Unformatted text preview: Last Initial
Section
FULL Name
Social Security Number FINAL EXAMINATION
21C, 10:3012:30, Monday December 10, 2001
Declaration of honesty: I, the undersigned, do hereby swear to uphold the highest standards of academic honesty, including, but not limited to, submitting work that is original, my own and unaided by notes, books, calculators, mobile phones, immobile phones,
muses or any other electronic device. Date Signature 1
2
3
4
5
6
7
8 9
10
11
12
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14
B ¡
¡
¢ ¡
¢ 1 ¨
5 and 2
. ¨ ¨ ¨ ¨
2 10 ¨ ¤ , and &(' !&%$" !
¨ #
, 43 , , . and ¡
¦ ©¨ , &)$"
¨ # c) ,
b)
a) ¦ ¤ ¡
§¥£ ¢ Question 1 (50 points)
Let
and . . Compute ¨ ¡ with , , and ¦ ¤
¦¤ ¢
§¥£¡ ¡
©¨
¡ and . ¦ ¤
¡
e) Now let for real and . Use your result to compute ©
Question 1 (continued)
d) Write in polar form all real. Show that ¨ ¡
2 ©¨
Question 2 (30 points)
Compute the moment of inertia of a solid, homogeneous sphere with radius
and total mass
about an axis through its center. (Hint: inﬁnitesimally
where is the distance from the axis of rotation and the mass density.)
! " % 3 ¡
1$# ¡ ( &
)# '¡ % Question 3 (20 points) satisﬁes ¦
§¡ ¤ ¥¤ £ ¡ ¡
¢§ . and does the following equation hold ©¨¡ ¤
¦ & & ¥¤ & & £ £ b) For which values of ¤ ¤ ¡ £ whenever a) Show that Question 4 (20 points) NO ( 1 & 0( # 948¡ ( & 7( # 543
6 3
6 MAYBE converge absolutely? NO 4 (Circle one) . Does ) &(
( 7'% ¦ $ #! &
"
1 ) &(
( 2 0'% YES converges and ) &(
0'% b) Suppose
verge? MAYBE £ YES . Does (( a) Suppose (Circle one) di MAYBE ) &(
( ¡ 7'% ¡ ¦ ¤
©
§£ §¥ ¡ )
( ( 0&( % converges when ¡
£ §
¡ YES NO MAYBE ¢ MAYBE e) The power series
lutely when
? (Circle one) converges and is any positive real
for all
? ¦ ¤
§¥ £
¨ ¦ ¤
¥§¥ YES NO ¡ d) Suppose we choose such that
number. Can we ﬁnd an such that converge? ¢ )
( 1 ¤ ( 0&( % YES converges. Does )
( 0&( % c) The series (Circle one) . Does it converge abso NO (Circle one) Question 5 (30 points)
Compute the integral
§¤ & ¢ #
& & &
¤ %#
$ " ¡
# && && & by ﬁrst writing
integral using polar coordinates. and calculating this double ¤ !
# ¤ ¡ Question 6 (30 points)
a) Clara the climber measures the northward slope of Mount Finality to be 3
and eastward slope to be 4. Use the total differential to estimate how much
5 higher/lower she will be by walking 50 feet southeast. b) Using coordinates where measures the eastward and the northward position
and measures height, write an equation for the tangent plane to Mount Finality
at Clara’s original position which you may assume to be the origin.
Question 7 (30 points)
The volume of a cone is
where is the area of its base and is the height.
Test this result by using an iterated integral to compute the volume of the following
cone: Draw a triangle in the
plane with corners
,
,
and then
connect every point on its edges to the point
. ¤ !§¥ ¢ ¡ ! ¢ £
¦¢
¢
¤¦ ¢ £ !£ ¦ ¢ £
¢ ¡¤ ¢
£ " Question 8 (40 points)
Consider the power series
(
£ ¤
& © ¥¥ "&
& &
©
© ©
! )7(
&¨ £ ¡ ¡ a) Find the radius of convergence . Discuss convergence at the upper end6 ¡ !
point . b) Compute the ﬁrst four terms of the Taylor series for the function
¤ £ ¡ £ ¡ ¡ and compare them to the power series above. Question 8 (continued)
c) The series expression for
is an alternating series when
. Compute
an accurate estimate for
by studying the second and third partial sums.
Express your answer in scientiﬁc notation
answer
error . (Answers
using simpliﬁed fractions are OK.) ¦
§ ¤¢
¥££ ¡ ¡ &
¤ & ¤ ¢¡
¥££ ¨
© £ ¡ £ ¡
¢
7 ¦
¡ Question 9 (50 points)
Let ¡ a) Sketch level curves for the function . is the function 8 ¡ ¢
£
¢
£ ¡ ¡ ! ¤¦ £
¢ ¢ Question 9 (continued)
c) For which domain of values and its traces in the ! ¢ ¤¦ !¦ ¢ £
¢
¢ b) Sketch the surface
planes. deﬁned? , and ¡ d) Compute the partial derivatives of
cal points of . needed to determine the Hessian and criti ¡ ¡ e) Find any extrema and critical points of and determine their type using the
Hessian where necessary to justify your answer. Question 10 (20 points) Decide, using any test you think is appropriate whether
the following series converge or diverge. Indicate also whether convergence is
absolute or conditional. £¢$ ¡ 7&( %
( )
¤ )
¤ 7&( % ¨ b) ¨ a) 9 ¤ ( 7&( %
) ¡ Question 11 (20 points)
Consider the sequence
. Prove that
such that
for all
. ¦ ¡ ( & 0( # 543
6 d) ( )
( " © 7&( % c) by proposing a value ¢ ( ¡ ( ¢
£ ¡ ©
§£ ( £ © ¦ ¢ Question 12 (30 points)
Use the derivative form of Taylor’s theorem with remainder to estimate how many
terms in the series ¤¦
¤ £ ¤ £ ¤ £ ¥£ ¤ £
¤ ¤ ¤ ¤
¦¦¥ ¢ would be needed to compute to three decimal places. (Note that you are not
asked to compute itself and may assume that
.) ¦ © ¢ 10 ¢ Question 13 (5 points)
In twenty words or less: Which part of the course did you ﬁnd most interesting
and why? ¢
£ ¡ ¢ £ 6 ¢ ¢ ¡ # 943
¢
¤ £ ¡ £ ¡ ¤ 5 ¤ & ¦ ¤
¥ ¨ 4
and in terms of ¦
§¤ ¢ ¦ ¨ ©§
¨ b) Express 11 ¦
¨¡ over . ¤ for &
& Question 15 (30 points)
a) Solve ¤ exists. & Explain whether & Question 14 (15 points)
Let . Use these expressions to prove the ¨
4 ¨ 4 ©§ ¨ ¨ § ¡ ¡¤ ¨ ©§ ¨ ¢ £ ¡ ¨ & 4
¤ ¨ & ¨
©§ Bonus Question (3 points)
In which years were Galileo, Leibniz and Newton born? 12 trigonometrical identities ...
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 Mount Finality

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