The Derivative and the Tangent Line Problem Section 2.1

The Derivative and the Tangent Line Problem

Section 2.1

After this lesson, you should be able to:

• find the slope of the tangent line to a curve at a point

• use the limit definition of a derivative to find the derivative of a function

• understand the relationship between differentiability and continuity

Tangent Line

A line is tangent to a curve at a point P if the line is perpendicular to the radial line at point P.

P

Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve.

The Tangent Line Problem

Find a tangent line to the graph of f at P.

f

P

Why would we want a tangent line???

Remember, the closer you zoom in on point P, the more the graph of the function and the tangent line at P resemble each other. Since finding the slope of a line is easier than a curve, we like to use the slope of the tangent line to describe the slope of a curve at a point since they are the same at a particular point.

A tangent line at P shares the same point and slope as point P. To write an equation of any line, you just need a point and a slope. Since you already have the point P, you only need to find the slope.

Definition ofa Tangent

0 0

0

( ) ( )limx

f x x f xm

x

• Let Δx shrinkfrom the left

Definition of a Tangent Line with Slope m

The Derivative of a Function

Differentiation- the limit process is used to define the slope of a tangent line.

Definition of Derivative: (provided the limit exists,)

0

( ) ( )'( ) lim

x

f x x f xf x

x

= slope of the line tangent to the graph of f at (x, f(x)).

= instantaneous rate of change of f(x) with respect to x.

This is a major part of calculus and we will differentiate until the cows come home!

Also,

Really a fancy slope formula… change in y divided by the change in x.

Definition of the Derivative of a Function

Notations For Derivative

y f(x)

'( )f x

If the limit exists at x, then we say that f is differentiable at x.

( )df x

dx

dy

dx ( )Df x 'y

Let

dx does not mean d times x !

dy does not mean d times y !

dy

dx does not mean !dy dx

(except when it is convenient to think of it as division.)

df

dxdoes not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

df x

dxdoes not mean times !

d

dx f x

y f x

y f x

The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xy

h

2y x

0lim 2h

y x h

0

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

Theorem 2.1 Differentiability Implies Continiuty

The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

( ) 2 1f x x




( ) 2 1f x x

'

0

0 0 0

2( ) 1 (2 1)( ) lim

2 2 1 2 1 2lim lim lim 2 2

x

x x x

x x xf x

xx x x x

x x




( ) 2 1f x x

'

0

0 0 0

2( ) 1 (2 1)( ) lim

2 2 1 2 1 2lim lim lim 2 2

x

x x x

x x xf x

xx x x x

x x




( ) 2 1f x x

'

0

0 0 0

2( ) 1 (2 1)( ) lim

2 2 1 2 1 2lim lim lim 2 2

x

x x x

x x xf x

xx x x x

x x




( ) 2 1f x x

'

0

0 0 0

2( ) 1 (2 1)( ) lim

2 2 1 2 1 2lim lim lim 2 2

x

x x x

x x xf x

xx x x x

x x




( ) 2 1f x x

'

0

0 0 0

2( ) 1 (2 1)( ) lim

2 2 1 2 1 2lim lim lim 2 2

x

x x x

x x xf x

xx x x x

x x

The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

2( ) 3f x x

a) x = 1

b) x = -2

a) x = 1:



2( ) 3f x x

a) x = 1

b) x = -2

a) x = 1:

2 2'

0

( ) 3 ( 3)( ) lim

x

x x xf x

x



2( ) 3f x x

a) x = 1

b) x = -2

a) x = 1:

2 2'

0

2 2 2

0

( ) 3 ( 3)( ) lim

2 3 3lim

x

x

x x xf x

x

x x x x x

x



2( ) 3f x x

a) x = 1

b) x = -2

a) x = 1:2 2

'

0

2 2 2

0 0

( ) 3 ( 3)( ) lim

2 3 3 (2 )lim lim

x

x x

x x xf x

x

x x x x x x x x

x x



2( ) 3f x x

a) x = 1

b) x = -2

a) x = 1: 2 2'

0

2 2 2

0 0

0

( ) 3 ( 3)( ) lim

2 3 3 (2 )lim lim

lim (2 ) 2

x

x x

x

x x xf x

x

x x x x x x x x

x xx x x



2( ) 3f x x

a) x = 1

b) x = -2a) x = 1:

2 2'

0

2 2 2

0 0

0

( ) 3 ( 3)( ) lim

2 3 3 (2 )lim lim

lim (2 ) 2

At X = 1, 2x = 2. So, the slope of the Tangent line is 2, and the equation of the tangent line is:

2

x

x x

x

x x xf x

x

x x x x x x x x

x xx x x

y x b

. When, x is 1, y is 4, so use (1, 4), to find 4 = 2(1) + b --> b = 2.

so, 2 2 is the equation of the tangent line at x = 1.y x


b) x = -2

Example: Given , find f ’(x) and the equation of the tangent line at:

2( ) 3f x x

2 2'

0

2 2 2

0 0

0

( ) 3 ( 3)( ) lim

2 3 3 (2 )lim lim

lim (2 ) 2

At X = -2, 2x = -4. So, the slope of the Tangent line is -4, and the equation of the tangent line is:

x

x x

x

x x xf x

xx x x x x x x x

x xx x x

y

+ b --> b = -1.4 . When, x is -2, y is , so use (4, 7), to find 7 = -2(-4)so, 4 1 is the equation of the tangent line at x = -2.

