MATH-UA 009-006: Algebra and Calculus
Midterm 1
Thursday, October 5, 2017
Name:
This exam is scheduled for 75 minutes, to be done individually, without calculators, notes,
textbooks, and other outside materials. You can detach the last sheet for scratch work; do
not detach any other sheets from this exam.
Show all work to receive full credit, except in multiple choice probelms.
Mark an “X” next to your section
X
Section
TA
Class Time & Location
007
Maiqi Ye
T 8:00am-9:15pm, 25 W 4th St, C-8
008
Maiqi Ye
M 8:00am-9:15am, 7 E 12th St, LL33
009
Rui Jiang
T 8:00am-9:15am, 25 W 4th St, C-13
010
Rui Jiang
M 9:30am-10:45am, 7 E 12th St, LL27
“We learned about honesty and integrity—that the truth
matters...that you don’t take shortcuts ... and success doesn’t
count unless you earn it fair and square.”-M.O.
I pledge that I have observed the NYU honor code, and
that I have neither given nor received unauthorized assistance during this examination.
Signature:
Problem
Points
MC
/36
FR 1
/6
FR 2
/12
FR 3
/5
FR 4
/6
Total
/65
Math-UA.009-006.A
Multiple Choice
(3 points each) Please clearly write your answer in the box next to the question.
You need not explain your answer. No partial credit will be given.
1. Which of the following is the domain of the function f (x) =
√
x2 − 1 ?
(D) [1, ∞)
(A) [−1, 1]
(B) (−∞, −1] ∪ [1, ∞)
(C) (−∞, −1) ∪ (1, ∞)
(E) None of the above
Problem 1
2. Solve the following equation:
2x2 + 20x = −32.
(A) x = −2, −8
(B) x = 2, 8
(C) x = 4
(D) x = −8, −32
(E) None of the above
Problem 2
2
Math-UA.009-006.A
3. Simplify the following expression:
50 − 2x2 x2 + 7x + 10
÷ 2
.
x+3
x + 5x + 6
2(5 + x)
x+3
1
(B)
2(5 − x)
(C) 2(5 − x)
(A)
(D) 2(5 + x)
(E) None of the above
Problem 3
4. The graphs of the functions f (x) and g(x) are given below. Using the graph, solve the
inequality
f (x) ≥ g(x).
y
(A) [−3, 3]
4
f (x)
3
(B) (−∞, −3] ∪ [3, ∞)
2
1
(C) x = −3, 3
x
0
−4
−3
−2
−1
0
1
2
3
4
−1
−2
(D) (∞, −1] ∪ [2, ∞)
g(x)
−3
−4
(E) None of the above
Problem 4
3
Math-UA.009-006.A
5. Use the graph of the function f (x) below to find the solution to f (x) = −3.
(A) x = −1, 2
y
4
f (x)
3
(B) x = −3
2
1
(C) [−1, 2]
x
0
−4
−3
−2
−1
0
1
2
3
4
−1
(D) (−1, 2)
−2
−3
−4
(E) None of the above
Problem 5
6. In which of the following intervals is the function h(x), whose graph is below, decreasing?
y
(A) (−∞, −2] ∪ [3, ∞)
4
3
(B) (−∞, −3] ∪ [2, ∞)
2
1
(C) [−2, 3]
h(x)
x
0
−4
−3
−2
−1
0
1
2
3
4
−1
−2
(D) [−3, 2]
−3
−4
(E) None of the above
Problem 6
4
Math-UA.009-006.A
7. Find the domain and range of the function g(x) whose graph is given below.
y
(A) Domain: [−3, 3]. Range: [−2, 3]
4
3
(B) Domain: [−2, 3]. Range: [−3, 3]
2
(C) Domain: [−1, 3]. Range: [−1, 3]
1
(D) Domain: [−3, −1]. Range: [−2, 2)
−3
−2
−1
(E) None of the above
x
0
−4
0
1
2
3
4
g(x)
−1
−2
−3
−4
Problem 7
8. Simplify the expression
(A)
y9
8x3
(B)
y6
6x
6x5 y −1
3x4 y 2
−3
.
y −6
8x−1
8x3
(D) 9
y
(E) None of the above
(C)
Problem 8
5
Math-UA.009-006.A
9. We would like to find the equation of a line that is parallel to the line described by
y + 7x = 1 and goes through the point (4, 1).
Which one of the following equation is the correct equation?
1
11
(D) y = − x +
7
7
(A) y = 7x − 27
(B) y = −7x + 29
1
3
(C) y = x +
7
7
(E) None of the above
Problem 9
10. Which one of the following is the graph of the line that corresponds to the equation
4x + 2y = −6 ?
(C)
(A)
y
y
4
4
3
3
2
2
1
1
0
−4 −3 −2 −1 0
−1
1
2
3
x
0
−4 −3 −2 −1 0
−1
x
4
−2
−2
−3
−3
−4
−4
(B)
1
2
3
4
1
2
3
4
(D)
y
y
2
2
1
0
−4 −3 −2 −1 0
−1
1
x
1
2
3
0
−4 −3 −2 −1 0
−1
4
−2
−2
−3
−3
−4
−4
−5
−5
−6
−6
−7
−7
−8
−8
−9
−9
−10
−10
x
(E) None of the above
Problem 10
6
Math-UA.009-006.A
11. Which one of the following is equivalent to the expression below, after factoring?
2x2 (x + 1)−3/5 − (x − 1)(x + 1)2/5 .
(D) (x + 1)−3/5 (x2 + 1)
(A) (x + 1)2/5 (x2 + 1)
(B) (x + 1)2/5 (2x2 − x + 1)
(C) (x + 1)−3/5 (2x2 − x + 1)
(E) None of the above
Problem 11
12. Let f (x) be a piecewise-defined function defined below.
x+3
if x < −1
f (x) =
−2x + 2
if x ≥ −1.
Which of the graphs below is the correct graph of f (x)?
(A)
(C)
y
y
9
4
8
3
7
2
f (x)
6
1
5
x
0
−4 −3 −2 −1 0
−1
4
3
−2
2
−3
1
1
2
3
4
−4
x
−4 −3 −2 −1 0
1
2
3
4
f (x)
(B)
(D)
y
y
4
9
3
8
2
7
1
0
−4 −3 −2 −1 0
−1
f (x)
6
x
1
2
3
5
4
4
−2
3
−3
2
−4
1
x
f (x)
−4 −3 −2 −1 0
1
2
3
4
(E) None of the above
Problem 12
7
Math-UA.009-006.A
Free Response
Please show all work and justification.
1. (6 points) Find all solutions of the equation
3
1
+
= 1.
x+1 x−3
If there is no solution, write “no real-valued solution”.
8
Math-UA.009-006.A
2. (a) (6 points) Solve the inequality
1 + |2x + 5| > 12.
(b) (6 points) Solve the inequality
x2 + x − 12
> 0.
x2 − 1
9
Math-UA.009-006.A
3. (5 points) Find the domain of the function
√
r(x) =
x2
x
.
+ 6x − 7
State the domain using the interval notation. Show all work.
10
Math-UA.009-006.A
4. (6 points) Rationalize the denominator in the expression below and simplify.
√
x−5
x2 − 9 − 4
11
Math-UA.009-006.A
Extra paper
12
Math-UA.009-006.A
Extra paper
13
Math-UA.009-006.A
Extra paper
14