MAC 1140 Final Exam Review
Instructions: The final exam will consist of 12 questions and be worth 200 points.
The point value for each part of each question is listed on the individual problem. The total number of points each question is worth is listed below. Only
a basic calculator or scientific calculator may be used, although a calculator is
not necessary for the exam. NO GRAPHING CALCULATORS OR CALCULATORS ON A DEVICE (SUCH AS iPOD, CELL PHONE, ETC.)
WHICH CAN BE USED FOR ANY PURPOSE OTHER THAN AS A
CALCULATOR WILL BE ALLOWED! The concepts and types of questions
on the final exam will be similar to the previous tests, previous reviews, and this
review, although the numbers and functions may be different on each question on
the exam. There are 3 questions from each of the previous four exams, and the
questions on the final exam will be taken from the previous four exams. Each of
the 12 questions can be found in the following places:
1. Exam #1, Problem #4 (16 points)
2. Exam #1, Problems #5 - 8 (10 points)
3. Exam #1, Problems #5 - 8 (10 points)
4. Exam #2, Problem #2 (16 points)
5. Exam #2, Problem #3 (12 points)
6. Exam #2, Problem #4 (12 points)
7. Exam #3, Problem #2 (18 points)
8. Exam #3, Problem #4 (16 points)
9. Exam #3, Problem #6 (26 points)
10. Exam #4, Problem #1 (32 points)
11. Exam #4, Problem #2 (16 points)
12. Exam #4, Problems #3 & #6 (16 points)
1. Let f (x) be the piecewise defined function
(
−x3
if x < 0
f (x) =
2x − 4 if 0 ≤ x ≤ 4
(a) Find f (−1). (2 point)
(b) Find f (2). (2 point)
(c) Sketch the graph of f (x). (8 points)
(d) Find the domain and range of f (x). (4 points)
2. Determine what transformations have been done to the function g(x) = x3
to get the function f (x) below. Explain how you know. Then graph
f (x). (10 points)
f (x) = −(x + 2)3 − 1
3. Determine what transformations have been done to the function g(x) = |x|
to get the function f (x) below. Explain how you know. Then graph
f (x). (10 points)
f (x) = 2|x − 3| + 2
4. Solve each inequality.
(a) (x − 1)2 (x + 1)3 (x + 4)4 > 0 (8 points)
(b)
(x−1)2
(x+2)(x−2)
≤ 0 (8 points)
5. Determine the y-intercept of the function. State what function the end
behavior resembles and graph the function which the end behavior resembles.
Then graph the function f (x). (12 points)
f (x) = x4 − 2x3
Hint: positive: (−∞, 0) ∪ (2, ∞); negative (0, 2)
end behavior:
function f (x):
6. State whether each point not in the domain of the function is a hole in
the graph or a vertical asymptote. Determine the horizontal and/or oblique
asymptotes of the function. Then graph the function. (12 points)
f (x) =
4(x − 1)2 (x − 4)
(x + 2)(x − 2)(x − 4)
Hint: You need to determine the x-intercept; the y-intercept is (0, −1);
positive: (−∞, −2) ∪ (2, ∞); negative: (−2, 1) ∪ (1, 2)
7. Suppose f is the one-to-one function
f (x) =
3x
x−4
(a) Find the inverse of f . (12 points)
(b) Find the domain and range of f and f −1 . (6 points)
8. Suppose
f (x) =
2x
2
; g(x) =
2x + 1
x
(a) Find f ◦ g and state the domain. (8 points)
(b) Find g ◦ f and state the domain. (8 points)
9. Solve each equation.
(a) 42x−1 = 3 (6 points)
(b) 22x − 3 · 2x+2 + 25 = 0 (10 points)
(c) log3 (x + 10) = 2 − log3 (x + 2) (10 points)
10. (a) Write down the first five terms of the sequence. (8 points)
(−1)n 3(n − 1)2
(b) Write down the first five terms of the sequence whose general nth term
is given. (8 points)
n!
an = n
3
(c) The given pattern continues. Write down the formula for the general
nth term of the sequence {an } suggested by the pattern. (8 points)
1 3 9
27 81
,− , ,− , ,...
4 8 12 16 20
(d) A sequence is defined recursively. Write the first five terms of the
sequence. (8 points)
a1 = 4; an = −an−1 + 4n
11. (a) Write out the series. (8 points)
6
X
5k + 2
k=2
4k−2
(b) Write the series using summation notation. (8 points)
2
4
8
16 32
1
+
+
+
+
+
10 15 20 25 30 35
12. Find the sum of each series (using the sum properties and the sum formulas).
(8 points each)
(a)
25
X
k 2 − 4k − 10
k=1
(b)
500 + 100 + 20 + 4 +
4
+ ...
5
Sum Formulas
(i)
n
X
c=c·n
k=1
(ii)
n
X
n(n + 1)
2
k=
k=1
(iii)
n
X
k2 =
k=1
n(n + 1)(2n + 1)
6
(iv)
n
X
n(n + 1)
k =
2
k=1
3
2
Sum of the 1st n terms of an arithmetic series
(a)
n
X
ak =
k=1
n
[2a1 + (n − 1)d]
2
(b)
n
X
k=1
ak =
n
(a1 + an )
2
Sum of the 1st n terms of a geometric series
n
X
a1 rk−1 = a1 ·
k=1
1 − rn
1−r
Sum of an infinite geometric series
∞
X
k=1
a1 rk−1 =
a1
1−r