1210-90 Exam 2
Name
Summer 2013
KY
Show all work and include appropriate explanations when necessary A correct answer
Instructions.
unaccompanied by work may not receive full credit. Please try to do all work in the space provided. Please
circle your final answers.
1. (25pts) For thisproblem, consider the function
3x
f(s)
—
3
5x
+
2.
(a) (4pts) Fin
c&)
Lc+_(ç
(b) (4pts) Find the three critical points of
f(z).
0= Ic— t6
ica point is a
(c) (4pts) Use the First Derivative Test to determine whether ea
a local maximum or neither.
,
x -L
L1_
x
0
—1
+
toce4I w-
xo
‘
(d) (4pts) Find
t
x or show that there are no inflection points. It is
(e) (4pts) Fin
acceptable to only list the x-values of the infection points.
x
C
)O
3o?
3
O—
—.
3O?c2_)
1
(f) (5pts) Find the minimum and maximum values of
C4
(
4
ç
f(x) on the interval [—1, 21.
i4
4
V4I14Q
0
1
I’.
“-
f” x.
,
11
1
inimum,
)
2. (8pts) A rocket launch is being watched by an observer 1000 meters away from the launch pad. When
the rocket’s height is 1000 meters, the observed angle of the rocket (labeled in the picture below) is
increasing at a rate of
radians per second. How fast is the rocket traveling at this time?
IL
“
.,
_
/
pj
3
I
ih
-
1:1
I
(
I
t
r.t
sc
rJ_
1 000m —I
L-
LAA’e44—
/
io 00
2-111-)
tODD
&
tDoo()()
i4)()
3. (5pts) Find the slope of the tangent line to the following curve at the point (1,2).
—
a1—s
f. 3(32
‘
2 +y
2xy
3
=
1
-
0
jc1
34
0
f!
3
4.
3
—
# /
0
(a) (4pts) Find the equation of the tangent line to theph of y
=
at the point (16,4).
Ct)
’
7
-f
-fo()
-?(
(b)(
Use part (a) above to estimate
5. (5pts) Find the value c guaranteed by the Mean Value Theorem for f(x)
[0,3]
=
2L3)’fLo)
2-
3c2
c-3
2-1—0
—
2x on the interval
3 of liquid. What are the radius
6. (8pts) A closed cylindrical can is being manufactured to hold 250ir cm
which
can
be made using the least amount
r
of
the
can
(labeled
below)
and height
and Ii in the picture
can
A
to a fixed volume v = 7rrh.
area
of
the
2irrh
subject
surface
=
2
2xr
Minimize
of material?
the
+
to
credit!!
calculus
You
must
use
Note:
get
A
-r- 2irrL
2c0
rZL
2
o
L -:
3A(r)
-hi-r-
OAtr)
Ct,.ec.4S
(E
(-
A (11
cI
t.
2r2irr()
—
—-
!!r r’qc
%..
d4_
hr-i-Iocowr =pi”
L
°
-
to
7. (l6pts) Find the indicated antiderivatives:
(a)
L
(d) (1pts)
3
2rr’t-
8.
(a) (5pts) Approximate
3
i
/
J—2
x dx using a Riemann sum with 5 subintervals of equal length and the
sample points being the right-endpoints of the subintervals.
1
A
ç..
1C
-
-1
-
I.
2
I
(b) (5pts) Find the exact value of
f
x dx. Hint: Use geometry and the area interpretation of the
—2
definite integral. What does the graph of y
=
x look like?
S
9. (l0pts) The pull of the Earth’s gravity is slightly less on the top of a tall building than it is on the
Earth’s surface. Suppose we drop a ball from the top of the Sears Tower (1730 feet). We estimate
that, as the ball drops, it experiences an acceleration of
a(t)
=
—31
I;
—
feet per second squared. The negative sign indicates that gravity acts downward.
(a) (5pts) Find v(t), the velocity of the ball in feet per second after t seconds.
S
) f1t) J(-3i-)e
L?4e4L
4j.O)
Vto)o
—
PLk.Pt,e4)
lvc
3‘
(b) (5pts) Find an equation whose ffition will give the irne it takes for the ball to hit the ground.
Do not attempt to solve.
t:).::O,
PL
S-,
10. (5pts) Compute
12,
Lt
o)z:I73o
JVlt)
-
the second approximation to the roo o
—2
=
using Newton’s Method with initial approximation x
0
=
1.
1
—
—
—
JLXhJ
-t
)
.
4
3.