Final Exam | Time: 3 hours |
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- If , and , give
**exact values**for:- .
- .
- .
- .

- Find the angle between the vectors and .
- Using the given sketch of the function ,
find the values for a, b, c, and d.
- 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.
- Find all angles between and that satisfy . Write your answers in the degree-minute-second format.
- Find the partial fraction decomposition of
- 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.

- Convert
- Verify the identity
Show all steps.

- 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).

- Eliminate the parameter and obtain a rectangular equation for
the graph described by
- Sketch the graph of . Include three cycles (periods) and label each asymptote.
- Find the sum of the sequence given below. Show your work:
- Find the all
**exact**solutions for on the interval . - 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?

- If , and ,
find
**exact**values for the following:- .
- .
- .

- Write an expression for the
*n*th term of the sequence. (Assume*n*begins with 1.)

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

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