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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