Review Final Exam

EXAM 1
 

1. Give two examples of
 (a) An integer that is not a natural number.
 (b) A rational number that is not an integer.
 (c) A real number that is not a rational number.

2. Solve the fol lowing equations

3. Solve the following inequalities . give your answer in interval notation:

4. The diameter of a circle is the segment with endpoints (-2; 2) and (4; -4). Find the equation of this circle .

5. Find the radius and center of the following circles:

6. Show the points (-2,-2), (3, 3) and (4, 0) form a right triangle.
 (a) By computing the slopes of the sides .
 (b) By using the distance formula.

7. Find the equation of the line that goes through the point (-2, 1) and is
 (a) parallel to the line 2x - 3y = 5
 (b) perpendicular to the line 2x - 3y = 5

8. Find the average rate of change for the following functions in the given interval
 (a) f(x) = 3x^2 - 3 on the interval [-1, 1]
 (b) f(x) = 3x^2 - 3 on the interval [-1, 0]
 (c) f(x) = 3x^2 - 3 on the interval [0, 1]

9. Find the difference quotient for the following functions and simplify:

10. Given f(x) = x + 2 and g(x) = 4  - x2 find the following:

11. Find the inverse of each of the following functions. Verify that you found the inverse by checking
that

12. Use transformation on a basic function to sketch the graph of the following functions. State which
is your basic function and describe the sequence of transformations you performed in order to
obtain your graph.

13. The functions f and g are given by the following tables:

 (a) Find
 (b) Find the domain of
 (c) Find
 (d) Find the domain of

EXAM 2

14. A local peanut butter manufacturer has a daily production costs of

  where x is the number of units produced and C(x) is the cost in dollars. How many units should
be produced each day to yield minimal cost?

15. Find two quadratic functions, one that opens upward, one that opens downward, whose graphs have the given x-intercept:

 (a) ( -2; 0); (3; 0)
 (b) (1; 0); (4; 0)

16. Find all the real zeroes of the following functions:

17. Find a polynomial equation with real coefficients that has roots 1 -i, 3, and 1.

18. Find a polynomial equation with real coefficients that has 2 as a zero of multiplicity 2 and -2 as a zero of multiplicity one.

19. Perform the following operations and write your answer in standard form.

20. Solve the following equations:

21. Knowing that -1 -3i is a zero of the polynomial

find the complete factorization of f(x)

22. Find a polynomial of degree 4 with real coefficients that has -1, 1 and 2i as zeroes.

23. Of all possible rectangles with perimeter equal to 200 ft, find the dimensions of the one with
maximum area.

24. Use either long division, or synthetic division to verify that (x -4) and (x + 2) are factor of the
polynomial

Then factor the polynomial completely.

25. Solve the following inequalities

26. Find the exact value of cos for an angle in standard position whose terminal side contains the point (1,-4)
27. From a point on the bank of a stream to the top of a 50 ft high cliff, directly across the stream, the
angle of elevation is How wide is the stream?
28. Find the exact value of each ex pression :

29. The radius of the earth is approximately 3950 miles. Find the distance over the surface of the earth
from the North Pole to the South Pole.

30. An airplane, flying at an altitude of 6 miles, is on a flight path that passes directly over you. If
is the angle of elevation from you to the plane, find the distance between you and the plane when
(a)= 30?, (b)= 90?, (c)= 120 ?/font>