Mathl2lO Midterm 3
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 use a scientific calculator for this exam (not programmable, not
graphing). 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)
2x
x 2 +1
1. Consider the following function:
.
Itis defined for the whole set of
real numbers.
a. (5 points)Compute the axis intercepts of the function.
>‘
XO
—
=>
2xo,e. >o
jo
0
x-intercept(s):
0
y-intercept:
b. (10 points)ln which interval(s) is the function increasing? And decreasing? List
all critical points.
(X)
1(14’
)
( :)
-.
-.1
X+
(I
x)
.
4
uC
0”’
‘x.4l
-
(1
k
jD/
c-
/jliC
LJj:
K-)
9
1():
.
) >0
Ow
cj?
xz_
/
-
)
(A
>0
j
lncr. If x lies in
9.
-‘
(—‘1 i)
decr. If x lies in (_. Cxj, -. I) (-1,4C.c)
Critical points:
1
()o
x.
(_;k
o+
)
s—
ci)
I
£:_____
C’..
c_><
,—
+
r
—---
CD
C
U
#—.-
x
0
.i
._
ci)
t
Th
)<
—
—‘,
.‘.j-
—
Ljr
U
C
C_—___.-
u
CD
-
><
f_
0_
0
ci)
0
-._i
U’-ct
-
.
0
.
U
U
0
I
D___o
.
><
—I-
-—---
-
0
\-C\jf\
-z:,N)
I
.
-—
0
D
(
-
v—c
I
U
>
.E
(
C
0
ci)
-
-‘
:2
CD
(1)
-c
4-,
I_fl
C
4-,
0
4-,
ci)
15
-c
U
ii;
-
-d
-Si
S7
-
D
S
—--
2. (10 points) Consider the following function: f (x)=x
+3x—20 Use Newton’s
3
method to approximate a solution to the equation f (x)=0 in the interval
2 as your starting point. Give an approximation with an
1
[2,3] Use the point x
accuracy of two decimal digits.
.
.
f(x)
Recall that Newton’s method is given by the iterative formula
2
x
x= 2
-
Solution:____________________________
3. (15 points)Find the two numbers whose sum is 19, and whose product is the
maximum possible. Justify your answer.
c
+&
c,J
P
o
.
x J
rxir4e
+0
x
x
W
4
P
So
P P(x.)
(A
CM±CQ
0
P(
1 x)
p
4E v (+
=
o ution:
—
o.
4. Compute the following indefinite integrals:
a.(1O points)
f (x2__)dx
3
—
3
I
xZci\x
-
J
-
4
—
3
Solution:
b.(1O points)
3
X
+4)
2
x(x
d
x
f5
,kk
)( 4(
Cl
2X
4C
kk
—
Solution:
I 2.
(j)’
(Z
5. Evaluate the following definite integrals. Follow the instructions given for each
one. In case no method is specified, use any technique you want. Show proof of
your work.
a.(1O points): Use the second fundamental theorem of calculus to evaluate
fxdx
j
3
I
1
L
Solution:___________________
b.(15 points): Use substituti: t: evaluate
)J- (o,x
e/r.(
)
in(x
dx
)
x(cos(x
s
3
)
f2
)
x) 2.x CIX
iU
‘1
—?
I
Solution:\
-
C
D
i-1-
C
Li,
1\
0
cx
>
S
4-
>
—
(N
c’)
Q
‘x
is.
r
I.
C
I-I
STUDENT—PLEASE DO NOT WRITE BELOW THIS LINE. THIS TABLE IS TO BE USED FOR
GRADING.
Problem
Score
1
3
/15
4
/20
1Li
J1J