Math 106 Preliminary Problems

The following problems cover material which you may recall from previous course work in algebra. 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 Calculus RESOURCE LIBRARY in the basement of the library outside Room 48 to help solve these problems if possible.
You might also look at these web resources. Review Algebra I and II materials (a web course)
Bring to class any questions you have on these problems - we'll use these as a foundation for our preliminary work.

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(1)                       iii) f(1+h)
            ii) f(h)                     iv) [f(1+h) - f(1)]/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. When possible, express the solution in interval notation:
A. | 3x - 6| < 9 .         B. |4 - x| > 5.             C. |x - 10000| < -1.

4. 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)

5. 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?

6. The price of a stock on Wednesday morning was approximately linear in its relation to the time. At 8 A.M. it was $40 and at 10 A.M. it was $46. Estimate this stock's price on Wednesday morning at 8:40 A.M.

7. 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.

8. 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.

i.. A = x ( 70 - x)                     ii.. A = 2 x ( 140 - x)
ii.. A = x ( 140 - x)                  iv.. A = x ( 200 - (l/2)x)
v. A = 2 x ( 70 - x)                  vi. A = 2 x ( 200 - x)

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