Midterm 1 Review
Answer each of the following:
√
1. Is {3, 57 , −271, 2} ⊆ Q? Why or why not?
2. What is {1, 7, 8} − {1, 8, 9}?
3. Is [0, ∞) ⊆ R − {π} a true statement? Why or why not?
4. Is [−17, ∞) ⊆ (−17, ∞) a true statement? Why or why not?
5. Find 3 things that are wrong with the following statement:
[2, −1] ∈ [−∞, ∞)
6. Suppose f : N → R is defined by f (n) =
1
n2
(a) n is an object of which set? (Rational numbers, Integers, Real numbers, or Natural numbers?)
(b) What is f (2)?
(c) What is f (−3)?
For #7-10, state if the sequence is arithmetic, geometric, or neither.
7. 3, 30, 300, 3000, ...
8. 1, 4, 9, 16, ...
9. 27, 21, 15, 9, ...
10
10. 40, 10, 10
4 , 16 , ...
For #11-14, find the indicated term:
11. If an = 3n2 − 2, find a8
12. If an = 3 − n2 , find a10
1
13. What is the 27th term in the sequence −10, −7, −4, −1, ...?
14. What is the 23rd term in the sequence 48, 24, 12, 6, ...?
For #15-23, find the indicated sum:
15.
12
P
4
i=1
16.
15
P
(−2i + 1)
i=1
17.
∞
P
i=1
18.
∞
P
4( 32 )i−1
( 43 )(4)i−1
i=1
19.
∞
P
i=1
4
3i−1
20. What is the sum of the first 83 terms for 91 + 86 + 81 + 76 + ...?
21. Find the sum of the following infinite series: 81 + 27 + 9 + 3 + ...?
2
22. Find the sum of the following infinite series: 25 + 15 + 9 +
23. Find the sum of the following infinite series:
1
2
27
5
+ ...
+ 1 + 2 + 4 + ...
Answer each of the following:
24. You’re on vacation and have brought with you 3 pairs of shorts, 4 shirts, and 2 pairs of shoes. How
many different outfits can you make?
25. There are 18 jelly-bellies in a jar, each are different flavors. How many different ways can you select 4
jelly-bellies?
26. There are 100 sales people at a large corporation. The president of the corporation has decided she
will select a head of sales, assistant to the head of sales, and secretary from the pool of 100. How many
different options does the president have?
27. If you have 4 different gifts and you want to give each one of your 4 best friends one of them, how many
different ways can you give your gifts?
28. You work for a large corporation that likes to give gifts to politicians. There is a pool of 12 different
gifts that you can give and you have to select a specific gift for three different people. How many ways
could you give the gifts?
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29. You just got your first credit card and need to choose a four-digit pin number.
(a) Assuming you must use 0 − 9 for each digit but there are no other restrictions, how many possible
pin numbers are there?
(b) You’ve decided that you want a pin number where the four digits are unique (no repeated digits).
How many possible pin numbers are there if you make this restriction?
30. There are 10 people interested in learning how to use a new interactive technology, but the instructor
can only train 3 people. How many ways might she choose the three people?
31. You’re at the gym and need to work out your arms, abs, and legs. There are 3 machines for arms, 2 for
abs, and 6 for legs. You only have time to use one machine for each area (arms, abs, and legs). How
many different ways could you select the machines you will use?
32. What is
12
5
?
33. Write (x − 3)4 in expanded notation using the binomial theorem.
34. If f (x) = 2x − 3 and g(x) = x2 − 3, then what is:
(a) f ◦ g(x)
(b) g ◦ f (x)
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For #35-36, state the implied domain:
35. f (x) = x2 − 3x − 10
36. h(x) =
x2 +5x−14
x+3
37. Below is the graph of a function f (x). Use the graph to answer the following questions:
(a) What is f (5)?
(b) What is the domain of f (x)?
(c) What is the range of f (x)?
(d) State all x−intercepts as ordered pairs.
(e) State all y−intercepts as ordered pairs.
38. Is the picture below the graph of a function? Why or why not?
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For #39-45 graph and state the domain and range of each.
39. f (x) = x
40. f (x) = 2
D=
D=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
41. f (x) = x2
D=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
R=
R=
R=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
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42. f (x) = x3
43. f (x) =
1
x
44. f (x) =
1
x
D=
D=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
D=
2with variable names? indicated units? labeled points with coordinates? titled your graph?
have you...labeled the axes
R=
R=
R=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
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45. g : [−1, 2) → R where g(x) = x2
D=
R=
Use graph transformations to graph #46-47 and state the domain and range of each. Make sure to explain
the transformation in words as well.
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
46. h(x) = − 12 x3 − 2
D=
R=
47. f (x) = (x − 2)2 + 3
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
D=
R=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
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