mth 251 – differential calculus chapter 3 – differentiation section 3.2 the derivative as a...
DESCRIPTION
The Derivative as a Function If we do not specify a specific value of x (i.e. use x instead of x 0 ) we get a function called the derivative of f(x). That is, the derivative of f(x) is the function … OR f(x+h) x x+h h f(x) f(x+h) – f(x)TRANSCRIPT
MTH 251 – Differential Calculus
Chapter 3 – DifferentiationSection 3.2
The Derivative as a Function
Copyright © 2010 by Ron Wallace, all rights reserved.
Review – The Derivative at a Point• The derivative was defined as the limit of the
difference quotient. That is …
• If x0 + h = z, then an alternate definition would be …
• Note that the result of this limit is a number. That is, the derivative at a specific value of x.
0 00 0
( ) ( )' lim
h
f x h f xf x
h
Remember: x0 refers to a specific value of x.
0
00
0
( ) ( )' lim
z x
f z f xf x
z x
The Derivative as a Function
• If we do not specify a specific value of x (i.e. use x instead of x0) we get a function called the derivative of f(x).
• That is, the derivative of f(x) is the function …
0
( ) ( )'( ) limh
f x h f xf xh
( ) ( )'( ) limz x
f z f xf xz x
OR
f(x+h)
x x+hh
f(x)
f(x+h) – f(x)
Derivative Notation
• All of the following can be used to designate the function that is the derivative of y = f(x)
'f
'( )f x
'y
( )d f xdx
( )xD f x
dydx
dfdx
Reminder: The results of these will be a function.
Derivative at x = a Notation
• All of the following can be used to designate the derivative of y = f(x) at x = a
'( )f a ( )x a
d f xdx x a
dydx
( )x aD f x
a
dfdx
Reminder: The results of these will be a number.
Examples …
• Determine the following derivatives …
1ddx x
d xdx
nd xdx
IMPORTANT!Memorize these 3 results.
Examples …
• Determine the following derivatives …
7
1
x
ddx x
9
d xdx
5d xdx
Sketching the Graph of f ’(x) using the Graph of f(x)
• Where is the derivative (i.e. slope) zero?
• Where is the derivative (i.e. slope) positive? Large or small positive?
• Where is the derivative (i.e. slope) negative? Large or small negative?
• Where is the derivative (i.e. slope) constant? Function is a line segment. Derivative is a horizontal line segment.
Sketching the Graph of f ’(x) using the Graph of f(x)
• Example – Sketch the graph of the derivative of the following function.
Left & Right Derivatives at a Point• If in the definition of the derivative at a point,
you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other.
• Right-Hand Derivative at x0
• Left-Hand Derivative at x0
• If these are equal, then …
0 00 0
' limh
f x h f xf x
h
0 00 0
' limh
f x h f xf x
h
00 0' ' 'f x f x f x
Left & Right Derivatives at a Point• Example:
24
' 2 ?
' 2 ?
f x x
f
f
Where does a derivative NOT exist?• Corner
left & right derivatives are different
22y x
Where does a derivative NOT exist?• Corner
• Cusp left & right derivatives
are approaching & –
21 52 4( 1)y x
Where does a derivative NOT exist?• Corner
• Cusp
• Vertical Tangent The derivative limit is
or –
31 1y x
Where does a derivative NOT exist?• Corner
• Cusp
• Vertical Tangent
• Discontinuity see the next theorem
3 2 65( 3)
x x xyx
Differentiability & Continuity
• If f ’(c) exists, then f(x) is continuous at x = c.
Proof …
Let x c h
Therefore: lim ( ) ( )x c
f x f c
0lim ( )h
f c h
0
lim ( ) [ ( ) ( )]h
f c f c h f c
0
[ ( ) ( )]lim ( )h
f c h f cf c hh
0 0 0
[ ( ) ( )]lim ( ) lim limh h h
f c h f cf c hh
( ) '( ) 0 ( )f c f c f c
as 0x c h
Differentiability & Continuity
• If f ’(c) exists, then f(x) is continuous at x = c.
Or … the contrapositive implies …
• If f(x) is NOT continuous at x = c, then f ’(c) does not exist.
• NOTES If the derivative does not exist, that does not
mean the function is not continuous. If the function is continuous, that does not mean
that the derivative exists. Example … the Absolute Value function.