Mathl2lO Midterm 2
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 nj 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.(1O points) Compute the derivative of the following function using the definition.
3
f(x)_x—x
(x)
(recall the definition of derivative:
()3
f’(x)=lim
h
)
3
.4x
(3xL) -3x-)
22
)X
‘I
Solution:
2. Compute the derivatives of the following functions. Use the rules you prefer, but
make sure to state them clearly. You don’t need to simplify your answers.
a. (lOpoints)
f(x)=X+l
x—3
L4lQ:
()
x(x- —(‘)
2(i-3)
Solution:
b.(1O points)
x- 6x
(x 3)
(
—
cos(2x)
2
f(x)=x
CO)(Z)
(2x2 (‘
()
2
LJ
Solution:
2>co,’(2.>:) — 2.x ,cjv(2.)()
3. For the following functions, compute first and second derivatives.
a.(15 points)
-CJJA
2
f(x)=2sin(x)—-
+2—
2
2CQAX 4
f’(x)
ci’
6
1
f”(x) :_______________
4. (15 points) Find the equation of the tangent line to the curve
point (—1,1) Make sure to provide proof of your work.
3 at the
—xy+y
2
x
=
.
ji.
1
y
11
2. x
-
y
-
xy
21
-x)
:
.
y4i
21%
(.4 / !)
-
-.
C
.3
4c
X
Solution:
V
2
estimated at
cLj.
if the error in xis
3 at
2+(sin(x))
5. (15) Estimate the error in computing
±0.02
?‘(x) cix.
lx=Ool
4 cL
3
!
.
1
0.02
.
\
,oo)
2.
O.Ol5
(Cu
A:-q.OO2
-
Solution:
± O.O
6. (20 points) Consider a hemisphere of radius 5 ft made of ice. The thickness of the
ice is decreasing at the rate of
-
in/hr. How fast is the volume of the ice changing?
3
Recall that the volume of a sphere of radius r is V=u r
.
-
;,
CL
,2.OO
H
Oft:
cI
V
C,vJ
! Z5
Solution:
-
ZOO
7. (10 points) Consider the function 4
—x Determine its maximum and
3
f(x)=4x
minimum in the interval [0,4] Show proof of your work.
.
.
1 cc
Ct
9
o
{
(x)
0,
/
x)
XO
o
X3
o
Maximum:
2.7
Minimum:
0
STUDENT—PLEASE DO NOT WRITE BELOW THIS LINE. THIS TABLE IS TO BE USED FOR
GRADING.
Problem
Score
1
/10
2
/20
3
/15
4
/15
5
/15
6
/20
7
/10
ZLZZ