Download Review for test 3

MATH 2414, CALCULUS II
Review for test 3, Fall 2003
Date of test
Thursday, November 6 in class
Material covered
Chapter 6, section 7
Ch 7 sections 1, 2, 3, 4
Ch 8 section 1
Allowable materials
Calculator (no TI-89 or 92)
3”5” index card of notes
Trigonometry formulas handout
Sample problems
1.
Which of the following are probability density functions?
1
a.
f  x 
2
1
xe x
2
-4
4
8
-1
b.
0
0.5 x  1

 
f  x   0.5
0.5  4  x 

0
x 1
1
1 x  2
2 x3
-1
3 x  4
x4
1
2
3
4
5
-1
2
c.
0

f  x    1  x2
0

x  1
1
1  x  1
x 1
-2
-1
1
2
-1
-2
page 1 of 6
MATH 2414, CALCULUS II
Review for test 3, Fall 2003
2.
3.
4.
Determine which of the given functions is a solution to the differential equation
dy
x
 x 2 cos x  y and show that it is a solution.
dx
a.
y  x   sin x  x
b.
y  x   x sin x
Find a family of solutions to each of the following separable differential equations
dy
y
dx
a.
x2
b.
dy
 e x y
dx
Find the exact solution to each of the initial value problems below..
a.
b.
dy
 2 xy
dx
x 2  1 dy
 2x
y dx
y  0  5
y  0  1
1
xk
5.
Find the orthogonal trajectories to the family of curves y 
6.
Sketch the orthogonal trajectories to the family of curves drawn, on the same graph.
page 2 of 6
MATH 2414, CALCULUS II
Review for test 3, Fall 2003
7.
State equlibrium solutions, if any, to the differential equation whose direction field is
drawn below.
3
2
1
-4 -3 -2 -1
1 2
3 4 5
-1
-2
-3
8.
The family of curves below represent solutions to a differential equation. Identify the
equilibrium solutions from the graph.
3
2
1
1 2 3 4 5 6 7 8 9 10
2
9.
dy  x y 
    is represented by the direction field below.
The differential equation
dx  3 2 




    









a.
Sketch the solution curve that passes through the point  0,1 .
b.
Use Euler's method with step size 1 and initial condition y  0  1 to estimate the
value of y when x = 2.
page 3 of 6
MATH 2414, CALCULUS II
Review for test 3, Fall 2003
10.
The density function for a normal distribution with mean 60 and standard deviation 4 is
drawn below. If this represents the probability distribution for a random variable X, then:
52
11.
56
60
64
68
a.
Write an integral that represents the probability that the outcome of X is between 58
and 60. Evaluate it with a calculator.
b.
On the above graph, draw a rough sketch of the normal distribution with mean 60
and standard deviation 2.
Sketch a direction field for the differential equation
2  x  2,  2  y  2
dy x  2 y

for
dx
5
2
1
-2
-1
1
2
3
-1
-2
-3
12.
The waiting time for a checkout line at a large department store is described by an
exponentially decreasing probability distribution. The median waiting time is 3 minutes.
a.
Give the density function that describes this distribution.
b.
What is the probability of waiting less than 4 minutes?
page 4 of 6
MATH 2414, CALCULUS II
Review for test 3, Fall 2003
13.
14.
15.
16.
17.
A 500 L aquarium is filled with a salt water solution of .02 kg of salt per liter. Fresh water
is poured in at a rate of 5L/min. The solution is kept thoroughly mixed and the tank is
drained at a rate of 5 L/min.
a.
Find an expression for the amount of salt in the tank after t minutes.
b.
How much salt is in the tank after 30 minutes?
Krypton-85 is a radioactive isotope of Krypton, with a half-life of 10 years.
a.
If 10 grams of Krypton-85 leak into a laboratory, give an equation for the amount of
Krypton that will be present after t years.
b.
How much will be present after 25 years?
A bacteria population doubles every 20 minutes.
a.
By what percentage will it have grown after 15 minutes?
b.
How long will it take the bacteria to grow by a factor of 10?
For each of the following sequences, determine whether it converges. If so, find the limit.
2n  1
5n  2
a.
an 
b.
an   1
c.
an 
n
n 1
2n
2n  1
n
Find an expression for an and determine whether the sequence converges.
a.
 4 9 16 25 
1, , , , ,...
 3 7 15 31 
b.
 2 5 10 17 26 
 , , , , ,...
 3 6 9 12 15 
page 5 of 6
MATH 2414, CALCULUS II
Review for test 3, Fall 2003
b.
ANSWERS:
1.
2.
a. Not a distribution because
f  x   0 for some values of x
b. Yes, it is a distribution
c. Not a distribution because area

under curve is
2
The solution is b.





11.
2
1
3.
a.
y  Ae1 x
-2
b. y  ln  e  C  , C  0
-1
1
2
3
-1
x
-2
4.
-3
a. y  5e x
b. y  x 2  1
2
5.
12.
0


a. f  t    ln 2  ln 2 3t
 3 e
b. 1  24 3  .6031
y3  3  x  C 
6.
t0
t0
13.
a. y  10et 100
b. 7.4 kg
14.
7.
y  1
8.
y  2

y  0
a. Kr  10e.0693147 t
b. 1.77 grams
15.
a. increase of 68%
b. 66.4 minutes
9.
16.
a.
2
5
b. diverges (oscillation)
c. diverges (infinite)
a. converges to




    









b. 2.1684
10.
60
a.
17.
n2
a. an  n
, converges to 0
2 1
n2  1
b. an 
, diverges
3n
2
1
 x  60  32
e
dx

58 4 2
probability  0.1915
page 6 of 6