Math 112 REVIEW

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Math 112 REVIEW
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Math 112 REVIEW
25 November 2011
Purpose of this Homework: To prepare for final exam
Answers: No answers will be posted. You are required to study, understand and do the questions. If you
provide step by step written answer to any question(s) I can check your answer.
th
Last Date: To discuss with me about this, the last date is December 6 , 2011
Topics related to questions in red will be covered in the class next week.
PROBABILITY

 ke 0.5 x ,
1. Find k if the function f ( x)  

0,
function.
0 x
is a probability density
otherwise.
Determine P(0  X  ln 4)
2.
1 x / 2
e
, x  0 be an exponential probability density function of a
2
random variable. Prove that the variance V ( X )  4
f ( x) 
3. The number of months between accidents in a factory is an exponential random
variable X with mean 2 months.
A) Write down the p.d.f.
B) What is the cumulative pdf?
C) Find P(1  x  3) using your answer in (B)
1  kx, if 0  x  2
4. Find the value of k for which f ( x)  
is a probability density
 0, otherwise
function
5. Find the mean of the probability density function f ( x)  6 x(1  x), 0  x  1
6. Find the cumulative p.d.f of a uniform probability density function in the
interval [0,8]. What is the expected value of X? Find P(4  X  7) .
7.
The hours spent per week on homework by Columbia College students is found
to be a random variable which is normally distributed with mean 4 hours and
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variance 2 hours. A) What is the probability that a student spends more than 6
hours per week on Math 112 homework?
B) What percentage of students spend more than 3 hours per week on math 112
homework?
8.
A normal distribution has mean 30 and standard deviation 7. Find
P(30  x  42)
9. Use the standard normal distribution to find P(0.25  x  0.25) . Use the table
given to you.
10. A continuous random variable x is normally distributed with mean 40 and
standard deviation 5 . Find P(45  x  52)
11. Find the average value of the function f ( x) 
1
x 2  5x  4
in the interval
5 x8
12. Using the method of least squares find the regression line for the points
(2,1), (1,1), (0,1), (1,2), (2,2) . Write down the two Normal Equations which
determine the constants in the equation of the line. Show all steps.
PARTIAL DERIVATIVES, MAXIMUM, MINIMUM AND LAGRANGE METHOD
2
2
13. Find the partial derivatives f xx (1,2), and f yx (1,2) for f ( x, y)  e x  y
14. Find all second order partial derivatives of each of f ( x, y)  ln(2 x 2  2 y 2  1)
15. Determine the equation of the tangent plane to the surface z  xy ( x  y) at
the point (2,3) .
16. Find the value of f (2,1) f x (2,1) and f y (2,1) where
f ( x, y ) 
3x 2  4 y 2  12 xy
( x  y) 2
17. Let z  e  x ln xy Show that
2z
2z

xy yx
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18. Find all the second order partial derivatives from f x 
19. Let f ( x, y ) 
x  y ln y
x2 y
fy 
y  x ln x
xy 2
,
x2  y2
. Simplify xf x  yf y
x y
20. Let C ( x, y)  xye y . Find the value of C y (1, 0) . Interpret clearly the meaning of
C x (1, 0) in simple and complete sentence.
21. Let C ( x, y)  32 xy  175x  205 y  1050 . A) Evaluate C x (80,20) B) Interpret
the meaning of C x (80,20)
22. Let f ( x, y)  x 3  y 3  6 x 2  3 y 2  9 y . Find all the critical points, local
maximum, local minimum and saddle point(s).
23. A company manufactures two products A and B. The price for A is
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p A  12  x, 0  x  24 and for B is p B  20  y, 0  y  20 , in thousands of
2
dollars. The cost function is C ( x, y)  9 x  16 y  xy  7 in thousands of dollars,
find the price of A and B which maximizes profit.
24. Determine the critical points of the function f ( x, y) 
x y
x  y2  8
2
25. Discuss the local extrema of the function f ( x, y)  x 2  x ln y  1,
.
y0
26. The revenue function for a company manufacturing x units of product A and
17000
y  5 xy . Find the
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number of units in product A and the number of units in product B which
maximizes revenue.
y units of product B is R( x, y)  3x 2  3 y 2  2000 x 
27. Find the relative max/relative minimum values of
f ( x, y)  y 3  x3  3x  12 y  5 using second derivatives test.
28. Let f ( x, y)  100 x 0.25 y 0.75 be the productivity function where x = the number
of units of capital, one unit of capital =$1000, y = the number of labor-hours.
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Find the marginal productivity of capital when the capital expenditure is$16,000
and the labor-hours is 81. Find the marginal productivity of labor when the
capital expenditure is$16,000 and the labor-hours is 81. What would you advise
to increase productivity at this level? Add a unit of additional capital or employ
additional unit of labor?
29. Using Lagrange multiplier method, find the maximum value of f ( x, y, z)  9 xyz
subject to the condition 6 x  4 y  3z  24
30. Maximize the volume of the rectangular box whose
height  length  width  120 centimeters.
31. Find the maximum of f ( x, y, z)  x  y  z subject to the constraint
x 2  y 2  z 2  9 using Lagrange’s multiplier method.
32. Find the maximum of f ( x, y, z )  4 x 2  y 2  z 2 subject to the constraint
2 x  y  z  4 using Lagrange’s multiplier method.
33. A cylindrical tank with open top is to be constructed with least surface area but
with a total volume of 64 cm 3 . Find the dimensions of the cylinder.
34. Determine the minimum value of f ( x, y)  2 x  3 y, subject to
2 ln x  3 ln y  10 using Lagrange Method.
INTEGRATION
35. Integrate the following:

