# K )1

```1210-90 Exam 1
)1
K
Name
Spring 2013
Instructions.
Show all work and include appropriate explanations when space is provided.
Correct
1. (2Opts) Compute the following limits. Be sure to show your work. Note: Answers can be values, +oo,
—oc, or DNE (does not exist)
(a) lun
1+z2
x-+O
COSX
ll”4.
-,LC-
teCXI
&gt;—o
k&amp;L
(b) urn
x2+z_
I
6
x—2
x-2
-‘
2
ti
ll
(X”L)(Xf 3)
5x2_5
urn
x—++ 3i
+x
ii s-)c--c
-II
—
.
1
r-+1 x2
—
1
2
IivkJ
)(-)1 x
-I
2
(e)
urn
z—*O
lIw’
)(-*10
C?
(d) urn
)(+3
•
x-,1-
X-z
)C -2-
(c)
—
cocx
x—,o
—
-
-
1
/A44JC
ci x’
2 •Lc
_.(_
x2
?c—L
—
•L
)1 X
V(-
I.
7N
)C-5o
1
L ve-.r ?-e4’
‘-.3
C”
N
C
H
JN
(12
+
C”
H
H
H
H
I
Ic’i
H
C.
HF
H
H
I-Th
1
—
4
x
.3
3. (lOpts) Compute the derivative of
compute the limit
f(x)
f’(x)
r(x)
X+k
1k
by using the definition of the derivative; that is,
=
=
f(x+h)-f(x)
urn
Ii
h—&divide;O
X
k -, 0
k
—k
k-’o
.
i!__
)hM
4. (lOpts) Find the equation of the tangent line to the curve determined by the equation
y+
2
x
—
p
=
8
at the point (1,2).
-
\i€k-
2?1
jv.
3
V
c cL.-s
r.
&divide;)(2
-)j 3x
.
2
)(-lj
tlb-L
e
1
tA1iJ 5(o
(I —2-
=:=9t,-l.
&amp;
Lr
—/
paSi
—i(?--/)
c
3
rn7k-
(iz-)
5. (lOpts) Use the graph of the function y
7...-
(a)
lirn
(b)
urn
(c)
—2 +
x-*-2
f(x)
f(x)
=
=
(d) The graph of
(e)
=
f(x)
provided below to fill in the blanks.
I,.
14J 6
f(x)
has a vertical asymptote at x
limf(x)=
6. (lOpts) An object moves along the x-axis in such a way that its position at time at time t is given by
s(t)
=
3 + 6t
—t
2
—
9t + 1.
Assume that the units of the axis are measured in meters and t is measured in seconds.
(a) Find the velocity (in meters per second) of the object at time t.
V(.L) S’(t)
—3-I74-
—7
(b) On what time interval is the object moving to the right? Fill in the blanks:
V)3-
&lt;t &lt;
3
-3(-+3) -3(3)/-i)
(,4A-44
/3
(c) When is the acceleration of the object negative? Fill in the blank: t&gt;
V ‘&amp;)
/
H) t1i
3
4
2.....
7. (lOpts) A snowball is being rolled to create the base of a snowman. Suppose the volume of the snowball
/s. How fast is the radius of the snowball increasing when the snowball
3
is increasing at a rate of lOx cm
Hint:
Assume the snowball remains perfectly spherical. Recall that the volume
cm
is 40
in diameter?
of a sphere is V = irr
.
3
v ir(
r
L,LO
t-f
lOlr
ij-(2-oj
8. (lopts) Consider the function
f(x)
=
(a) Find the linear approximation to the function f(x) = / at x
approximation is the same as the equation for the tanent line.
‘ix’=
LA1k4-
pc
—
-
X
7