MATH171-15c
Time Allowed: 2 hours
MATH 171, FALL SEMESTER 2015
Version B
Name (print):
Signature:
Section No:
GUIDELINES
1. In Part 1 (Problems 1–14), mark your responses on your ScanTron form using a
No: 2 pencil. For your own record, mark your choices on the examination paper as
well . ScanTrons will be collected at the conclusion of the examination; they will not
be returned.
2. Calculators should not be used throughout the examination.
3. In Part 2 (Problems 15–19), present your solutions in the space provided. Show
all your work neatly and concisely, and indicate your final answer clearly. You
will be graded, not merely on the final answer, but also on the quality and correctness
of the work leading up to it.
4. Be sure to write your name, section number, and version letter of the
examination on the ScanTron form.
1
Part 1 – Multiple Choice ( 56 points)
1.
Consider the following curves:
x2 + x − 2
x2 − 1
(I) y =
(II) y =
x2 − x − 2
x2 − 1
(III) y =
x
sin(πx)
The line x = 1 is a vertical asymptote for
(a) all three
(b) (II) and (III) only X
(c) (I) only
(d) (III) only
(e) (I) and (III) only
2.
3.
sin2 (3x)
=
x→0
3x2
1
1
(a)
(b)
(c) 3 X
(d) 9
9
3
1
2
Compute lim
− 2
.
x→1 x − 1
x −1
lim
(a) 1
(b) +∞
(c) −∞
(e) Does not exists
(d) 0
(e) 1/2 X
√
2t2 − t − 2
.
t→−∞
2t + 1
√
(a) does not exist
(b) 1/ 2
4. Calculate lim
√
(c) −1/ 2 X
(d) 1
(e) −1
5 . A diver jumps off a diving board that is 16 feet above the water. The height of the
diver at time t seconds is given to be s(t) = −8t2 − 8t + 16. What is the velocity of
the diver when he hits the water?
(a) −32ft/s
(b) −24ft/s X
(c) −8ft/s
2
(d) −16ft/s
(e) 0ft/s
6.
(a) 3i−4j
7.
3
y = x + 2015?
4
(d) None of these X
(e) 8i−6j
Which of the following vectors is paralell to the line
(b) 4i+3j
(c) 6i−8j
Find the scalar projection (component) and vector projection of v = 5i − 12j onto
w = 4i + 3j.
64
48
vector projection = 169
i + 169
j,
64
48
vector projection = − 25 i − 25 j, X
64
vector projection = 25
i − 48
25 j,
64
48
vector projection = 25 i − 25 j,
64
48
vector projection = − 169
i − 169
j,
16
(a) scalar projection = 13
(b) scalar projection = − 16
5
(c) scalar projection = − 16
5
(d) scalar projection = 16
5
(e) scalar projection = − 16
13
8.
If 7x = 2y − 3 is the equation of the tangent line to the graph of f (x) at x = 5 then
f (x) − f (5)
=
lim
x→5
x−5
−7
7
2
(a) −7
(b)
(c)
X
(d)
2
2
7
(e) Not enough information to be determined
9.
Find the distance from the point (1, 3) to the line y = 3x − 3
19
(a) √
37
10.
9
(b) √
10
19
(c) √
10
3
(d) √ X
10
3
(e) √
37
The equation x5 + x2 − 2x = 3
(a)
(b)
(c)
(d)
(e)
has
has
has
has
has
a solution on
a solution on
a solution on
a solution on
no solution
[1, 2]
[0, 1]
[1, 2]
[0, 1]
11. Compute lim+
x−2
.
|2x − 4|
(a) −1/2
(b) 1/2 X
x→2
by
by
by
by
the
the
the
the
Squeeze Theorem,
Squeeze Theorem,
Intermediate Value Theorem, X
Intermediate Value Theorem ,
(c) −1
(d) 1
3
(e) does not exist
12. Consider the function
f (x) =
x2 cos(1/x) if x 6= 0
0
if x = 0.
