# 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.

1
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.

2
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

3
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

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

4
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.

5

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