1210-90 Exam 2
v/
Name
Spring 2014
Instructions.
Show all work and include appropriate explanations when necessary. A correct answer
unaccompanied by work may not receive full credit. Please try to do all work in the space provided. Please
circle your final answers.
1. (l6pts) For this problem, consider the function
f(x)=j—.
(a) (4pts) Find
f’(x).
(i-x)(c)
—
±
(I
f(x).
(b) (4pts) Find the two critical points of
-ç
‘ii— x
‘
O f’
-
.
,
ic
-fcx) Xo
a
V.-o
C.
Ox-x(-)
(c) (4pts) Fill in the blanks: f(x) is decreasing on the intervals (
).
,
) and (
,
Note: + are acceptable answers.
(d) (4pts) Use the First Derivative Test to classily each of the critical points you found above as
a local maximum or a local minimum.
jL 1Dt4 wt
H
__t_
)
2—
0
2. (lOpts) Now consider the function
2
4
f(x)=x0+5x
(a) (4pts) Find
f”(x).
-
42Ox
4
5c
3
.f
(b) (4pts) Find the inflection point(s) of
f(x).
o
‘
1(
+
+
3)
.-
°
(c) (2pts) Fill in the blanks:
acceptable answers.
f(x) is concave down on the interval ( E
1
,
r3 ).
Note:
+00
are
3. (Gpts) Find the maximum and minimum values of the function
[0.2].
f(x)
=
—
3x2
+ 2 on the interval
ccecLt
f’&: I2e-?c
.i
\‘Iaximurn value=
(p
c
7/
Minimum value=
f)
x(-i)
L
2
A
(ç
/
_.,
L
/s in a large conical pile. The size of the pile is growing,
3
4. (Spts) Sand is being dumped at a rate of 30x ft
but the radius of the base is always equal to the height (ii = r in the picture below). Consequently,
h = irh
2
. How fast is the height of the pile growing when
3
the volume of the sand in the pile is V irr
the height is 10 feet?
h
;i:f1r
)
3cb-r
3c31D
7
SD/’
tV
5. (8pts) Find the equation of the tangent line to the following curve at the point (1, 0).
cos(y)
2
zy+x
TtfAS44hL1”L
‘A’A
.Y
1.
=
cA.-.s
S’
1- ZX€O
tL4
“
2-
6
-
ci
e
it
6. (5pts) Compute
12,
the second approximation to the root of
—
2x ± 3x
using Newton’s Method with initial approximation
—
xi =
1
=
0
0.
C
—
-
xh-- 3
(o)
2
7. (lOpts) Find the side lengths, labeled x and y in the picture below, of the right triangle of area 2 with
the smallest hypotenuse. Instead of minimizing the length of the hypotenuse, it is easier to minimize
2 + y
the length of the hypotenuse squared, H = x
, subject to the area constraint xy = 2. Note:
2
credit.!!
You must use calculus to get
xl
()
+
x+
32
32
-
x
1°
or
So
a4 ‘%f
S. 2-
1’
LOCA.L
1(.
&
I.c-41
64.L6:
i,V’iY-
LM(i6.
X2(—Z
-zf.
8. (l6pts) Find the indicated general antiderivatives: Remember +0!
(a) (4pts)
f(x2
—
4x + 9) dx
1 C
(1
(b) (4pts)
f
dx
(c) (4pts) f(sinx
—
1) dx
cf
(d) (4pts) f(x3
—
9)7(32)
dx
—)
3
V(x
j
3 2x
(31
(3jox
3
jç Io
9. (6pts) Approximate
f
2
(x
—
5) dx
using a
Riemann sum with 4 subintervals of equal length and the
sample points being the left-endpoints of the subintervals.
4.
7 (
2—fit
()L X
1
Li+t3) +(c
-I
+
-
10. (8pts) Use the graph of y
the followipg:
9
=
f(x) below and the area interpretation of the definite integral to evaluate
(a) j°f(x)dx=
f(x)dx=
3
(b) f
11. (7pts) Find the solution to the initial value problem
=3cosx+sinx+5
y(O)=3
Note: What this means is that. ou are looking for a function y
the above and the whose value at x 0 is 3.
çtut.
4SI’.
j SC3S
3
.
c)
j
=
f(x) whose derivative is given by
A.:
=
—
3c+s.tx*c.
-
jte
c(b4
c-f
4
C
3
(