Review for ENM 503 Final Exam covering Chapters 8-17
1. The probability of rolling a 5 before a 7 in throwing a pair of fair
dice is __.
4/( 4+6) = 4/10
2. How many diagonals does an octagon have?
(- (comb 8 2) – 8)
3. What n-gon has the same number of sides as diagonals?
Pentagon
4. Calculate the probability of getting 4 of the same rank in 5-card
poker.
13 * 48/(comb 52 5) or [xxxx y]
5. Write the 5th row of Pascal's triangle.
(comb 5 0) (comb 5 1) …
6. In tossing a fair coin, Heads and Tails are mutually exclusive
(T or F);
independent
(T or F);
Collectively exhausted (T or F)?
1
7. Compute the expected value of a fair die toss.
8. Disease in a population is 5% and both positive and negative
tests are 95% accurate. A person takes the test with positive
results. Calculate probability of actually having the disease.
9. Find the probability that a fair die shows face 2 given that the
event EVEN occurred.
P(A|B) = P(AB)/P(B)
10. Compute the density distribution for the number of heads in
tossing a fair coin 3 times.
X
0 1 2 3
P(X) 1/8 3/8 3/8 1/8
HTT
11.Given Markov matrix [0.7 0.2; a
0.3 0.8], b
a
b
find the steady state and market distribution given that the
initial distribution
is [ ¼ ¾].
(solve '((-.3 .2 0)(1 1 1)))  (0.4 0.6)
(expt-matrix #2A((0.7 0.2)(0.3 0.8)) 20)  #2A((0.40000
0.4)(0.6 0.60000))
2
(M* * #2A((0.3)(0.7)))  #2A((0.4)(0.6))
12. Find limit of sin x / x; (1 + x) 1/x and Ln x / x as x  0.
1 e 
13. What number less its square is a maximum?
F(x) = x – x2
F' = 1 – 2x = 0 when x = 1/2
14. T or F Differentiability implies continuity.
15. Find the derivative of y = Ln x.
y' = 1/x
16. At what point(s) does (do) the tangent to y = 2x3 – 3x2 + 1
have slope 12?
Y' = 6x2 -6x =12
17. If f"(x) = 12, f'(1) = 2 and f(2) = 15, find f(x).
18. Find area under curve y = x2 bounded by x-axis and x = 1.
3
19. Find area under curve y = x2 - 4 bounded by x-axis and y = 1.
20. Which is better: take a 10% discount or take two 5%
discounts?
21. Find the rate of change the area of a circle with respect to its
radius when the radius = 5.
A = pi r2
x-1
22. y = 1/x; y' = _____
23. Show that y and Ln y have the same critical points for a
polynomial function.
y = x2 + 5x + 6 and ln y = ln (x2 + 5x + 6)
24. Where do x and Ln x intersect if anywhere?
25. Evaluate dy/dx at (1, 1) for x3 + 5xy + 6y2.
Implicit differentiation
26. If e2x = 4, then x =
.
4
27. If
z
a
2dx
2
= 5, then a = .
28. The rate of change of the surface area S of a cube with
respect to x the length of a side is ______.
29. (f + g2)’ (x) =
when f(x) = x2 - 2x and g(x) = x.
30. If f ‘(x) = ex and f(0) = 4, then f(1) =
.
31. If d/dx(5x2 + x3) = f(x), then f(2) =
.
5
.
32. If x = ln xy, then y’ = .
1 = 1(xy' + y)/xy
33. d(uv) = udv – vdu => udv = d(uv) – vdu or  udv  uv   vdu
Integrate by parts I = integral of f(x) = x2
Let u = x2 and dv = dx
du = 2xdx; v = x and I = x3 – 2I => I = x3/3
34. d(u/v) = u * 1/v
35. The function f(x) = 3x2 - 12x + 5 is increasing over the interval
________.
6x – 12) > 0
True or False
36. The equation x99 - 39x88 + 33 = 0 has a root between 0 and 1.
37. ln (1/x) = - ln x.
38. The equation x = ln x has no solution.
39. ln (a + b) = ln a + ln b.
40. y = e2x is a solution to y” - 3y’ + 2y = 0.
6
.
41.. The function y = x4 + 1 is increasing for x < 0.
