MA140-01
2/8/09 Major Quiz 2 Review Problems
1.) Find
dx
dy
for each of the following.
(a) y = 3x
3
− 5x
2
+ 4x +13 (b) y = −3(x
5
+ 2x
4
) (c)
2
3
3
5
x
x
y+=
(d)
54
23
x
x
y
−=
(e)
2
1
2
1
2
1
2
−
−= xxy (f)
1
44
2
−
+−
=
x
xx
y
(g) y = (4x
2
− 1)(2x
3
+ x + 5) (h)
1
1
3
3
−
+
=
x
x
y (i) y = (2x
4
− 3x)(5x
2
− 2x + 5)
(j)
6
3
2
+
=
x
x
y (k) y =
5
2
13
+
−
x
x (l) y =
2
3
4
5
4
3
5
3
3
2
x
x
x
x−+−
2.)
Use the definition of derivative to find f'(x) for f(x) = 3x
2
+ 5x − 1.
3.)
Prove the following theorem.
"If
f
and
g
are functions and if k is the function defined by
( ) ( ) ( )
k x f x g x
= +
then if
'( )
f x
and
'( )
g x
exist,
'( ) '( ) '( )
k x f x g x
= +
."
4.)
Find an equation of each of the lines through the point (1, 2) that is tangent to the curve
y = 2x
2
+ 8.
5.)
Find the values of a and b such that f is differentiable at 2 if f(x) =
≥−
<+
212
2
2
xifx
xifbax
.
6.)
Find an equation of the tangent line and normal line to the curve y = 3x
4
− 12x at the
point (1, −9).
7.)
Use the definition of derivative to find f'(x) for f(x) = 2x
2
− 3x + 1.
8.)
Find
'( )
f x
for each of the following.
(a) f(x) = 5x
3
+ 2x
2
− 3x (b) f(x) = (3x + 5)(2x
2
− 3x +1) (c) f(x) =
7
3
32
5
−
+
x
x.
(d) f(x) =
24
53
x
x
− (e) f(x) =
4
2
13
2
+
−
x
x
9.)
Use the definition of derivative to find
'
f
for
2
( ) 3 5
f x x
= −
.
10.)
Use the Intermediate Value Theorem to determine if
2
( ) 29
f x x
= −
has a zero in the interval
[5,8].
11.)
Find the derivative of
2
–9 + + 7
( ) x x
f x x
=
.
12.)
Using the position function 7
( ) – 9
s t t
=, find the acceleration function.
13.)
Find the third derivative of
5
2
( ) 3 + 7 –
f x x x
x
=.
