Download lim m m = ( ) f x x ( ) 2 f x x

Sec 11.3
The Tangent Line Problem
Definition of the slope of a graph
The slope, m of the graph of f athe the point (x, f(x)) is equal to the slope of its tangent line at, (x, f(x)), and is given by
m = lim msec
h →0
= lim
h →0
f ( x + h) − f ( x )
provided the limit exists.
h
Ex: Find the slope of the graph of f ( x) = x 3 at (2, 8)
this question is asking us to find:
lim
h →0
f (2 + h) − f (2)
since we know a specific value of x.
h
EX:
Find the slope of f(x) = -3x + 5
Ex:
Find the formula for the slope of the graph of f ( x=
) x 2 − 2 . What are the slopes at (-3, 7) and (1, -1)?
The Derivative of a function is a fancy word for the slope of the line tangent to a function.
If a function is called f(x), then its derivative is called f’(x). f’(x) is pronounced “ f prime of x”
Definition of Derivative:
The derivative of f at x is given by:
f '( x) = lim
We have been finding derivatives all year.
h →0
f ( x + h) − f ( x )
h
Ex:
Find the derivative of f (=
x) 4 x 2 − 5 x
Ex:
Find f’(x) for f (=
x)
Ex:
Find f’(x) if f ( x) =
EX:
Use the derivative of f ( x) = x 2 − 6 x + 4 to determine any points where the line tangent to f is horizontal.
x + 1 . Then find the slopes of the graph of f at the points (4, 3) and (9, 4).
1
x−2
Hint: what do we know about horizontal lines?
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