MA140-01
Spring 2009
Calculus Fun [12 points possible]
The following problems are to be completed on separate paper and turned in for a grade. The completed problems
are due Friday (4/24/09) at the beginning of class. This assignment is to be your work and your work alone.
1.) Find each of the following limits if they exist. [5 points, 1 each]
(a)
2
4
2 24
lim
4
x
x x
x
→
+ −
−
4 4
( 6)( 4)
lim lim( 6) 10
4
x x
x x x
x
→ →
+ −
= + →
−
(b)
2
3 5
lim
2
x
x
x
→
+
−
( )
2 2
3 5 3 5 9 25 7
lim lim undefined
2 0
3 5 ( 2) 3 5
x x
x x x
xxx x
→ →
+ − − −
⋅ = → →
−−− −
(c)
3 2
3 2
5 4 13
lim
2 7 3 4
x
x x
x x x
→∞
+ −
+ + −
3 2
3 3 3 3
3 2
2 3
3 3 3 3
5 4 13 4 13
5
5
lim lim 7 3 4
2 7 3 4
2
2
x x
x x
x x x x x
x x x
x x x
x x x x
→∞ →∞
+ − + −
→ →
+ + −
+ + −
(d)
3
4
0
1
cos 1
3
lim
x
x x
x
→
− +
0
0
2
3 2
0 0
sin 0 0 cos 2 1
lim lim undefined
4 0 12 0
x x
x x x x
x x
→ →
− − + − + −
→ → →
(e)
5
1 ln
0
lim
x
x
x
+
+
→
5'
1 ln
0
5
5 5ln
lim ln ln ln 5
1
1 ln 1 ln
L H
x
x
xx
y x x
x x
x
+
+
→
= = = = =
+ +
5
5
1 ln
0 0
lim ln 5 lim
x
x x
y x e
+ +
+
→ →
= → →
2.)
Find each of the following. [3 points, 1 each]
(a)
( )
4 2 3
3 5 2 1
dx x x x
dx
+ − + −
3 2
4 6 15 2
x x x
+ − +
(b)
( )
3
sin 5
x
d
e x
dx
(
)
3 3
3 (sin5 ) ( )(5cos 5 )
x x
e x e x
+
(c)
( )
3
cos(sin 4 ) 5sec 7
d
x x
dx
−
2 3
sin(sin 4 ) (4 cos 4 ) 15 sec 7 sec 7 tan7 7 4cos 4 si n(
sin 4 ) 105 sec 7 tan 7
x x x x x x x x x
− ⋅ − ⋅ ⋅ = − ⋅ −
3.)
Find the second derivative of
3 3
2 4
x y
+ =
. Simplify as much as possible. [4 points]
2 2
2 2
2 2
3
3 6 0
6 3
dy dy x x
x y
dx dx y y
− −
+ = → = =
2 2
2 2
2 2 2 2
( 2 )(3 ) ( ) 6
3 (3 )
dy
x y x y
dy x d y
dx
dx y dx y
− − −
−
= → =
2
2 2
2 2
2 2 4 1
2
2 4 4 4
6 6
6 6 6 2
3
9 9 9
x
dy xy x y
xy x y
d y xy x y
y
dx
dx y y y
−
−
− +
− + − −
= = =
