @
@@
(a) Use a geometric series to find the area covered by the squares after the N th^ step.
(b) Use your answer to part (a) and your knowledge of series to find the total area covered by the infinitely many squares.
(c) How do you know your answer to part (b) is the correct one?
- (13 points) We shall investigate a well-known physical phenomenon, called the “Doppler Effect”. When an electromagnetic signal (e.g. a ray of light) with frequency Fe is emitted from a source moving away with velocity v > 0 with respect to a receiver at rest, then the received frequency Fr is different from Fe. The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law:
Fr =
1 − v/c 1 + v/c
Fe , where c is a constant, the speed of light.
For this problem, you might find useful to know that the third order Taylor polynomial for the
function
1 + x 1 − x
near x = 0 is 1 + x +
x^2 2
x^3 2
(a) On Earth, nearly all objects travel with velocities v much smaller than the speed of light c, i.e. the ratio v/c is very small. Use this fact to obtain the Doppler Law for slow-moving emitters:
Fr '
v c
Fe.
(b) The relationship in part (a) is not exact, and an error is made when it is used to approximate the Doppler Law. Find an expression for the “error”, in terms of v, c and Fe. Is the approximation accurate within 1% of Fe if the velocity is at most 10% of the speed of light c? Explain.
