## Precalculus2 Practice Exam

Final Exam Time: 3 hours

1. If , and , give exact values for:
• .

• .

• .

• .

2. Find the angle between the vectors and .

Answer: The angle is approximately .

3. Using the given sketch of the function , Find values for a, b, c, and d.

• a=-2.
• b=3.
• .
• d=-1.

4. A kennel raises Doberman pinschers and German shepherds. The kennel can raise no more than 40 dogs and wishes to have no more than 24 Doberman pinschers. The cost of raising a Doberman pinscher is \$50, the cost of raising a German shepherd is \$30, and the kennel can invest no more than \$1500 for this purpose. Find a system of inequalities which describes the number of Doberman pinschers and German shepherds that can be raised by the kennel.

Answer: Let x be the number of Doberman pinschers, and y be the number of German shepherds. The system of inequalities is:

5. Find all angles between and that satisfy . Write you answers in the degree-minute-second format.

Answer: The angles are , and .

6. Find the partial fraction decomposition of

7. Let .
• Convert z to its trigonometric form.

• Using DeMoivre's theorem, write as a complex number in standard form (a + bi).

• Use DeMoivre's theorem to solve the equation

Write your solutions in standard form.

8. Verify the identity

Show all steps.

9. Consider the parametric equations , and .
• Eliminate the parameter and obtain a rectangular equation for the graph described by x and y above.

• Using you observations from the problem above, find a set of parametric equations for the ellipse with vertices at (4,7) and (4,-3), and foci at (4,5) and (4,-1).

Answer: The set of parametric equations is

10. Sketch the graph of . Include three cycles (periods) and label each asymptote.

11. Find the sum of the sequence given below. Show your work:

Answer: The formula for finding the sum of an infinite geometric sequence is . In this case a=7 and r=-3/7; thus the sum of the given sequence is

12. Find the all exact solutions for on the interval .

13. A manager must select 4 employees for promotion: 12 employees are eligible.
• In how many ways can the 4 be chosen?

Answer: The 4 employees can be chosen in 495 different ways.

• In how many ways can 4 of the 12 employees be chosen to be placed in 4 different jobs, if they are all well qualified for any of the 4 positions?

Answer: The 4 employees can be chosen in 11880 different ways.

14. If , and , find exact values for the following:
• .

• .

• .

15. Write an expression for the nth term of the sequence. (Assume n begins with 1.)

Answer: The nth term has the form

16. Find the equation of the tangent line to the parabola given by at the point (-1,1).

Answer: The equation of the tangent line is .

17. Find a unit vector in the direction of v= .

Answer: The unit vector is .

18. Use mathematical induction to prove the given formula for every positive integer n:

• When n=1, the formula is valid, because

• Assuming that

You must show that

To do this, write the following:

Combining the results from the two parts, it is concluded by mathematical induction that the formula is valid for every positive integer.