1210-90 Exam 1
Summer 2014
Name
/
Instructions.
Show all work and include appropriate explanations when space is provided. Correct
answers unaccompanied by work may not receive full credit. Please circle your final answers.
1. (22pts) Let
1/
41—4
—
(x-i)(-3)
Answers below may be values. DNE (does not exist), or ±oo. You must show your work.
—
f(x)
(a) (4pts) Compute lim
‘-+3
Sc)
X
flt
£4-
LU J’CV-
3 ji.*:.-
p I-ife
>°
Ve4fhLCLj
t’t
(x-i)
ttf
x
Y
Lct,v-eL1-c’r
L
y
f/x
+
f(x)
(b) (4pts) Compute lim
(.A4,I-&
L.+ c
ftI
>3J
•3) : <0 -f(x
(c) (4pts) Compute lim
x—3
f(x)
/‘
x)
(d) (4pts) Compute
/i
lim
f(x)
(- )
-* )
x
-
-
(e)
(2pts)
f(x) has a horizontal asymptote at y
(f) (2pts)
f(x) has a vertical asymptote at x
(g) (2pts)
f(x) is continuous everywhere
=
I
-i
(x) y
/‘
(.
=_____
3
except x
1
i
=
I
(list all values).
JEE
0
2. (l6pts) Compute the following limits. Be sure to show your work.
2.
LI
—2
2
x
(a) (4pts) lim
x—o5cosx
çLoS()
cLCS(O)
/,
x
+
2
x6
x-2
(b) (4pts) urn
x-+2
x-O
Xt3
X2-
sinzcosx
(c) (4pts) lim
(2
/i-
/
.
I
X
i
/(
(3)2_9
(d) (4pts) urn
/
—
X+O
)‘
x
X
3. (lOpts) Use the definition of the derivative to compute the derivative of
compute the limit
f(x + Ii) f(x)
f’(x) = lirn
h—O
h
-
Hint: Use the algebraic identity
ç
—
1
—
a
—
C
‘“-‘
—
-
-)o
(o
,—;c’
-
1i
+
72
)
—
—
a-b
c(+)
X+
X
f(x)
=
/; that is,
=
0
(I)
ci)
rj5
H
ci)
0
0
+
cjD
I
I
—
k
—
+c
N,
4
II
SE’
7-
/
/7
/
/
—
JN
+
H
A
-
(I)
ccci
ci)
=
0
ci)
-0
C
L
CN
+
=
‘
‘.1
s_)
‘_)
Ii
—‘I
H
0
1L1
1
--:-‘--
II
II
—
-
w
6. (l2pts) Consider the function
(a) (4pts) Find
f(x)
2+
f’(x).
2))_
L7L
=
7-
()()
I
J
(b) (4pts) Find the equation of the tangent line to the graph of y
/o)
=
f(x)
=
0.
frfy-c)
/
‘1o)
L
at x
j2fr i(<-o)
(c) (4pts) At what points x is the tangent line to the graph of y
=
f(x)
horizontal?
-
(7P
,
J
C
(i t?( 2j 2-
-
7. (9pts) An object moves along a horizontal coordinate line so that its position (in meters) at time t
(measured in seconds) is given by
s(t) = (t i) 12t + 6.
—
—
(a) (3pts) Find the velocity of the object at time t.
v(fls(t-)
3
3
3Z
(&-ij —i
-
-
3(-z-3
(b) (3pts) Find the acceleration of the object at time t.
aL
3
(c) (3pts) At what time is the object the farthest to the left? In other words, when is the object’s
position the smallest (i.e. the most negative)?
vft).
—
I
1
v”.oab
=
f(x)
is given below. Use it to answer the following questions:
f(x)
I
(a)
True (T) or False (F):
2
(b)
True (T) or False (F): f’(l)
f(x)
L, u-
3
8. (5pts) The graph of y
(c)
‘-K
i’s.
is continuous on (—4,4).
>
is not differentiable at x
4
f(3).
-