Math 109 Review Problems
The following problems cover material which you may recall from previous course work in algebra and trigonometry. You may not recall all of these topics or how to do the problems precisely. Use your texts or notes from previous courses or the MATH 109 RESOURCE LIBRARY in the basement of the library outside Room 48 to help solve these problems if possible. Bring to class any questions you have on these problems - we'll use these as a foundation for our review.

1. The line L goes through the point (3,2) with a slope of -4.
   A) Give an equation for L.
    B) What is the X intercept of L ?
2. For this problem let f be defined by f(x) = 5x2 + 3.
    a) Find the following. Simplify you answer when possible.
            i) f(l)                       iii) f(l+h)
            ii) f(h)                     iv) [f(l+h) - f(l)]/h
    b) Find any number(s) z where f(z) = 23.
    c) For which values of x is f(x) < 23? Express your answer as an interval.

3. (15 points). USING INTERVAL NOTATION, express the largest set of real numbers that can serve as the domain of each of the following functions:
        a) f(x) = (4 - x2)/(x + 2)
        b) g(x) = ln[(4-x2)]

4. Suppose that f is defined by
        f(x) =  -1     when x < 0
                    2x       when 1 < x
Sketch a transformation figure and a complete graph of  f.    Determine the domain and the range of f.

5. Solve for x:
a) 3 x-2 = 3 7-2x
b) 4 3x = 8
c) log 3(x - 5) = 2
d) ln ( 30 ) - ln ( x ) = ln ( 6 )

6. Suppose that sin t = 2/5 and tan t > 0.    Find the following exactly:         a) cos(t)                 b) tan(t)

7. Assume that N(t) is the number of bacteria present in a culture after t minutes and that N(t) can be determined by the equation N(t) = Aekt. Suppose initially ( t = 0) there where 100 bacteria present and that after 4 minutes the number of bacteria present was 250.
                a) Find k ( approximately)
                b) Estimate the number of bacteria present after 10 minutes.

8. Let f(t) = cos(p t) and g(t) = sec( p t).
        a. What is the period of f? Explain your answer briefly.
        b. Sketch of the graph of f(t) and g(t), indicating the coordinates of any X-intercepts and vertical asymptotes on the interval [-2,2]

9. Prove the following identity

sin 3(t)     +     sin(t) cos2(t)   =    tan(t)
10. Evaluate the following exactly:
a) arccos(-\/ 2)         b) cos( sin-1 (2/5))            c) tan(arccos(2/3))

11. Suppose A is an acute angle with sin(A) = 5/13.
a Find cos ( A ).         b) Find sin (2A).         c) Find cos ( A + 30°)

12. When possible, express the solution in interval notation:
a) | 3x - 6| < 9 .         b) |4 - x| > 5.             c) |x - 10000| < -1.

13. When possible , find the slope of the line connecting each of the following pairs of points. [ Write "NO SLOPE" if appropriate.]
a) (2,-2)   (3,1)            b) (4,1)    (-3,-1)             c)  (-4,1)   (-3,-1)

14. Boyle's law states that , for a certain gas P*V = 320, where P is pressure and V is volume.
    (a) Draw a complete graph representing this situation.  Label your axes and write an equation for each asymptote.
    (b) If 8 < V < 40, what are the corresponding values of P?

15. The temperature on Wednesday morning was approximately linear in its relation to the time. At 8 A.M. it was 40 degrees fahrenheit and at 10 A.M. it was 64 degrees. Estimate the temperature on Wednesday morning at 8:40 A.M.

16. Let f(x) = x2 + 4x - 5.
        A. Find the axis of symmetry and the vertex of f.
        B. Sketch a graph of f labelling clearly the coordinates of the vertex and the X- and Y- intercepts.

17. Sketch a graph of the functions 1og2(x) and 2x. Indicate the coordinates of at least three points on each graph.

18. Old MacDonald has a farm ,and on that farm she has some sheep and a pasture with a 200 meter long stone wall. She wants to enclose a rectangular section of the pasture for a small sheep pen using the wall for one side and 140 meters of fencing she was given by her uncle Milo for the other three sides.

A. Let x denote the length of the fence that will be attached to the wall used as a side for the pen. Which of the following equations express the area of the pen, A, as a function of x.
a. A = x ( 70 - x)                     b. A = 2 x ( 140 - x)
c. A = x ( 140 - x)                   d. A = x ( 200 - (l/2)x)
e. A = 2 x ( 70 - x)                  f. A = 2 x ( 200 - x)

B. Find the dimensions of the pen she can make that will enclose the largest rectangular area.

19. Circle the polynomials below which have a factor of X - 2.
A. 2X2 - X - 6                     B. X5 -X4 - 2X2 + 3X - 5
C. X4  - X2  -12                   D. X20 + 1
E. X20 - 2                             F. (l/2)X20  - 219

20. Use some graphing utility to sketch a graph of the function f(x) = x4 - 4x3 - 2. On what intervals is f increasing? Give the coordinates of any local maximum points on your graph.