Calculus I 1210-90 Final Exam
Summer 2014
Name_________________________
Instructions. Show all work and include appropriate explanations when necessary. Answers unaccompa
nied by work may Hot receive credit. Please try to do all work iii the space provided and circle your final
answers.
1. (lfipts) Find the fr)llowing derivatives. Show your work below and circle your final answer.
(a) (4pts) D(x
3
—
9x + 6)
(b) (4pts) D
.(sinxcosx)
1
.
(Si
)(_sIt)L)
csx —srn5
(c) (4pts) D()
2+-qx’
—
2
lYx
(d) (4pts) D(sin (3x
4
—
x))
2. (8pts) Compute the following limits. Answers may be values, ±oo, or ‘DNE’. Show your work below
and circle your final answer.
(a) (4pts)
urn
(b) (4pts) lim
E+O
T
2
1
x
s
)y.
)(...4
)IL4A
2_
Xô
bc1
—
a l.wCu.(c
o
LIL4A.
yosihii’e.
S
!
x
x-o_ l’,cl
1
3. (6pts) Find the equation of the tangent line to the graph of y
-f-)
=
(x
—
i) at the point (2, 1).
=-.i.
(
)
1
t
f)
4-
‘a) ()
-
1 i-q(x----)
=
4. (6pts) Use the definition of the derivative to compute the derivative of the function fQr)
that is. compute
f(x+h)-f(x)
inn
h
=
6:
jg
i’v..”
L
1I
-
2-L +
—
k-go
5. (lOpts) Billy Joe wants to build a rectangular pig pen of area 8 square meters up against the side of
his cabin; only three sides of the enclosure are fence since the side of his cabin will form a fourth wall
(see picture below). What dimensions (labeled x and y in the picture below) should Billy Joe make
the pen to use the least amount of fence? Minimize the perimeter P = y + 2x subject to the constraint
xy = 8. Note: You must use calculus to get credit!!
CABIN WALL
M’’i-
).,
XI
A-g
y
F’
0
c.
1 —)
—
Ct
2
-=‘
so
2
x2
6. (2Opts) Consider the function
f(x)
(a)
(3pts) Find
= 3x
—
2Ox + 8
1(x).
N
1 O
(b) (3pts) Find
—‘°
f”(x).
(c) (3pts) Find the critical point(s) of
3
f.
Lcx-&o
-
Z 2(d) (2pts) Fill in the blank by circling the cect aEelow: f(x) is
2—
INCREASING
1.
f ?)
ASING
(e) (2pts) Fill in the blank by circling the correct answer below:
CONCAVE UP
at x
f(x)
is
at x = 1.
.f ‘()
VE DOWN
—
.
o
(f) (4pts) Classify each critical point you found in part (c) as a local minimum, a local maximum, or
neither.
f f’ Lz
2.-.
ic
0
X2-
ve.Lfla.cf
‘i-.
1(
&
bt4(
(g) (3pts) Find the inflection point(s) of
i
f.
_ l2-ôc
3
(,o
=
-
2.-)
1
I;, J.
0
-
i%’f
c-€4A
7. (6pts) An object is thrown upwards off a 100 foot tall building. Its velocity after t seconds is
v(t) —32t + 32 feet per second. How far off the ground is the object after 3 seconds?
Sv1)L
loo
-3+3)
C
SL-) -13,o
)—
3
(3)
-i() 3l(3)too
given
by
—l+jC
8. (6pts) Use the graph of y
f(x) below to compute
J f(x) dx
=___________
I t1o-7
3
_Lt
yf(x)
-4
9. (8pts) Find the following antiderivatives. Remember: +C!
(a) (4pts)
/(6x2
—
—
c51L1
5x +2) dx
(b) (4pts) /cos2xsinx dx
)
Note: 2
cos x is the same as (cos x)
.
2
10. (8p1s) Evaluate the following definite integrals using the Fundamental Theorem of Calculus.
(a) (4pts)
•/2(x3)
dx
I’
2.
-r
(b) (4pts)
/
I
——
L22
x(x
—
i) dx
‘
1
u
(o)
(if
11. (6pts) Evaluate the Riemann sum for f(x) = x
3 x on the interval [—2,2] using the partition of 4
subintervals of equal length with the sample points being the left-endpoints of each subinterval.
—
£x’
Co
2—
0
—,
i
2-
fig)
Zj
x
3
Y
4
12. (2Opts) Consider the region R in the first quadrant bounded by y
Figure A below is a rough sketch of the region R.
= Vx
2
+ 1, the i-axis, and x
(a) (lOpts) Find the volume of the solid obtained by rotating the region R around the i-axis
the washer nOUiod
=
2.
using
2..
V
()4
K)
=
]r(f+z)
1
r
fr
(b) (lOpts) Find the volume of the solid obtained by rotating the region R around the y-axis using
the method of cylindrical shells.
f
V
0
ç
-
__-I
(ix)
(x ‘ )‘
0
I,
3
13. (8pts) Consider the region S bounded by the curves y = 2x
. the i-axis, and x = 1 sketched in Figure
2
B below. Each integral below is the volume of a solid obtained by rotating S around a particular axis.
Match the correct axis with the expression for volume by writing the appropriate letter in the blank
provided. Each answer is used exactly once.
D
f
) dx
2
2x(2x
2 +
j x((2x
f
3)2
A. x-axis
—
32)
dx
B. y-axis
2(x + 3)(2x
) dx
2
C. x
1
2
iJ’
0
r(2x-)
dx
=
—3
D.y=—3
I
Figure A
Figure B
0
14. (6pts) Find the arc length of the parametric curve x
and t 4.
LJj4 LLt4*,)
)(‘Ui.::
2 + 4t + 9, y
2t
+ 3t
=
—
6 between t
=
0
tf3 3(t#’)
4-
(h_____
j
=
i (t’)H (tti)
X
f
t
t)
I
•1
(.1 +e
‘0
0
15. (6pts) A metal rod is located between i = 0 and x
density of the rod at location x is given by p(x) = 4
center of mass of the rod.
3 on the x-axis (units are centimeters). If the
—
x grams per centimeter, find the location of the
(XZ/
3
= i2-
0
0
jxe) -)()
Mr
1fr4S
(1%)
—
16. (l0pts) A tank is 4 feet tall, 4 feet wide, and 4 feet long; when viewed from the side, the tank has the
shape of a right triangle (see the picture below). This tank is filled with water which has a density of
60 lbs/ft
. Use an integral to determine much work is required to pump the water out over the top
3
edge of the tank. Give your answer in foot-pounds.
IL
ft
I
VoI
4ff
-tq’
‘E tEe F
2
q
ok&
I
L
4)
j14
Z b k (- k) ctL 4
)o
Lf
14’
W
(oL(-L)L
Jo
Jo
lo
6
—
°
=
(-o)(3z
(32%
zc o
(,Lj
3
+