Precalculus 2 Practice Exam

Final Exam Time: 3 hours

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

2. Find the angle between the vectors and .
3. Using the given sketch of the function , find the values for a, b, c, and d.

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

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?
• 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?
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.)

16. Find the equation of the tangent line to the parabola given by at the point (-1,1).
17. Find a unit vector in the direction of v= .
18. Use mathematical induction to prove the given formula for every positive integer n:

Mon Jul 21 15:30:58 MDT 1997