21 Tangent, Slope and Derivative

8/12/2019 21 Tangent, Slope and Derivative

1/22

Aim: The Tangent Problem & the Derivative Course: Calculus

Do Now: What is the equation of the linetangent to the circle at point (7, 8)?

Aim: What do slope, tangent and the

derivative have to do with each other?

5 10

10

8

6

4

2

-2


2/22


Tangents & Secants

A tangent to a circle is a line in the plane of

the circle that intersects the circle in exactly

one point.

O

A

A secant of a circle is a line that intersects the

circle in two points.

B

C


3/22


-1

Tany

x1-1

1radius = 1

center at (0,0)(x,y)

cos , sin

cos

tan lengthof theleg oppositelength of the leg adjacent to

x

y

cos

sintan

1

slope


4/22


slope is

steep!

slope islevel:

m= 0

Tangents to a Graph

(x1, y1)

(x2, y2)

(x

3,y

3)

(x4, y4)slope is

falling:

mis (-)

3

2

1

-1

2A

Unlike a tangent to a circle,

tangent lines of curves can

intersect the graph at more than

one point.

3

2

1

-1

2A


5/22


Finding the Slope (tangent) of a Graph at a Point

10

8

6

4

2

5

h x = x2

1

2

21

2 x

y

mslope

This is an approximation. How can we be

sure this line is really tangent to f(x) at (1, 1)?

(1, 1)


6/22


7/22


3

2.5

2

1.5

1

0.5

1

Slope and the Limit Process

x, f(x)

h

f(x + h)f(x)

x

ymslope

sec

(x + h, f(x + h))

h is the change in x

f(x+ h)f(x)is thechange in y

As (x + h, f(x + h))

moves down the curve

and gets closer to

(x, f(x)), the slope of

the secant moreapproximates the

slope of the tangent at

(x, f(x).


8/22


3

2.5

2

1.5

1

0.5

1


x, f(x)

h

f(x + h)f(x)

(x + h, f(x + h))

What is happening to

h, the change in x?

Its approaching 0,

or its limit at x as h

approaches 0.

x

ymslope

sec

h is the change in x

f(x+ h)f(x)is thechange in y


9/22



3

2.5

2

1.5

1

0.5

1

As h 0, the slope of the

secant, which

approximates the slope

of the tangent at (x, f(x))

more closely as (x + h, f(x+ h)) moved down the

curve. At reaching its

limit, the slope of thesecant equaled the slope

of the tangent at (x, f(x)).

sec0

tan lim mmslopeh

h

xfhxfmslope

)()(sec


10/22


Definition of slope of a Graph

3

2.5

2

1.5

1

0.5

1

The slopem

of the graphof fat the point (x, f(x)) ,

is equal to the slope of its

tangent line at (x, f(x)),

and is given by

provided this limit exists.

sec0

tan lim mmslopeh

h

xfhxf

m h

)()(

lim0

difference quotient


11/22


Model Problem

Find the slope of the graph f(x) = x2at the

point (-2, 4).

h

xfhxfm

h

)()(lim

0

h

fhfm

h

)2()2(lim

0

set up difference

quotient

h

hm

h

22

0

)2()2(lim

Use f(x) = x2

h

hh

m h

444

lim

2

0

Expand

h

hhm

h

2

0

4lim

Simplify

h

hhm

h

)4(lim

0

Factor and divide out

Simplify)4(lim0

hmh

4)4(lim0

hm h Evaluate the limit

8

6

4

2

Slope ED = -4.00

D: (-2 .00, 4.00)

q x = x2

E

D


12/22


Slope at Specific Point vs. Formula

h

xfhxf

m h

)()(

lim)1( 0

h

cfhcfm

h

)()(lim)2(

0

What is the difference between thefollowing two versions of the difference

quotient?

(1) Produces a formula for finding the

slope of any point on the function.

(2) Finds the slope of the graph for the

specific coordinate (c, f(c)).


13/22


Definition of the Derivative

The derivative of fat xis

h

xfhxfxf

h

)()(lim)('

0

provided this limit exists.

The derivative f(x) is a formula for the

slope of the tangent line to the graph offat the point (x,f(x)).

The function found by evaluating the limit of

the difference quotient is called thederivative of fat x. It is denoted by f (x),

which is read f pr ime of x.


14/22


Finding a Derivative

Find the derivative of f(x) = 3x22x.

0

( ) ( )'( ) lim

h

f x h f x f x

h

h

xxhxhxxf h

)23()](2)(3[lim)('

22

0

h

xxhxhxhxxf

h

2322363lim)('

222

0

h

hhxhxf

h

236lim)('

2

0

h

hxh

xf h

)236(

lim)(' 0

factor out h

)236(lim)('0

hxxfh

26x


15/22


Do Now:

Find the equation of the line tangent to

Aim: What is the connection between

differentiability and continuity?

( ) 2Find the slope of tangent at = 9f x x

x


16/22


6

5

4

3

2

1

0.5 1 1.5 2 2.5

f(x) is a continuous function

Differentiability and Continuity

What is the relationship, if any, between

differentiability and continuity?

(c, f(c))

(x, f(x))

f(x)f(c)

xc

xc

Is there a limit as x

approachesc? YES

x c

f x f c f c

x c

( ) ( )'( ) lim

alternative

form of

derivative


17/22


18/22


19/22


Graphs with Sharp TurnsDifferentiable?

2.5

2

1.5

1

0.5

-0.5

1 2 3 4 5

f(x) = |x2|

Is this function continuous at 2?

m= 1m= -1

x

x2

lim | 2 | ?

x

x2

lim | 2 | ?

YES

One-sided limits are not equal, fis therefore notdifferentiable at 2. There is no tangent line at (2, 0)

2 2

2 0( ) (2)lim lim 1

2 2x x

xf x f

x x

x c

f x f c

f c x c

( ) ( )

'( ) lim

alternative

form of

derivative

2 2

2 0( ) (2)lim lim 1

2 2x x

xf x f

x x

Is this function differentiable at 2?


20/22


Graph with a Vertical Tangent Line1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1 1

f(x) = x1/3

Is f continuous at 0?

YES

x

f x f

x0

( ) (0)lim

0

x

x

x

1

3

0

0lim

x

x

20 3

1lim UND

x

x

13

0

lim ?

Does a limit exist at 0?

NO

fis not differentiable at 0; slope

of vertical line is undefined.


21/22


Differentiability Implies Continuity

a b c d

fis not

continuous at a

therefore not

differentiable

f is

continuous at

b& c, but not

differentiable

corner

vertical

tangent

fis continuous at

dand

differentiable


22/22


Summary

Documents

21 Tangent, Slope and Derivative