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The Tangent Line Problem
Finding the tangent line at point P is the same as finding the slope of the tangent line. We can approximate the slope of the tangent
line by using a line through the point of tangency and a second point on the curve. This creates a secant line.
P
)( , )(fc c
Q
)( )( , fc x c x
x
( ) ( )x cf c f
sec
ym
x
( ) ( )f fc x cc x c
sec
( ) ( )cfm
x cx
f
In 1637, mathematician Rene Descartes said this about the tangent line problem.
I dare say that this is not only the most useful and
general problem in geometry that I know, but even that I
ever desire to know.
Definition of The Tangent Line with Slope m
If f is defined on an open interval containing c, and if the limit
0 0lim l
( ) (im
)x x
y f cx
mc x
xf
exists, then the line passing through (c, f (c)) with slope m is the tangent line to the graph of f at the point (c, f (c)).
The slope of the tangent line to the graph off at point (c, f (c)) is also called the slope of the graph of f at x = c.
The Slope of a Linear FunctionFind the slope of the graph of f(x) = 2x – 3 at the point (2, 1).
I know just by looking at the equation that the slope is 2, but let’s see if we get the same answer when we use the definition of
the slope of a tangent line.
To find the slope of f when c = 2, we can apply the definition of the slope of a tangent line as follows.
0
(m
(l
)i
)x
c cxx
f f
0
( )lim
( 2 3 2 3)
x x
2 2 x
0
(2lim
( )2)x
xx
f f
0
4 2m
3 4 3lix
xx
0
2limx
xx
0lim 2x
2
Let’s find the slope of the same graph at the point (5, 4)
0
(5lim
( )5)x
xx
f f
0
( )lim
( 2 3 2 3)
x x
5 5 x
0
10 2 3 1 3lim
0x
xx
0
2limx
xx
0lim 2x
2
Hey, it’s Sam Ting!
Tangent Lines of Nonlinear FunctionsFind the slopes of the tangent lines to the graph of 2 1( )xf x
at the points (0, 1) and (-1, 2).
0
(m
(l
)i
)x
x xxx
f f
Let (x, f (x)) represent any point on the graph. Then the slope of the tangent line at (x, f (x)) is given by
2
0
( ) ( )i
1l mx x
2 1xx x
0
2 2 2( ) ( 1i
)2 1l mx
xx xx
x x
0
22 (lim
) ( )x
xx xx
0
(lim
)2x
x xxx
02lim
xxx
2x
Therefore, the slope at any point (x, f (x))on the graph of f is m = 2x.
At the point (0, 1), the slope is
2(0) 0
At the point (-1, 2), the slope is
2( 21)
That was easy
Homework
Page 104: 5 – 10 All
Finding the Equation of a Tangent LineFind the equation of the tangent lines to the graph of f at the given point.
2 5 1) ,4 1(f x x 0
(im
) ( )lx
xm
f x xx
f
2
0
( ) ( )li
5m
x x
2 5xx x
0
2 2 2( ) ( 5i
)2 5l mx
xx xx
x x
0
22 (lim
) ( )x
xx xx
0
(lim
)2x
x xxx
02lim
xxx
2m x
At the point (4, 11), the slope is
( )2m
( ) ( )( ( ))xy
4 8m
11 8 4
8 311 2y x
218y x
That was easy
Finding the Equation of a Tangent LineFind the equation of the tangent lines to the graph of f at the given point.
2 3( ) 810 ,3x x xf 0
(im
) ( )lx
xm
f x xx
f
2
0
( ) ( 3 10 ) ( )limx x
x x x x 2 3 10x x
2 2 2
0
2 3 3 10 3( )li
) ( ) 1( 0m
x
x x x x xx x xx
0
2(2 ) ( )i
( )3l mx
x x xx
x
0
(2i
)m
3lx
xx xx
02li 3m
xx x
2 3m x
At the point (3, 8), the slope is
)2 3(m 3 9m
( ) ( )( ( ))xy 8 9 3
9 78 2y x
199y x
That was easy
Homework
Page 104: 25 – 29Part A Only
The Derivative of a FunctionThe limit used to define the slope of a tangent line is also used to define the Derivative of a Function.
Definition of the Derivative of a Function.
The Derivative of f at x is given by
x
ff x
xx x xf
0
( ) (' m( i
)l) provided the limit exists.
