1210-90 Final Exam
Name
Summer 2013
1< Y
Instructions. Show all work and include appropriate explanations when necessary. Answers unaccompa
nied by work may not receive credit. Please try to do all work in the space provided and circle your final
answers.
1. (l6pts) Find the following derivatives. Show your work below and circle your final answer.
(a) (4pts) D(x
3
—
2 + 9)
4x
-
t (x ‘toc
(b) (4pts) D(/cosx)
l..
(/
o_X.Z,ix
zX
COSXJLktX
2-J5E
(c) (4pts) D x (
((1)
(d) (4pts) D(cos
7
/
(2))
=
—
-
Note: cos
7 (z
) is another way of writing (cos 7
2
))
2
(x
.
) (_ 5(
7
xosy
ñtx vs (x
i-))
(2)
‘t()
2. (8pts) Compute the following limits. Answers may be ±oo or ‘DNE’. Show your work below and circle
your final answer.
-
(a) (4pts) lim
X400
x 2 —x
4 5.2
iit
JZ)
°
x
-
-
-c
—
2
)(>o)
W
_s
r
—
00
x—,
2
-Sx
?(‘
Y
x
-
X-)
I”M
(b) (4pts) lim
x—*o
)(( I
11144
lxi
)(
(UL&
I(I
—
—
X(i
‘C.
l1kk
1J
.\‘Z-
—)‘-1
24
3. (6pts) Use differentials to estimate the increase in the area of a circle when the radius increases from
.
2
4 cm to 4.25. Remember, the area of a circle is A = rr
A-trr
-7
2—rrr
=
2)(.ic)
4. (6pts) Use implicit differentiation to find the slope of the tangent line to the ellipse at the point (1, —3).
i
Din-. øotvu
9
±
y
9
=
2
tdts
x-i)j
-o
2-.
K+
-3
2-
c)+ (-
0
2 with the smallest perimeter is the square with sides of
5. (lOpts) Show that the rectangle of area 100 cm
minimizing
the
perimeter
P = 2x + 2y assuming that the area A =
= 100
length 10 cm. Do this by
calculus
to
get
credit!
. Note: You must use
2
cm
x
P1x* Zj
—
co
1’-
zo
cvthcd
o
Ib
‘(=2
x
-
a
t
4 P
—
2
2
6. (l3pts) Consider the function
f(x)
(a) (4pts) Find
=
4+X2
f’(x).
-
(I
(c)(x)
*
(b) (4pts) Find the critical point(s) of
f(x).
-
LU
0
9 xj
-
f(x) is increasing on the interval (
(c) (2pts) Fill in the blanks:
acceptable answers.
F’
/3
I-
).
Note: +oc are
+
-2-
(d) (3pts) Which critical point is a local minimum?
2-
7
7. (6pts) Consider the function
(a) (4pts) Find
.
y
f”(’).
(b) (2pts) ill in t e blanks:
acceptable answers.
o e
f(x) is concave up on the interval (Zf
, f ). Note: +oc are
6
t’4ic
strt
4”
0
2 on the interval [—1, 5] using the partition of 3
8. (6pts) Evaluate the Riemann sum for f(x) 4 x x
subintervals of equal length and the sample points being the left-endpoints of each subinterval.
—
—
-I
t
2..
A
3
)
t3))
<ED
I
3
)
9. (l2pts) Suppose F(t) is an antiderivative of the function f(t)
2—
(a) (2pts) Then F’(t)
=
_S
=
cos
(/).
()
(b) (2pts) According to the Second Fundamental Theorem of Calculus,
x
cos(/)dt=F(
)
)-F(
1
.
(c) (3pts) Use the above or the First Fundamental Theorem of Calculus to find
rx
d
cos(./)dt
37
-
(d) (5pts) Find
fcos() dt
10. (l5pts) Find the following antiderivatives. Remember: +C!
(a) (5pts)
(b) (5pts)
(x
f
x(x
+ 2) dx
+
9)6
dx
A-2X
4u-
=
-M
X c)(
L+C
-l
-
(c) (5pts)
f
cos xVi + sin x dx
—
Z.
1L
u— L+s’x
4
1-c
11. (26pts) Consider tile region R ill tile first quadrant bounded by y
a rough sketch of the region R.
4x and y
=
x2.
(a) (5pts) Find the coordinates (x. y) of the point of intersection of the curves y
labeled P in the sketch below.
Figure A below is
=
4x and y
=
x2
(b) (5pts) Find the area of the region R.
c
A
-x
I
—
-
=32--
(c) (8pts) Find the volume of the solid obtained by rotating the region R around the x-axis.
V
J(-)j
T (i-j
(d) (8pts) Find the volume of the solid obtained by rotating the region R around the
2-
liEl.
2-’
x-I
DOct Vp’,k4
40.
p
)
Figure A
5
12. (8pts) Find the arc length of the curve defined by the parametric equations
I
y
=
=
1
(2t + 1)3/2
forO<t<2.
L c+
-
(iL
13. (6pts) A 1 meter-long spring requires a force of 5 Newtons to compress it 5 centimeters. How much
work is required to compress the spring from it natural length to a length of .8 meters? Remember:
springs satisfy Hooke’s Law: F = ki, where k is the spring constant.
toc.
=
14. (l2pts) A cylindrical tank is filled with water. The tank is 5 feet tall and has a radius of 2 feet at its
rim. Use an integral to determine much work is required to pump the water over the top edge of the
tank? To simplify your calculations, use the symbol p to denote the density of water in lbs/ft
.
3
2ft
4
c(4,
Rf
[bc--tui&.
L
5 ft
WjL
4-
sJ, ‘4:
Dts--
W
LAL
6
[ob-c4-L
L.