Math Fall Midterm Sample Questions Write the equation of by takemehome

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									Math 135 - Fall 2008                                                  Midterm Sample Questions

  1. Write the equation of this line in point-slope and slope-intercept forms.
     (Integer points have been emphasized with a dot)
                          y
                           5
                           4
                           3
                           2
                           1

        −5 −4 −3 −2 −1          1   2   3    4   5
                                                          x
                           −1
                           −2
                           −3
                           −4
                           −5


  2. Express the shaded region as the union of two intervals, then as the intersection of
     two intervals. Finally, suppose that x = −3 is included in the region and express this
     new region using absolute values, i.e. find c and d such that |x − c| ≥ d represents
     the region (−∞, −3] ∪ [5, ∞).

       −7 −6 −5 −4 −3 −2 −1         0   1    2   3    4   5   6   7

  3. Determine the radius and center of a circle from its equation:

                                            2x2 + 4x + 2y 2 = 0

     Does the point (0, 0) lie on the circle? Is the point (2, 1) an x-intercept? Find the
     y-intercepts.

  4. Find the equation of the line perpendicular to the line pictured in problem 1 and
     passing through its x-intercept.




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5. Find the domain of                             √
                                                 3 3−x
                                         g(x) =    √
                                                2− x+1
6. Find the domain of
                        y

                        5
                        4
                        3
                        2
                        1
                                                      x
     −5 −4 −3 −2 −1          1   2   3   4    5
                        −1
                        −2
                        −3
                        −4
                        −5


7. How many distinct real roots does this quadratic equation have?
   (Hint: compute the discriminant)

                                                          7
                                             x2 + 3x =
                                                          4

8. Compute the distance between (−1, 2) and (5, 3) and find the coordinates of the
   midpoint of these two points.

9. Write the equation in problem 7 in the form a(x − k)2 = h; that is, complete the
   square.




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10. Which of these are functions?




      A               B                         A             B



       A               B
11. Which of these are functions?
               y                                       y




                           x                                      x




              y




                           x




12. Several fruit flies (Drosophila melanogaster) have found their way into your kitchen
    and plot to reproduce exponentially. You first count only 5, but after 2 days you find
    15. Find a linear function f (x) which expresses your kitchen fruit fly population at
    time x. After how many days can you expect to be overwhelmed by a swarm of no
    less than 100 flies?

13. The half-life of cocaine is 1 hour. Supposing that you have ingested the minimum
    lethal dose of 1.2 grams, how long will you feel the effects of the drug, i.e. how long
    before you metabolize all but the usual effective dose of 0.080 grams? Round your
    answer to the nearest minute.




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14. Pictured is the graph of f (x). Find the x-coordinates of the relative maxima. Is there
    a global minimum? If so, at which x value does it occur? Is there a global maximum?
    If so, at which x value does it occur?
                         y

                          3
                          2
                          1
                                                    x
      −5 −4 −3 −2 −1           1   2   3    4   5
                          −1
                          −2
                          −3
                          −4
                          −5


15. What is h(3)?                         
                                           −3, −3 ≤ x < 0;
                                   h(x) =   x3 , 0 < x < 3;
                                             x, 3 ≤ x < 10.
                                          

    Is h(x) a function? Find its domain and range.
                √
16. Let f (x) = 3x2 − 11 and g(x) = x3 . Compute f (g(x)) and g ◦ f (x). Compute the
    difference quotient of g(x).

17. Solve                                           √
                                            x−3=        x−1
    (Hint: remember to check your answers)

18. Simplify:
                                                 1
                                                x+y
                                                    +x 2
                                                 1   1
                                                 x
                                                   +y

19. Write
                                               x3
                                            f (x) = √
                                              1 − x3
    as a composition of two or more simpler functions.

20. Solve and express the answer in interval notation:

                                           −5 ≤ −x + 4 ≤ 11




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21. Solve using the key number method:

                                                4x
                                                         >0
                                          (x − 1)(x + 3)

22. Given is the graph of
                                                      4x
                                      f (x) =
                                                (x − 1)(x + 3)
                          y
                     10
                      8
                      6
                      4
                      2

      −5 −4 −3 −2 −1          1   2   3    4    5
                                                    x
                   −2
                     −4
                     −6
                     −8
                    −10

   Use the graph of f (x) to verify your answer to the previous problem.




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