1210-90 Exam 1
Fall 2013
Name
Show all work and include appropriate explanations when space is provided.
Instructions.
answers unaccompanied by work may not receive full credit. Please circle your final answers.
Correct
1. (24pts) Compute the following limits. Be sure to show your work. Note: Answers can be values, +,
—cc, or DNE (does not exist). An answer of DNE requires some explanation!
(a) llm/z2+4
L/.
(x-3)(-)
(x-
x2_7x+12
(b) lim
x-3
x-43
‘
=
x—
‘1
(c) urn
-o
/ ,tu.(2c\f2..%\
_1
L —)(
xj
sin(2x)
/h
(d) lim
x—*1
i—f
(‘)()
%4O
(y-3b
—
z
—
1
)
1
X7
t”- L)(-1)
Ltu
i-I)
X-( t o
l’W
1
•--i )c—)U
Xt) I-II1-
>
_(4
412+9
(e)
lirn
x—+--oc
2
x
tL4
Ls.11
X’
•ii
-
(i’
I
‘
—
liM
2-
(f) urn
x-—
—4
—
Lii
?o5CiVC
1
o
4JI4Ll€
1
x
tL
2. (l2pts) Suppose
c is a constant and consider the piecewise-defined function
9
ç(
(a) (4pts) Compute urn
r— 1
9
(
j z-—4x+c,
—3cx+1,
x<1
x1
f(x)
ltLê..4
—
3+c
T
(b) (4pts) Compute
urn f(x)
x—* 1 +
I
I
—
(c) (4pts) Find the value(s) of
f(x)
d
l Em.,h’4u
—3
*
c that make f(x) continuous at x
o’
*t
3e-
I
4-
I.
=
1.
-
3c.
-3-i
I
-
3. (lOpts) Use the definition of the derivative to compute f’(l) for
the limit
f(1 + h) f(1)
f’(l) = urn
hO
h
0
f(x)
=
f4’c)
1
ii
3 + x; that is, compute
x
—
/
-
h
(+)
L))
J’-I
-
-H
3
l:
34
+I,i.
3
k
L
2
12-.’
4. (20 pts) Compute the following derivatives. There is no need to simplify.
(a) D(x
7
—
5
5x
—
.
iL
7
(b) D((x
3 +)_-
—
—
cos (Tx
(d) 1
D
(
2 + 9))
(e) D((j
)
5
-—
-
/
5. (6pts) Find
jf y
53
+x
—
sinx.
I5Xi) Co5X
3’x ks1L&
(33O+ctS)
—--
3
xt3x2
6. (8pts) Find the equation to the tangent line to the graph of the function
-x)
-
f(x)
at the point
=
-
(x
-3
)3
(-i)
X
4*
-ç-
:1
fta)
-
-ç 7cL)
+
‘
vik be
,
,
Z
-1.
7. (8pts) Let
)
19
f(x)=(
(a) (4pts) Find the derivative
1x)
f’(x).
7 L’1
O-1x)7
=
f(x)
Q-x
f(x)
(i-
below and fill in the blanks.
\
—3
3
(a) 4
lim f(x) =
(b) 1im_
+ f(x) =
4
(c) List all values of x, —6 <x < 5, where f(x) is
True (T) or False (F): f’(l) > 0.
(d)
F
not continuous.
True (T) or False (F): f’(—2) > f’(—3)
True (T) or False (F): f is differentiable at z
4
=
4.
x
horizontal?
2v€4V)
0-
(e)
(f)
—
I
‘e*4 (stqe
//
I
-.-
(b) (4pts) At what three points x is the tangent line to the graph of y
8. (l2pts) Examine the graph of the function
XC1—
-1
1