AP Calculus Unit 2 Day 4
Warmup (Concept Type Question—Quiz/Test)
f ( x)  f (2)
1.If f ( x) is a function such that lim
 5, which
x2
x2
of the following MUST be true?
A) The limit of f ( x) as x approaches 2 DNE.
B) f ( x) is not defined at x  2
C) The derivative of f ( x) at x  2 is 5
D) lim f ( x)  5
x 2
E) f (2)  5
2.If f ( x) is a differentiable function, then
f ' (a) is given by which of the following:
f ( a  h)  f ( a )
I. lim
h 0
h
f ( x)  f (a )
II. lim
x a
xa
f ( x  h)  f ( x )
III. lim
xa
h
A) I only
B) II only
C ) I and II
D) I and III
E ) I , II , and III
Do you really understand limits?
1 h

lim
10
h 0
1
h
10
?
d x10

dx
x 1
d 10
x  10 x9 using the power rule
dx
10 1  10
9
evaluate at x  1
1 h

lim
10
h 0
h
 110
 10
8
8
1

1
8  h   8 
2
2



1. What is lim
?
h 0
h
1
A)0
B)
C )1
2
D)The limit does not exist.
E) Cannot be determined.
1969 AB TEST---YEP been around ALONG time! 
Product Rule
Given f (x) = u(x)iv(x) ,
f ( x ) is the product of two functions u ( x) and v ( x ) ,
dv
du
v
then f '( x )  u
dx
dx
(“First times the derivative of the second plus the second
times the derivative of the first”)
Example
f ( x)  (9 x 2  4 x)( x3  5 x 2 )
f '( x)  (9 x 2  4 x)(3x 2  10 x)  ( x3  5 x 2 )(18 x  4)
Different Type Example
f (3)  4
f '(3)  2 g (3)  1 g '(3)  5
d
f ( x) g ( x)
dx
x 3
d
f ( x) g ( x) 
dx
x 3
f ( x) g '( x)  g ( x) f '( x) x 3 
f (3) g '(3)  g (3) f '(3) 
4(5)  (1)(2)  22
Quotient Rule
u ( x)
Given f ( x) 
,
v( x)
f ( x ) is the quotient of two functions u ( x) and v ( x )
then
du
dv
v u
f '( x)  dx 2 dx
(v )
(“Lo d-Hi minus Hi d-Lo, All over low squared”)
Example
x  3x
f ( x)  2
x  7x
5
4
( x 2  7 x)(5x 4  12 x3 )  ( x5  3x 4 )(2 x  7)
f '( x) 
( x 2  7 x) 2
Different Type Example
f (3)  4
f '(3)  2 g (3)  1 g '(3)  5
d f ( x)
dx g ( x) x 3
d f ( x)
g ( x) f '( x)  f ( x) g '( x)

2
dx g ( x) x 3
 g ( x) 
g (3) f '(3)  f (3) g '(3)
 g (3) 
2

x 3
(1)(2)  (4)(5)

 18
2
(1)
Always simplify a quotient first
2( x  3)  ( x  3) x x  2

2
( x  3)
x 3
Easier to apply
quotient rule to
this side.