the derivative. def: the derivative of a function f at a number a, denoted f’(a) is: provided this...
TRANSCRIPT
The Derivative
Def: The derivative of a function f at a number a, denoted f’(a) is:
Provided this limit exists.
0
( ) ( )limh
f a h f a
h
If that limit looked familiar, it should! It is the same limit as the one for finding the slope of the tangent line to a function at a point.
0
( ) ( )limh
f a h f a
h
Some Books:
Provided this limit exists.
0 0
0
( ) ( )limx
f x x f x
x
EX: For what function and at what point would this limit represent the derivative?.
3
0
(2 ) 8limh
h
h
3( )f x x 2at x
EX: For what function and at what point would this limit represent the derivative?.
0
4 2limh
h
h
( )f x x 4at x
EX: Suppose the equation of the tangent line to a function f(x) at x=4 is y=2x+3.
What is f’(4)?
'(4) 2f slope
EX: Suppose the tangent line to a function f(x) at (3,2) also passes through the point (0,-1)
Find: f(3) and f’(3)
(3) 2f 2 1'(3) 1
3 0f
Alternate form:
Provided this limit exists.
( ) ( )'( ) lim
x a
f x f af a
x a
EX: For what function and at what point would this limit represent the derivative?.
2 2
sin 1limx
x
x
( ) sinf x x 2at x
EX: For what function and at what point would this limit represent the derivative?.
2
1
1lim
1x
x
x
2( )f x x 1at x
EX: For some functions this limit is easier to evaluate:
2
1
1lim
1x
x
x
2
0
(1 ) 1limh
h
h
They both will show f’(1)=2
The derivative function:
Provided this limit exists.
0
( ) ( )'( ) lim
h
f x h f xf x
h
The Domain of the Derivative Function:
How can a function not have a derivative at a point ?
Clearly if the function is not defined at a point then no derivative exists there.
4
2
-2
-4
-6
-5 5 10
g x = 1
cos x
The Domain of the Derivative Function:
-Would only consist of x values that were also in the Domain of the function (How can there be slope without a curve?)
How can a function not have a derivative at a point?
6
4
2
-2
-4
-5 5 10
f x = x
1
3
There is a vertical tangent line at x=0. The slope and the derivative at x=0 are undefined.
The Domain of the Derivative Function:
-But may not exist for all of those values.
-Would only consist of x values that were also in the Domain of the function (How can there be slope without a curve?)
How can a function not have a derivative at a point ?
Also if the function is discontinuous at a point then no derivative exists there. Here, there is no derivative at x=0 since it would matter from what side x approaches zero as to what slope you’d get.
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6 8
How can a function not have a derivative at a point?
Even though this function is continuous, the derivative at x=0 does not exist. It matters from what side of zero x approaches as to what the slope is. (There is an abrupt change of slope at x=0, not a gradual one.)
4
2
-2
-4
-6
-5 5
Notations:
'( )
'
( )
( )x
f x
y or y
dy
dxd
f xdxD f x
Notation:
2
2
'(2)
( )
x
x x
f
dy
dx
D f x
Terminology:
T
• The derivative (n.)
• Instantaneous rate of change (n.)
• Slope of the tangent line (n.)
• Derive (v.)
• Differentiate (v.)
• Differentiable (adj.)
Old Terminology:
T
• Average rate of change
• Slope of the secant line
2 1
2 1
y y
x x
y
x
( ) ( )f x h f x
h
Def: A function f is differentiable at a if f’(a) exists. It is differentiable on an open interval
if it is differentiable at every number in the interval.
( , ) ( , ) ,... ,a b or a or
Theorem:
If f is differentiable at a, then it is continuous at a.
Is the converse true?
NO!
Counterexample:
( )f x x
If f is continutous at a, then it is differentiable at a.
NO!
NO!
NO!
NO!
NO!
Examples:
In the next screens you will be asked to describe the derivative values (slopes) on parent functions by answering: <, > or =
Fill in the blank with: <, =, or >
[sin ] 0 ____ 0xD x at x >
Fill in the blank with: <, =, or >
[cos ] ____ 02xD x at x
<
Fill in the blank with: <, =, or >
[ ] 0 ____ 0xxD e at x >
Fill in the blank with: <, =, or >
11 ____ 0xD at x
x
<
Fill in the blank with: <, =, or >
2
[ 1 ] 0 ____ 0xD x at x =
Fill in the blank with: <, =, or >
[sec ] 2 ____ 0xD x at x =
Fill in the blank with: <, =, or >
[sin ] ____ 0xD x at x <
Fill in the blank with: <, =, or >
[ ] 5 ____ 0xxD e at x >
Fill in the blank with: <, =, or >
3[tan ] ____ 0
4xD x at x
>
Fill in the blank with: <, =, or >
1[ ] ____ 0
2x
xD e at x >
Fill in the blank with: <, =, or >
[ tan ] 0 ____ 0xD Arc x at x >
Fill in the blank with: <, =, or >
[ cos ] 0 ____ 0xD Arc x at x <
Fill in the blank with: <, =, or >
[ sin ] 0 ____ 0xD Arc x at x >
Fill in the blank with: <, =, or >
2 1[ 1 ] ____ 0
2xD x at x <
Fill in the blank with: <, =, or >
3[ ] 0 ____ 0xD x at x Undefined
Given the function:
Find the points on the graph where there is a horizontal tangent line.
3 2( ) 2 3 12f x x x x
Using a Calculator to graph a derivative function:
1
2 ( 1, , )
Y The function you want
Y nDeriv Y x x
Math (8)Vars
FunctionY1