The function f (x)
(a) is neither continuous, nor differentiable at x = 0,
(b) is not differentiable, but continuous at x = 0,
(c) is differentiable, but f 0 (x) is not continuous at x = 0, X
(d) is differentiable, and f 0 (x) is continuous, but not differentiable at x = 0,
(e) is differentiable, and f 0 (x) is continuous and differentiable at x = 0,
13.
Let f (x) and g(x) be two functions and h(x) = f (x) + g(x). Consider the following
statements:
I If limx→a f (x) and limx→a g(x) do not exist then limx→a h(x) does exist,
II If limx→a f (x) exists and limx→a g(x) does not exist then limx→a h(x) does not exist,
III If limx→a f (x) = −∞ and limx→a g(x) = ∞ then limx→a h(x) does not exist,
IV If limx→a f (x) = −∞ and limx→a g(x) = ∞ then limx→a h(x) does exist,
(a) None of the statements is true,
(b) Only I is a correct statement,
(c) Only II is a correct statement, X
(d) Only I and IV are correct statements,
(e) Only II and III are correct statements.
Hint: First go through all the statements I,II,III, and IV and decide whether they are
true or false, then answer a,b, c etc
1
d
=
2
dx sin (x2 )
14.
(a)
1
cos(x2 )
(b)
−2x cos(x2 )
sin2 (x2 )
(e) None of the above answers
(c)
X
4
−2x cos(x2 )
sin2 (x2 )
(d)
−1
cos(x2 ) sin(x2 )2x
Part 2 – Work Out (54 points)
In this part you need to show your work leading to the answer: NO WORK
(understandable to the reader), NO CREDIT.
15. a) (6 pts) Let f (x) be a function. State what it means that f (x) is differentiable at
a point a and define in this case the derivative of f (x) at the point a.
√
b) (8 pts) Using the definition of (a) compute the derivative of f (x) = x − 1. (No
credit will be given for using any other method, correct answer notwithstanding)
√
0
f (x) = lim
h→0
x+h−1−
h
√
x
√
√ √
√
( x + h − 1 − x)( x + h − 1 + x)
√
√
= lim
h→0
h( x + h − 1 + x − 1)
(x + h − 1) − (x − 1))
h
√
√
= lim √
√
h→0 h( x + h − 1 +
h→0 h( x + h − 1 +
x)
x − 1)
= lim
= lim √
h→0
1
1
√
= √
.
2 x−1
x+h−1+ x−1
c) (6 pts) Using the definition of (a) write down the the derivative of sec(x) at x = 0
as limit (do not compute that limit).
5
16. Using the laws of differentiation, compute:
a) (5 pts.)
d
x3 − 1
dx x3 + 3x2 − 5
d
x3 − 1
3x2 (x3 + 3x2 − 5) − (x3 − 1)(3x2 + 6x)
.
=
dx x3 + 3x2 − 5
(x3 + 3x2 − 5)2
b) (5 pts.)
d 3
x tan(1 + x4 )
dx
d 3
x tan(1 + x4 ) = 4x3 tan(1 + x3 ) + 3x6 sec2 (1 + x3 ).
dx
17. (8) Prove using the limitlaws lim x2 cos
x→0
Proof:
Since
1
= 0. Name every law you are using.
x3
1 cos 3 ≤ 1,
x
it follows that
−x2 ≤ x2 cos
1
≤ x2 , forallx
3
x
and since limx→0 x2 = 0 and limx→0 −x2 = 0, it follows from the Sqeeze Theorem that
1
lim x cos 3 = 0.
x→0
x
2
6
18.
a) (6 pts) State the exact definition of limx→a f (x) = L.
b) (5 pts) Using that definition prove that limx→1 2 − 3x = −1.
Proof: Let ε > 0 be given. We need to find a δ > 0 “which works”:
Choose δ = ε/3 then:
If |x − 1| < δ it follows that
|(2 − 3x) − (−1)| = | − 3x + 3| = 3|1 − x| < 3δ = ε.
19. (5 pts.) State the Squeeze Theorem.
7
Rough(Scratch) Work
This blank page is meant for rough work only; work presented here will not be graded.
8
QN
PTS
1–14
15
16
17
18
19
TOTAL
9