First exam practice sheet
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?
a. [pic]
b. [pic]
c. [pic]
2. Determine which of the given functions is a solution to the differential equation [pic] and show that it is a solution.
a. [pic]
b. [pic]
3. Find a family of solutions to each of the following separable differential equations
a. [pic]
b. [pic]
4. Find the exact solution to each of the initial value problems below..
a. [pic] [pic]
b. [pic] [pic]
5. Find the orthogonal trajectories to the family of curves [pic]
6. Sketch the orthogonal trajectories to the family of curves drawn, on the same graph.
[pic]
7. State equlibrium solutions, if any, to the differential equation whose direction field is drawn below.
[pic]
8. The family of curves below represent solutions to a differential equation. Identify the equilibrium solutions from the graph.
[pic]
9. The differential equation [pic] is represented by the direction field below.
[pic]
a. Sketch the solution curve that passes through the point [pic].
b. Use Euler's method with step size 1 and initial condition [pic] to estimate the value of y when x = 2.
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:
[pic]
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.
11. Sketch a direction field for the differential equation [pic] for [pic]
[pic]
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?
13. 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?
14. 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?
15. 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?
16. For each of the following sequences, determine whether it converges. If so, find the limit.
a. [pic]
b. [pic]
c. [pic]
17. Find an expression for [pic]and determine whether the sequence converges.
a. [pic]
b. [pic]
1.
Answers:
2.
a. Not a distribution because [pic] for some values of x
b. Yes, it is a distribution
c. Not a distribution because area under curve is [pic]
3. The solution is b.
4.
a. [pic]
b. [pic]
5.
a. [pic]
b. [pic]
6. [pic]
7.
[pic]
8. [pic]
9. [pic]
10.
a.
[pic]
b. 2.1684
11.
a. [pic]
probability [pic]
b.
[pic]
12.
[pic]
13.
a. [pic]
b. [pic]
14.
a. [pic]
b. 7.4 kg
15.
a. [pic]
b. 1.77 grams
16.
a. increase of 68%
b. 66.4 minutes
17.
a. converges to [pic]
b. diverges (oscillation)
c. diverges (infinite)
18.
a. [pic], converges to 0
b. [pic], diverges
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