MATH 204 Final, Fall 2006
American University of Beirut · Mathematics for Social Sciences II · 9 pages
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Time: 2 hours Math 204 24/01/07
Final Exam
Fall Semester 06/07
Instructor: Mrs. Muna Jurdak
Section 9: Thurs 3:30 p.m. Section 10: Tues. 2:00 p.m.
Section 11: Tues. 12:30 p.m. Section 12: Tues. 11:00 a.m.
Instructions:
1. Write your name and ID number clearly where indicated.
2. Circle your section number above, according to the time of the problem
solving session in which you are enrolled.
3. Solve the problems on this, the white question sheet. Use the colored
sheets for scratch work only. You may use the back of a white sheet to
complete the solution of a problem.
4. Write your name on the colored scratch sheet also.
5. Show your work in all the problems.
6. If you fail to write your name, ID number, or to circle your section
number, you will lose grades.
Problem Grade Problem Grade
7
1
8
2
9
3
10
4
11
5
12
6
Final Exam Grade /100
(8%) 1. Find the coordinates of all critical points of the following
function and determine the nature of each point.
f(x,y) =x? — y? -3x+4y -5
(7%) 2. Find the 4 second-order partial derivatives for the function
_ 5x
F(%y) ~ >
y
a
(8%) 3.Given the integral ee +1dx.
(a)Approximate the above integral using the Rectangle Rule, with the
number of subintervals n = 4.
(b) Approximate the above integral using Simpson’s Rule, also with
n=4,
(12%) 4. Evaluate each integral:
(a) [x*(x' -1)"dx
(b) (eee
(6%) 5. Find g(x), if g"(x) = 12x?+ e*, g'(0) =2, and g(0) =5.
(8%) 6. The solution to a system of equations having the matrix form
AX = B can be found by the matrix multiplication X -( ° a) . What
was the original system of equations? (Write the answer as a system of 2
equations with 2 unknowns.) Show your work.
(13%) 7. A fair die is rolled 3 times. Let X be the number of 6’s obtained.
(a) Construct the probability distribution of X.
(b) Find the probability that no more than two 6’s are obtained.
(c ) Find the mean and standard deviation of the probability
distribution of the variable X.
(7%) 8. Given the function z = f (x,y) = 25x’ y -10xy+ 200x — 500.
(a) Find f (10,1).
(b) Using a certain derivative, estimate the expected change in z, ify
increases by | unit, and x remains constant.
(c) Compare the expected change found in (b) with the actual change
in z, if y increases by | unit.
(9%) 9. Referring to the following figure, set up the combinations of definite
integrals that would compute each of the following areas:
44\
(a) Area of region P.
(b) Area of region Q.
(c) Area of region R.
(8%) 10. A box contains a set of 30 balls: 20 red, 5 blue and 5 green in color.
Three balls are selected at random from the box, one at a time,
consecutively. What is the probability that the three balls are of three
different colors,
(a) If the selection is done with replacement? Explain.
(b) If the selection is done without replacement? Explain.
(6%) 11. Find the area of the region bounded below by the curve y =x’ — 4,
above by the line y = 2 — x, and to the left by the y-axis. Draw a sketch.
(8%) 12. The scores on a test are normally distributed, with a mean of 100
and a standard deviation of 15. If a personnel manager of a company, wishes
to select from the top 20% of the applicants who take the test, find the cutoff
score (lowest score that would be considered for selection).
646
TABLE 14.26 | LIAL SL Area Under the Standard Normal Curve 7
Figure 14.9
CHAPTER 14 PROBABILITY DISTRIBUTIONS
“9g”
Area = 0.3413
.00 .01 .02 .03 04 05 .06 .07 .08
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319
0.1 .0898 .0438 .0478 .0517 ~©.0557 .0596 .0686 .0675 .0714
0.2 .0793 .0832 .0871 .0910° .0948 .0987 .1026 . .1064 .1103
0.3 .1179 1217 1255 .1298 1831 1868 .1406 .1443 .1480
0.4 .1554 1591 .1628 .1664 1700. .1736 .1772 .1808 .1844
0.5 .1915 .1950 .1985 .2019 .2054 .2088 2123 .2157 .2190
0.6 2257 2291 2324. 2357 2389. 2422 2454 2486-2518
0.7 .2580 .2612 .2642 .2673 .2704 2734 .2764 .2794 .2823
0.8 .2881 .2910 .2939 .2967 .2995 3023 .3051 .3078 3106
0.9 .3159 .8186. .3212 .3238 .3264 3289 .3315 3340 .3365
1.0 .3413 .3438 3461 .3485 .3508 .3531 3554 3577 .3599
1.1 3643 3665 3686 .3708 +=©.3729' «3749-8770 =«.3790~—-.3810
1.2 .3849 3869 .3888 - .3907 .3925 .3944 . 3962 +.3980 ~ .3997~
1.3 .4032 .4049 .4066 .4082 .4099 .4115° 4131 4147 .4162
1.4 4192 4207. 4222. .4236- .4251 .4265. -.4279 .4292 .4306
1.5 .4382 .4345 - 4857 1.4370 "4882 .4394-- 4406 4418 4429
16 .4452 4463 4474 .4484 .4495 .4505 .4515 .4525 .4585
1.7 .4554 4564 4573 .4582 4591 .4599 .4608 .4616 .4625
18 .4641 .4649 4656 .4664 4671 4678 4686 4693 .4699
1.9 .4713 .4719 4726 .4732 47388 4744 4750 .4756 .4761
2.0 4772 .4778 4783 .4788 .4793 .4798 .4803 .4808 .4812
2.1 4821 .4826 .4880 .4834 .4888 .4842 .4846 .4850 4854
2.2 4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887
2.3 .4893 .4896- .4898 .4901 .4904 ~~4906 . 4909. .4911 4913
24 .4918 .4920 .4922 .4925 .4927 .4929 4931 .4932 4934
2:5 .4938 4940 .4941 .4943 4945 4946 .4948 4949 4951
26 4953 .4955 .4956 .4957 .4959 4960 .4961 .4962 .4963
“2.7 4965 4966 ° .4967 .4968 .4969....4970'- .4971 .4972 .4973
“2.8 ,4974 .4975' .4976° .4977 .4977 .4978 .4979 .4979 4980
4981 .4982 .4982° .4983 4984 .4984 .4985 .4985 .4986
3.0- 49865 14987" .4987 ".4988° .4988 4989 - 4989: 4989 4990