Unit 2
Applications of the Derivative
Relative Extrema/Absolute Extrema
Let f(x) be defined on an open interval I
containing c
1. If f(x) f(c) for all x in I, then f(c) is a maximum of f(x) on I,
i.e. f(c) is a relative maximum of f(x)
2. If f(x) f(c) for
Example : differentiate
y = x ln(5 x )
2
Derivatives of other logarithmic and
exponential functions
1.
dx
a = (ln a )a x
dx
d f ( x)
2.
a
= a f ( x ) (ln a) f '( x)
dx
d
11
1
3.
log a x =
or
dx
ln a x
x ln a
d
1
1
4.
log a f ( x) =
f '( x) or
dx
ln a f (
Definition
x
An exponential function is of the form f(x) = b ,
with b > 0 and b 1.
The natural exponential function is of the form
f(x) = e
x
Logarithmic Functions
If b > 0 and b1,
the logarithmic function with
base b is defined by:
t=log b r means b t =r
Definition
x
An exponential function is of the form f(x) = b ,
with b > 0 and b 1.
The natural exponential function is of the form
f(x) = e
x
Logarithmic Functions
If b > 0 and b1,
the logarithmic function with
base b is defined by:
t=log b r means b t =r
Definition
x
An exponential function is of the form f(x) = b ,
with b > 0 and b 1.
The natural exponential function is of the form
f(x) = e
x
Logarithmic Functions
If b > 0 and b1,
the logarithmic function with
base b is defined by:
t=log b r means b t =r
The Quotient Rule
d N ( x) D ( x)g '( x ) N ( x)g '( x )
N
D
=
2
dx D ( x)
[ D( x)]
Example : Find f(x)
4x 3
f ( x) =
5x + 2
The Chain Rule
(extended power rule)
For y = [ f ( x ) ] , y ' = k [ f ( x ) ]
k
k 1
gf '( x )
Example : Differentiate
f ( x ) =
c. f
35
( x) = 3 x
4
d . f ( x) = 3
x
e) f ( x ) = 2 x 5 x + 7 x 4
3
2
f ) f ( x) = x + 3
x
3
2
Velocity
For an object in motion with position function
s(t), the instantaneous velocity at any time t
is:
s (t + h) s (t )
lim
= s '(t ) = v(t )
x 0
h
Exampl
Definition of the Derivative
f ( x + h) f ( x )
f '( x ) = lim
h 0
h
NOTATIONS
dy
f '( x), Dx f ( x), Dx y,
, y'
dx
Example: Find f '( x)
f ( x) = x 2 + 4 x 3
For the previous example, find
f(4)
f(0)
Applications of Derivative:
Slope of a Tangent Line
Introduction
to
Limits
Average Rate of Change
Avg rate of change
between 2 points,
(a, f(a) and (b, f(b)
is:
f (b) f (a )
ba
Example
Textbook, page 174 number 4
Instantaneous Rate of Change
f ( a + h) f ( a )
lim
h 0
h
Example
Textbook page 175, number
How to find a limit:
If f(x) is defined at x = a, the limit is f(a)
If f(x) is undefined at x = a, but f(x) can be
algebraically manipulated to make it defined at
x = a, do the algebra first, then calculate f(a) for
the remaining expression
Find each li
Basic Curves
1. y = x 2
2. y = x3
3. y = x
Transformation/Translation
To translate y = f(x) into y = f(x) + c,
move the points on y = f(x) c spaces up
if (+) c, down if () c.
To Translate y = f(x) into y = f(x + c),
move the points on y = f(x) c spaces
ri