Mathl2lO Midterm 1
Name
Spring, 2016
Franco Rota
uid:
Instructions:
• Please show all of your work as partial credit will be given where appropriate,
and there may be no credit given for problems where there is no work shown.
•
All answers should be completely simplified, unless otherwise stated.
•
You may not use a scientific calculator for this exam. No other electronics are
allowed on this exam. Make sure all cell phones are put away and out of sight. If
you have a cell phone out at any point, for any reason, you will receive a zero on
this exam.
•
You may ask for scratch paper, but please transfer all finished work onto the
proper page in the test for us to grade there. I will fl grade the work on the
scratch page.
•
Notice that the space left for each question is sufficient, but possibly not
necessary, to answer the question.
(This exam totals 105 points, I will enter the grades as a total of 100 to allow for some
room for arithmetic mistakes)
1. Compute the following limits. Write DNE if the limit does not exist.
urn
a. (8 points)
x-.-2
(XL)
x
—
2
4
x+2
X-2
LA.
x
—)
-
(x41)
a
)(
-
—ç
L
C”
2
Solution:
b. (8 points) urn
x—.0
3:
x
X <0,
X
LJ
xl-x
0.
(2
X<.O
-
—
x
2.
t.
Solution:
c.(1O points) lim
x—*O
2
-
—
-k1
)
2
x+sin(x
X
_—
X
,AiJv(X
(2
4.
JM
X
X-)O
—
—
-
x
1.
/A/v
=
A- Q;jw’x
,í(X
.tfY’A
x -°
0
Solution:
-i
)
A
J
-
0.W’’- X_
-
/
I’
ri’,
x
Ij
3
3
-.
C
0
0
‘I
<0
r
-
4L
I
x
0
D
0
V
(I
><
‘I
v
I
><
ci
I
I
‘C
+1
3
D
—
2. Consider the function
f (x)=3x
a. (6 points) Compute hlITl f(x) (You’ll need the answer to this question in the
following parts, your answer should be a number, call that number L).
c
L 3
(1
L=
14
of 4,
b. (8 points) Fill in the blank in the following sentence: “If x is within
f(x) is within 0.003 of L’
then
LI K OOO
<ooo
<)
IIjx4I<0003
ix-ti <OOO
c
=
OOOl
3
2
it
Solution:
c. (8 points) Write down using
F>o
0<
>o
3
CAJJQc.{
ikc
/}1
4RA±
3—L
-f
3:
-I
c—ö language what it means that
9Xi/ti
zzz;>
:) •D)
1I
.
-
d)
lim3xL
-
Ic
9
:
—3_4
C
‘I’
0
C.
E
-
‘I
ic)
L
.
‘—I
I—
a
,cD
.
-ç
x
II
&
I I
H.
S
0<
-
R
‘K
L 3
o
ai
tDsçcs\
A
r
LJ’
II
x
C
F-.
C
0
4J
D
0
CM
0
CD
C
C)
-c
4J
C
0
If)
C
0
a
0
U,
CD
C)
4J
CD
C)
-o
CD
-J
><
0
44-
r
a
CD
C
0
C)
-c
•1-
CD
C
U
U,
4I
C
0
a
N
-d
—
—
al
&
cS3
c
‘
r
C
—
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III
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I
II
II
II
I
II
II
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II
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III
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F—F—F—f-f-f- —+—+--H—4--H
II
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Cs
4. (6 points) Evaluate sin(t), cos(t) and tan(t) at t=
.4
cos(t)t
sin(t)=
tan(t)=________
L
5. (3 points)RecalI that
Q denotes the set of rational numbers, i.e. it contains
those numbers that have the form
1,
0,
,
where a,b are two integers. Let
if xeQ
ifxEQ
Circle the letter corresponding to the correct statement (there’s only one):
cR
(f is not continuous at any point
a. f(1)=O;
d. f is not continuous at 0, but it’s continuous at
every other point of its domain.
b. f is continuous at 0;
2(1)-/1
4
:;=)
)
ciQ9
C’i”JiJ
_uJv%j
LLV.
(
Ai
JtY’
-> c.X “rrcj
LAAt 1Ats?A
-
fcQr
0
(iu
,vottn
A C&i”
JWJJW 3
.
STUDENT—PLEASE DO NOT WRITE BELOW THIS LINE. THIS TABLE IS TO BE USED FOR
GRADING.
‘
,L.
(e.g.
Problem
Score
-10
1
/52
2
/24
3
/22
_;_f
-
4
5
x
I3
;A
fA
Total Score:
/105
)