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Unformatted text preview: Final Exam. MTHU 142 _ Spring 2007. Name: FOR FULAZIL CREDIT, SHOW ALL YOUR WORK! !
1. (3 points each) Find the following integrals: (a) fﬁ+§+edx (b) f(:c2 — 69:)“3 (a: — 3) d2: (0) f 562111622) dzz: (d) j? sin3(x) dz: (6) [04 £126"? 223;); dy d3: '2. (a) (4 points) Find the area between the curves 3; = $2 — 4 and y = 2 — :3. (b) (6 points) Find the volume of the solid generated by revolting the region enclosed by the lines y = m, y = 2a: and the vertical lines a: = 1, :1: = 2
about the x—axis. 3. (a) (5 points) Use Simpson’s Rule with n = 4 to approximate the integral /4 1
01+x2’ (b) (5 points) Solve the differential equation (6” + 1)dy = ye“’d:1:,, y(0) = 2. Express your solution in the form y = f Hint: Separate the variables. (0) (6 points) Solve the following initial value problem by using an integrating factor:
7 4. An oil tanker is leaking oil at the rate given in barrels per hour by _ 80in(t+1)
_ t+1 ‘ Where t is the time in hours after the tanker hits a hidden rock (when t = O). 4 wt) (a) (5 points) Find the total number of barrels that the ship will leak on the
ﬁrst day. (b) (5 points) Find the total number of barrels that the ship will leak on the
second day. (0) (5 points) What is happening over the long run to the amount of oil leaked
per day? 5. Given the function f (11;, y) : f (m, y) = 273; _ 9x3 + 6xy2 _ 21/3 (a) points) Identify the four critical points of this function, Where fCC (as, y) =
fy (33, y) = 0. Verify that two of the four critical points are P(3, 6), Q(—3, —6). (b) (4 points) For each critical point decide Whether it is saddle point, maxi mum or minimum of f(:v, Show clearly your decision process. '6. Determine Whether the following integrals converge, if it converges, determine
its value. (a) (5 points) Find f2°° % d:c (b) (5 points) Find jg” ﬁg; dz 7. (a) (6 points) Set up a double integral to evaluate the signed volume under .2 = ye“?2 above the rectangle in th xy—plane given by 2 S m S 3,. and
O S y S 2. Thenlevaluate this volume. (b) (6 points) A portion of a blood vessel is measured as having length 7.9 cm
and radius 0.8 cm. If each measurement could be oﬁ by as such as 0.15 cm, estimate the maximum possible error in calculating the volume of the
vessel. 8. The height of a particular plant can be predicted based on past data. The probability density function is given by the function f = gégxg Where it is
between1 cm and 8 cm. (b) (3 points) What is the probability that a plant is shorter than 4.5 cm? (0) (3 points) Find the expected value of the plant’s height. ((1) (3 points) Also, set up a formula for the variance of the height, in this
particular case. Math U142 Final Exam April 25, 2007 Name FOR FULL CREDIT, SIMPLIFY YOUR ANSWERS, REDUCE ALL FRACTIONS & SHOW
ALL WORK!! USE 3 DECIMAL PLACES” GOOD LUCK” 1. Compute the following: (a) /3\/E—g+edm (b) / (t _ 3)(t2 — 6t)‘3dt (c) / 11:2ln(3:v) d2: 7r/2
(d) /0 sin3 cos dzc 2. Find the area between y = m2 3. Find the volume of the solid
2 4. —4andy=2—:I:. generated by revolving f = 6x — 5 about the m—axis between :1: = 1 and AWJJ‘I‘X'W“'umwnﬂimfmruvwnmwyn>‘nwv'»p.»Lr'h‘u .. WM“. NW,“ N. _ . , . . Lanmmuarmmxemmmmmnnw 4
4. Approximate / 1
0 dy 4y d2: :1:+:1: 1 +m2 dzv usiﬁg Simpson’s rule with n = 4. (b) Find the total number of barrels that the ship will leak on the second day. ‘ (c) Find the average number of barrels that the ship will leak on the second day. anvruurm emu,“ .« L ,... A («rumba mm. .Muwmwwmmrﬂ a E
g
a; 7. Find all critical points, relative minima, relative maxima and saddle points for ﬂat, y) = 273: — 9:103 + 6:131:12 — 23/3 (Hint: There are 4 critical points!) \\ V, 4 r 41'
9. (a) Compute/ / 2mg dy dx
0 2:2 (b) Write an equivalent double integral with the order of integration interchanged. 10. A portion of a blood vessel is measured as having length 17.9 cm and radius 0.8 cm. If each measurement can have an' error of i015 cm, estimate the maximum possible error in calculating the volume of the
vessel. 11. Find the volume under 2 = yemﬂ’2 and above the rectangle 2 S :1: S 3, 0 S yg 2. 12. The height of a particular plant can be predicted based on past data. The probability density function is given by f = $19”, Where a: is between 1 cm and 8 cm. (a) Find the probability that the given plant is between 3 cm and 6 cm tall. (b) Find the probability that the plant is shorter than 4.5 cm. (0) Find the expected value of the plant’s height. (d) Find the variance in the plant’s height. v.3 ,:..m’wmitzu‘zxmnarwwvﬁmma. Wm an. ...
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This note was uploaded on 04/17/2008 for the course MTH U142 taught by Professor Schulte during the Fall '08 term at Northeastern.
 Fall '08
 Schulte
 Calculus

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