The process of finding the derivative of a function is called Differentiation.
A function is differentiable at x if its derivative exists at x and is differentiableon an open interval (a, b) if it is differentiable at every point in the interval.
Derivative NotationsThe most common notations for the derivative of a function and the proper way of reading them are as follows:
f'( x ) f prime of x.
d ( y )d ( x )
The derivative of ywith respect to x.
y' y prime.
( )( )d
f xxd
The derivative of f(x) with respect to x.
xD y The derivative of ywith respect to x.
Finding the Derivative of a Function
Find the derivative of3( 2)x xf x
0li
( ) ( )( m' )
x
f xx xx
xf
f
3
0
( ) ( ) ( 2 )limx x
3 2 x xx x x x
3 2 2 3 3
0
( ) ( ) ( )lim
(3 3 2 2 2)x
x x x xx
x x xxx x
0
2 2 3( ) (li
) (3 3 2m
) ( )x
x x x xxx
x
0
2 23 3 ( )l
( )m
2ix
x xx x x
x
0
2 23 3 2m )l (i ( )x
x x x x
23 2x
This takes a while,
but it’s pretty easy.
Using the Derivative to Find the Slope at a Point
Find the derivative of ( )f x x
Then find the slope of the graph at the points (1, 1) and (4, 2).
Discuss the behavior of the function at (0, 0).
0li
( ) ( )( m' )
x
f xx xx
xf
f
0
( ) ( li
)m
x x
x xx x xx x
xx
0limx
xx x
x xx x
0
1limx x xx
1
2 x
At (1, 1), the slope is:1
2 1
1
2
At (4, 2), the slope is:
1 1 1
22 4 2 4
At (0, 0), the slope is:
1 1undef ed
02 0in
0
1li
2m
x x The graph has a vertical
tangent line at (0, 0)
More DerivativesFind the derivative with respect to t of y
t
2
t
y f ftt t
td
ty
d
0
( )m
(i
)l'
t t
0lim
tt t
2 2
0
( )
( )
2
(l
2
i)
mt
t tt t t t
tt t
t
0
(2 2
lim(
2 )
)t
t t
tt
ttt
0
(im
2 2
)l
2
t
tt
t
tt t
t
0
2 1im
( )lt
tt tt t
0 (li
) )(m
)
2
(t
tt tt t
0lim
2
( )t tt t
( )
2
0t t
( )
2
t t
2
2
t
Homework
Page 104: 13 – 20 All
Differentiability and Continuity
You can find the derivative of a function at a specific point.
The derivative of f at c can be found by using
l(
i) ( )
(' m)x c
f fxc
x cf
c
In order for the derivative to exist at x = c, the limit and the derivative from the left must be equal to the limit and the derivative from the right.
( ) ( ) (lim i
)l m
( )x c x c
c cc
f f fx x fx x c
Graph with a Sharp Curve
( ) 2f x x The function is continuous at x = 2, however, let’s see if it is differentiable at x = 2.
( ) ( ) (lim i
)l m
( )x c x c
c cc
f f fx x fx x c
2 2
( ) ( )lim lim li
( 2) (
2
) 0m 1
2
2x c x x
xx xcf f f fx xc x
2 2
( ) ( ) (lim lim l
2 02) (i 1
)m
2 2x c x x
f f f fcc
xx xx x x
They are not Sam Ting.
Xf is not differentiable at x = 2 and the graph does not have a tangent line at the point (2, 0)
Graph with a Vertical Tangent Line
1
3( )f x x
The function is continuous at x = 0, however, let’s see if it is differentiable at x = 0.
l(
i) ( )
(' m)x c
f fxc
x cf
c
0
( ) ( ) ( )lim
( )li
0m
0x c x
cc
x xx
f f fx
f
0
1
3
lim0
x
xx
0
1
3
limx
xx
2
3
0limx
x
0
2
3
1limx
x
Since the limit is infinite, there is a vertical tangent line at x = 0, therefore, f is not differentiable at x = 0.
Summarizing Differentiability and Continuity
Differentiability implies Continuity.
If a function is differentiable at x = c, then it is continuous at x = c.
It is possible for a function to be continuous at x = c and not differentiable at x = c.
Continuity does not imply Differentiability.
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