Math 107.04 Practice Final Exam

1. Solve the following inequalities . Express the solution set in interval notation.

2. (a) Find an equation of the line passing through (-2, 3) and (3,-7).
(b) Find an equation of the line passing though (1, 2) that is perpendicular to the line in (a).
(c) Is the line y = 2x + 3 parallel to the line in (a)? Why or why not?

3. The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce
300 chairs in one day. Assume there is a linear relationship between cost and the number of chairs produced.

(a) Find an equation that expresses cost as a function of the number of chairs produced.
(b) What is the y- intercept of this line , and what does it represent? Hint: It is a fixed cost.
(c) How many chairs must be produced in one day for the manager to break even?

4. Consider the following piecewise-defined function:

	(a) Sketch the graph of f , and state its range:
	(b) State intervals on which f is increasing, decreasing, or constant:

5. Find the domain of each function. State your answer in interval notation where appropriate.

6. Let f(x) = -x² + x + 1.

(a) Does f have a maximum or minimum value? Why?
(b) Complete the square to find the vertex of f.
(c) Find the average rate of change of f from x = -1 - h to x = -1 (for h > 0).

7. List the transformations applied ( in order ) to the second function below to obtain f.

8. A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the
equation for the final transformed graph. You do not have to sketch the graph of f.

(a) f(x) = sin^-1 x, shift right 4 units, shrink vertically by a factor of 2, reflect in the x-axis, and shift down 1 unit.
(b) f(x) = 3^x, reflect in the y-axis, stretch vertically by a factor of 5, reflect in the x-axis, and shift up 3 units.

9. A gardener has 200 ft of wire fencing. He wants to enclose a rectangular garden plot adjacent to his house, and then
divide it into seven smaller plots with fencing parallel to one side of the rectangle, as shown in the figure.

	(a) Find a function that models the total area of the seven plots:
	(b) Find the largest possible total area of the seven plots:

10. Express the function in the form f o g.

11. Let g(x) = x + 1 and h(x) = 2x² + 4x - 1. Find a function f such that f o g = h.

12. For each function f, compute the inverse function f ^-1.

13. (4 pts) Use long or synthetic division to find the quotient and remainder.

14. Find a polynomial P with real coefficients which satisfies the given conditions. If no such polynomial exists, explain
why. You do not need to multiply out your polynomial .

(a) P has degree 7, zeros 2, i, 3 - 5i, and leading coefficient 8.
(b) P has degree 4, zeros -1 (of multiplicity 3) and 1 + i, and constant coefficient 13.
(c) P has degree 5, zeros 1, 1 - i (multiplicity 2), and leading coefficient -6.

15. Which of the following polynomials exhibit the same end behavior and zeros as in the graph? Justify your selection(s).

(e) None of the above:

16. Find all zeros of the following polynomials. State the corresponding multiplicities.

17. Which, if any, of the following rational functions exhibit the same intercepts and asymptotes as in the graph? Justify
your selection(s).

(g) None of the above:

18. Write an equation representing each graph which has the given form.

19. Simplify :

20. Expand the following logarithmic expressions.

21. Combine the following logarithmic expressions . Evaluate the resulting logarithm (if possible).

22. Solve the following equations for x.

23. A certain type of bacteria thrives when immersed in water. If a population of 30, 000 of these bacteria are immersed for
30 seconds, their population grows to 120, 000. Assume that the population of bacteria follows the model: ,
where t is measured in seconds.

(a) Construct a function that models the bacteria population after t seconds of immersion.
(b) How long must the population be immersed to grow to 240, 000 bacteria?.

24. Find the six trigonometric functions of the following values. Use a trigonometric identity if necessary.

25. If where µ is in QIII, find the other five trigonometric functions of θ.

26. A man 6-ft tall is standing 5 ft from a lamppost. If the angle of depression from the lamppost to the tip of his shadow
is 60°, how tall is the lamppost?

27. For each function, find (i) amplitude, (ii) period, (iii) phase shift, (iv) domain, (v) range, and (vi) vertical asymptotes.

28. Consider all possible triangles ABC which satisfy the given conditions, and determine the indicated parameter. Use
trigonometric identities if necessary to evaluate associated functions. If no such triangle exists, explain why.

(a) A = 30°, B = 45°, c = 3, find a
(b) A = 30°, a = 100, b = 100, find c
(c) a = 9, b = 5,C = 120°, find c
(d) a = 24, b = 15, c = 30, find B
(e) a = 12, b = 6, , find A
(f) C = 30°, b = 22, c = 10, find a

29. To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured
to be 32°. One thousand feet closer to the mountain along the plain, the angle of elevation to the top is 35°. Find the
exact height of the mountain.

30. A tree on a hillside casts a shadow 50 ft down the hill. If the angle of inclination of the hillside is 22° to the horizontal
and the angle of elevation of the sun from the tip of the shadow is 52°, find the height of the tree.

31. A car travels along a straight road heading due east for 1 hour, then travels for 30 minutes on another road that leads
northeast. If the car has maintained a constant speed of 30 mph, how far is it from its starting position?

32. Prove the following identities.

33. Find the exact value of each expression , if it is defined.

34. Sketch a triangle and/or use an appropriate trigonometric identity to evaluate the following expressions.

35. Find all solutions of the following equations.