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.