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Unformatted text preview: MATH 141, Final Examination 1:30—3:30 PM
Monday, December 15, 2008 Instructions. Answer each question on a separate answer sheet, labeled with the problem number. Do
not put the answers to two different problems on the same answer sheet. If you need extra room, continue
on the back or ask for an extra answer sheet. Show all your work. A correct answer with no justiﬁcation
may not receive full credit. Be sure that you have copied and signed the honor pledge at the top of the ﬁrst
answer sheet. The point value of each problem is 18 points unless indicated. The exam is worth a total
of 200 points. In problems with multiple parts, whether the parts are related or not, the parts are graded
independently of one another. Be sure to go on to subsequent parts even if there is some part you cannot do.
Please leave answers such as 5\/§, 37r, or e2 in terms of radicals, 7r, or e, and do not convert to decimals.
No notes or calculators allowed. 1. Let R be the region between 31 : w1/2 and y 2 2931/4. Find the volume V of the solid obtained by
revolving R about the sic-axis. 2. A tank of water is in the shape of an inverted cone (point down) with height 10 feet and radius 3 feet.
It is ﬁlled to the top with water. Find the work necessary to pump all but 2 feet of water to a point 7
feet above the top of the tank. You may assume water weighs 62 pounds per cubic foot. 3. Find the length of the graph of the function f (11:) : ln(cos a") for 0 g a: g g. 4. Find the integrals:
(a) / 1 da:
£82+4£12+7 ' a:
/m2+2r—3dm. lim 1082(1082W».
“H00 10g2(.’13) (b) 5. Find the limit: 6. Solve the differential equation: 7. Calculate the integrals: (a 3
/ (111 m)2 dw.
1 (b) °° 1
/2 x(ln ac)2 dx. 8. (20 points, 10 points per part) Find the sum of each series. You do not need to check for convergence
or give a rigorous justiﬁcation. (e
w) m A
gnaw Hint: What function f (at) is represented by the power series 22°21 311:? If you can’t remember,
compute f ’ (51:) and see if that helps. 9. Find the radius of convergence R for the power series 00
71:0 10. Find the Taylor series or the function f(06)=\/1+$ about a: = 0. If possible, ﬁnd an explicit formula (in terms of n) for the coefﬁcient of :13". If you can’t
do that, you can get 3 points of partial credit per coefﬁcient, up to a maximum of 12 points, for ﬁnding v
the coefﬁcient of ac” for each value of n from 0 up to 3. 11. (a) Write the complex number
3 + i lw’i in the form a + bi, with a and b real. (b) Find all square roots of ~3 + 42‘ in both we“) form and a + bi form. ...
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- Spring '07