42. A horizontal asymptote of y = (x2 - 4)/(x2 + 3x) is y = 1.
43. The function f(x) = ln x has a relative minimum at x = e.
44. Limits are never indeterminate.
45. Find the area under the curve y = x2 bounded by the x-axis
for x on [0, 1] using both vertical and horizontal elements.
46. Find expected value of density function f(x) = 2x on [0, 1].
1
E(X2) = 0 x2 (2x)dx
V(X) = E(X2) – [E(X)]2
=
½ - (2/3)2
For problems 47-50 let f(x) = 2x on [0, 1].
47. E(X) =
48. E(X2) =
z
1
xf(x)dx =
2/3 .
z
1/2 .
0
1
0
x2f(x)dx =
49. V(X) = E(X2) - E2(X) = 1/18
.
7
.
50. V(X)
z
1
0
[x  E(X)]2f(x)dx =
.
4
4
x3 4 x 2 4 x
0 ( x  2 / 3) 2 xdx  0 ( x  3 x  9 )dx  3  6  9
1
1
2
2
1 1
51.   4xydxdy =
.
0 0
2x 2y 4xy
1
52.  4xydy =
f(x) = 2x .
0
1
53.  4xydx =
f(y) = 2y .
0
1 1
54.   xy(4 xy)dxdy =
.
0 0
n
55.  ( xi  x) =
.
i 1
56. Use the Lagrange multiplier to find critical points for f(x, y, z)
= x2 + y2 + z2
subject to 3x2 + 2y + z = 4.
8
57. Given f(x) = x3 - ax2 + bx + 10, find a, b and c given that the
critical x-values occur at x = -1 and x = 3 and that the maximum
value is -29,
f'(x) = 3x2 – 2ax + b
3 + 2a + b = 0 or 2a + b = -3
27 – 6a + b = 0 or -6a + b = -27
8a = 24 => a = 3
b = -9
58. Find relative extrema and identify for F(x, y) = x3 + 3xy2 – 3x2
–
3y2 + 4
Fx = 3x2 + 3y2 – 6x = 0; Fy = 6xy – 6y = 0 => x = 6y (0, 0)(1,1)
x2 + y2 – 2x = 0 and y(x – 1) = 0; x = 1, y =  1; (1, 1), (1, -1)
(0, 0)(2, 0)
Fxx = 6x – 6
Fxy = 6y
= (6x – 6)2 – 36y2
Fyx = 6y
Fyy = 6x – 6
(0, 0), D > 0 and Fxx < 0 => relative maximum;
9
for (1, 1), D < 0 => saddle, for (1, -1), D < 0 => saddle point
for (2, 0), D > 0 and Fxx > 0 => relative minimum.
59. Integrate
x 1
x  2x  3
2
A
X+1
60. A part exponentially distributed has a mean time to fail of 5.
Find probability that 2 of 5 of these parts will survive for 7 years.
K =  = 1/5


7
0.2e0.2 x dx
(binomial 5 (U-exponential 1/5 7) 2)
61. A projectile is fired at an angle of 60 with the horizontal at a
initial speed of 1496 ft/sec. How high will it go, how far and when
will it hit the ground?
62. Integrate a) (2x + 5)4 dx
b) x(2x2-1)1/2dx c)
x 5
x 5
A
B



x  5 x  6 ( x  1)( x  6) x  1 x  6
2
10
63. Given RV X ~N(50, 4) find
a) P(X < 52)
(normal 50 4 52)
b) P(48 < X < 52) (del-normal 50 4 48 52)
c) X at the 90th percentile.
(inv-normal 50 4 0.9) 
52.563456
64. The total cost of making x items per day is x2/4 + 35x + 25
and the price per item is 50 – x/2 both in dollars. Find minimum
production for maximum profit.
Profit = x(50- x/2) - x2/4 - 35x - 25
P' = 50 -x – x/2 - 35 = 0 when x = 10 items per day.
Cost to make a set is C = x/4 + 35 + 25/x and C' = ¼ - 25/x2 = 0
when x = 10
11
65. Given x + xy + y = 5, evaluate y'' at point (2, 1)
1 + xy' + y + y' = 0 => y' = -(1 + y)/(x + 1) = -2/3 at (2, 1)
y'' = (x + 1)(-y') + (1 + y) / (x + 1)2
= [3(2/3) + 2]/9
= 4/9
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