x by x


Example: Find f ’(x) and the equation of the tangent line at x = 2 if 1

( )f xx



( )f xx

'

0

1 1

( ) limx

x x xf xx



( )f xx

'

0 0

1 1( )

( ) lim limx x

x x xx x xx x xf x

x x



( )f xx

'

0 0

20

1 1( )

( ) lim lim

1lim

( )

x x

x

x x xx x xx x xf x

x xx

xx x x x

Example-Continued

'

0 0

20

1 1( )

( ) lim lim

1lim

( )

x x

x

x x xx x xx x xf x

x xx

xx x x x

If x = 2, the slope is, -¼. So, y = 1/4x + b. Going back to the original equation of y = 1/x, we see if x = 2, y = 1/2. So:

1 1 12 1, and the equation is 1

2 4 4b b y x

Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative


3 3'

0

2( ) 3( ) (2 3 )( ) lim

x

x x x x x xf x

x

Derivative


3 3'

0

3 2 2 2 2 3 3

0

2( ) 3( ) (2 3 )( ) lim

2( 2 2 ) 3 3 2 3lim

x

x

x x x x x xf x

x

x x x x x x x x x x x x x x

x

Derivative


3 3'

03 2 2 2 2 3 3

02 2

0

2( ) 3( ) (2 3 )( ) lim

(2 2 4 4 2 2 ) 3 3 2 3lim

(6 6 2 3)lim

x

x

x

x x x x x xf xx

x x x x x x x x x x x x x xx

x x x x xx

Derivative


3 3'

03 2 2 2 2 3 3

02 2

02

2( ) 3( ) (2 3 )( ) lim

(2 2 4 4 2 2 ) 3 3 2 3lim

(6 6 2 3)lim

6 3

x

x

x

x x x x x xf xx

x x x x x x x x x x x x x xx

x x x x xx

x

Derivative

Example: Find for ( )f x x'( )f x

Derivative


'

0

( )( ) lim

( )x

x x x x x xf x

x x x x

Derivative


'

0

0

( )( ) lim

( )

lim( )

x

x

x x x x x xf x

x x x x

x x x

x x x x

Derivative


'

0

0 0

( )( ) lim

( )

1lim lim

( ) ( ) 2

x

x x

x x x x x xf x

x x x x

x x x x

x x x x x x x x x

THIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Example-Continued

Let’s work a little more with this example…

Find the slope of the graph of f at the points (1, 1) and (4, 2). What happens at (0, 0)?

1, (1, 1) the slope is 1/2.

2at (4, 2) the slope is 1/4. However, at (0,0)

this slope is undefined!!!!!!!!!!!!!!!!!!!!!

atx

Example-ContinuedLet’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1.

Graph the function on your calculator.

( )f x x

(I changed my window)

Now, hit DRAW

Select 5: Tangent(

Type the x value, which in this case is 1, and then hit

1

2

3

4

Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

Differentiability Implies Continuity

If f is differentiable at x, then f is continuous at x.

Some things which destroy differentiability:

1. A discontinuity (a hole or break or asymptote)

2. A sharp corner (ex. f(x)= |x| when x = 0)

3. A vertical tangent line (ex: when x = 0)3( )f x x

• Where a Function is Not Differentiable:

• 1) A function f(x) is not differentiable at a point x = a, if there is a “corner” at a.

2.1 Differentiation Using Limits

of Difference Quotients

• Where a Function is Not Differentiable:

• 2) A function f (x) is not differentiable at a point• x = a, if there is a vertical tangent at a.


of Difference Quotients

3. Find the slope of the tangent line to at x = 2. 2f x x

4

2

-2

-4

-5 5

This function has a sharp turn at x = 2.

Functions are not differentiable at

a. Discontinuities

b. Sharp turns

c. Vertical tangents

Therefore the slope of the tangent line at x = 2 does not exist.


of Difference QuotientsWhere a Function is Not Differentiable:

3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a.

Example: g(x) is not continuous at –2,so g(x) is notdifferentiable at x = –2.

4. Find any values where is not differentiable. 1

3f x

x

This function has a V.A. at x = 3.

Theorem:

If f is differentiable at x = c,

then it must also be continuous at x = c.

Therefore the derivative at x = 3 does not exist.

4

2

-2

-4

-5 5

Example

Find an equation of the line that is tangent to the graph of f and parallel to the given line.

f(x) = x3 + 2 Line: 3x – y – 4 = 0

Example

to 3 4 0, implies the slope of the tangent line must be the same as

the slope of the line. So, solving for y, we get

3 4, so the slope = 3

Parallel x y

y x

23x

1

Find an equation of the line that is tangent to the graph of f and parallel to the given line.

f(x) = x3 + 2 Line: 3x – y – 4 = 0

Taking my word for it, the derivative of the function is

This is 3 when x is

If x = 1, y = 3. If x = -1, y = 1. So there are two possible lines.

The first is 3 = 3(1)+b 3

The second is 1= 3(-1) +b y = 3x + 4

y x

Definition of Derivative

• The derivative is the formula which gives the slope of the tangent line at any point x for f(x)

• Note: the limit must exist– no hole– no jump– no pole– no sharp corner

0 0

0

( ) ( )'( ) lim

x

f x x f xf x

x

A derivative is a limit !A derivative is a limit !

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The Derivative and the Tangent Line Problem Section 2.1