1
2
 ln( x  2)dx
0




3
( x 2  2)( x  1)
10
dx
2x  1
dx

2 x3
3
 x x  1dx
1
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x2  x  2
 x 2 ( x 2  1)dx
2
3


2x  5
dx
 2
0 x  4x  8
1
1
1


0 1  9e

∫

∫

∫(
x
(
dx
(
Answer:
)
)
(
) (
)(
)
)
36. Prove that ∫
. Do not integrate. Use derivative.
37. Using substitution method integrate: ∫(
)√
1
38. Find the value of the constant a where  ( x 2  2 x  a)dx  2
0
39.∫
(
)
 x 1
40. Is the integral 
dx convergent or divergent? Prove your answer
2x3
41. ∫
⁄
42. Find the cost function C (x) if the marginal cost
C ( x)  
x3
 10 x 2  120 x, 0  x  45 given that the fixed cost is $120.
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43. The marginal Cost function is C ( x)  3x 2  6 x  100, 20  x  30 . What is the
increase in Cost if the production level increases from 25 units to 26 units.
44. Find the equilibrium price and quantity, consumer surplus and producer surplus
where D( x)  0.03( x  50) 2 is the demand function and S ( x)  0.03x 2  x  3 is
the supply function, 0  x  50
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45. Find the particular solution of the differential equation xy 2 
 1 dy
, where
x  2 dx
( )
46. The population of Canada is 30 million at present. The population is growing at
an annual rate of 2.5%. Also in addition, new immigrants arrive at the rate of
50000 per year. Write a differential equation model. Solve the equation. How
long will it take for the population to reach 80 million?
47. Determine the logistic growth model function given by the differential equation
dN
 0.05 N (1  N ) where N (0)  0.8 , where t is measured in years. Without
dt
using calculator show that when N  0.9 , t  20 ln(1.25)
(
) ( )
48. Solve the logistic growth model
. Sketch the graph
of the solution.
49. Find the particular solution of the unlimited growth model
dM
 0.12M , M (0)  120 . Also, find the value of t when M  10173 .
dt
0.05t
dP
50. The weekly profit of a company is given by
 10e 
, 0  t  52 where t is
dt
measured in weeks, and P is measured in dollars. The profit when
is zero.
Find the profit function in terms of t. What is the profit when t=15 weeks.
51. A company installs a new computer which saves $M during the first t years. The
marginal savings is given by M (t )  20000e 0.02t . Find the total savings M from
2 to 5 years.
52. Find the present value of a permanent endowment fund needed to generate
annually $2000 for ever at a continuous interest rate of 12%.
( )
(
)
53. Find the approximate sum
54. Sketch the graph , shade the area and find the area enclosed between
f ( x)  ( x  2) 2  8
and y  x  4 .
55. Sketch the graphs and shade the area between the curves
f ( x)  x 2  2 x and g ( x)  6 x  x 2 . Find this area.
56. Determine the area enclosed between f ( x)  5x  x 3 and y  x . The graph is
given below.
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