MIT 18.01 Fall 2006 Single Variable Calculus Practice Exam 3, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Practice Exam 3

Problem 1.

a) (10) Derive the trigonometric formula cos 2 x = 1 − 2 sin

2 x and use it to evaluate

sin

2 xdx.

e

b) (10) Differentiate x ln x, and use your answer to evalute ln x dx.

1

Problem 2. (15) K-mart is selling at half-price its left-over Great Pumpkins— thin orange plastic

shells filled with half-price Halloween candy.

A Great Pumpkin has the shape of the curve x 2

• y 4 = 1, rotated about the vertical axis, i.e., the

y-axis. This curve is symmetric about the x-axis and the y-axis — it looks something like a circle, but

somewhat flatter at the top and bottom.

Using units in feet, how many cubic feet of candy will it take to fill a Great Pumpkin? Give the exact

answer, then tell if 5 cubic feet will be enough.

x 2

Problem 3. (20: 3,7,5,5) The function F (x) = t

2 e −t dt is not elementary; it comes up in 0 calculating the standard deviation of The Curve of normal distribution. (In the following, (a) and (b) go

together, but (c) and (d) are both independent questions.)

a) Find F � (x).

b) Find the critical point(s) of F (x), and determine their type(s) by studying the sign of F � (x) when

x is near a critical point.

9 −u c) Express

ue du in terms of values of F (x). 0

3 x d) Estimate F (x) by showing that F (x) � , if x > 0. 3

Problem 4. (15: 12, 3) The end portion of a boneless AllSoy SmartHam of length a has approxi

mately the shape of the region under the curve y =

x, 0 � x � a, rotated about the x-axis.

a) When it is sliced vertically into thin slices, what is the average area of a slice?

b) Where on the SmartHam is there a slice having this average area (i.e., how far from the tip)?

Problem 5. (15: 7,8) You only have time to look at the newspaper on Sunday, but the first thing

you turn to is the baseball statistics from Saturday’s game, to see how many hits your favorite ball-player

Pepe LeMoko got. In September (which started on a Saturday) he had a slump in the middle, but came

out of it. His record on the five successive Saturdays was

Day: 1 8 15 22 29

No. hits: 3 2 0 1 3

Suppose there was a game every day; estimate the total number of hits he got during those 29 games

by using

a) the trapezoidal rule

b) Simpson’s rule