1210-90 Exam 1
)1
K
Name
Spring 2013
Instructions.
Show all work and include appropriate explanations when space is provided.
answers unaccompanied by work may not receive full credit. Please circle your final answers.
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
>—o
k&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—÷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.
÷)(2
-)j 3x
.
2
)(-lj
tlb-L
e
1
tA1iJ 5(o
(I —2-
=:=9t,-l.
&
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-
<t <
3
-3(-+3) -3(3)/-i)
(,4A-44
/3
(c) When is the acceleration of the object negative? Fill in the blank: t>
V ‘&)
/
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
(b) Use your answer above to estimate the value of
3
T
ii
5
=
64.
Recall: The linear
£J
.L