Aim: What do slope, tangent and the derivative have to do with each other?

Aim: The Tangent Problem & the Derivative Course: Calculus

Do Now: What is the equation of the line tangent 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


Tangents & Secants

A tangent to a circle is a line in the plane of the circle that intersects the circle in exactlyone point.

O

A

A secant of a circle is a line that intersects thecircle in two points.

B

C


-1

Tan y

x1-1

1radius = 1center at (0,0)

(x,y)cos , sin

cos

sin

tan length of the leg opposite

length of the leg adjacent to

xy

cossintan

1

slope


slope is steep!

slope is level: m = 0

Tangents to a Graph

(x1, y1)

(x2, y2)

(x3, y3)

(x4, y4)slope is falling: m is (-)

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


Finding the Slope (tangent) of a Graph at a Point10

8

6

4

2

5

h x = x2

12

212

xy

mslope

This is an approximation. How can we be sure this line is really tangent to f(x) at (1, 1)?

(1, 1)


Slope and the Limit Process

3

2.5

2

1.5

1

0.5

1

3

2.5

2

1.5

1

0.5

1

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

A more precise method for finding the slope of the tangent through (x, f(x)) employs use of the secant line.

xy

mslope

sec

h

h is the change in x

f(x + h) – f(x)

f(x + h) – f(x) is the change in y

hxfhxfmslope )()(

sec

This is a very rough approximation of the slope of the tangent at the point (x, f(x)).

x, f(x)


3

2.5

2

1.5

1

0.5

1


x, f(x)

h

f(x + h) – f(x)

xy

mslope

sec

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

h is the change in xf(x + h) – f(x) is the change in y

As (x + h, f(x + h)) moves down the curve and gets closer to (x, f(x)), the slope of the secant more approximates the slope of the tangent at (x, f(x).


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?It’s approaching 0,or its limit at x as h approaches 0.

xy

mslope

sec

h is the change in xf(x + h) – f(x) is the change in y



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 the secant equaled the slope of the tangent at (x, f(x)).

sec0tan lim mmslopeh

hxfhxf

mslope)()(

sec


Definition of slope of a Graph3

2.5

2

1.5

1

0.5

1

The slope m of the graph of f at 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.

sec0tan lim mmslopeh

hxfhxf

mh

)()(lim0

difference quotient


Model Problem

Find the slope of the graph f(x) = x2 at the point (-2, 4).

hxfhxf

mh

)()(lim0

hfhf

mh

)2()2(lim0

set up difference quotient

hh

mh

22

0

)2()2(lim

Use f(x) = x2

hhh

mh

444lim2

0

Expand

hhh

mh

2

0

4lim

Simplify

hhh

mh

)4(lim0

Factor and divide out

Simplify)4(lim0

hmh

4)4(lim0

hmh

Evaluate the limit

8

6

4

2

Slope ED = -4.00

D: (-2.00, 4.00)

q x = x2

E

D


Slope at Specific Point vs. Formula

hxfhxf

mh

)()(lim)1(0

hcfhcf

mh

)()(lim)2(0

What is the difference between the following 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)).


Definition of the Derivative

The derivative of f at x is

hxfhxf

xfh

)()(lim)('0

provided this limit exists.

The derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x,f(x)).

The function found by evaluating the limit of the difference quotient is called the derivative of f at x. It is denoted by f ’(x), which is read “f prime of x”.


Finding a Derivative

Find the derivative of f(x) = 3x2 – 2x.

0

( ) ( )'( ) limh

f x h f xf x

h

hxxhxhx

xfh

)23()](2)(3[lim)('22

0

hxxhxhxhx

xfh

2322363lim)('222

0

hhhxh

xfh

236lim)('2

0

hhxh

xfh

)236(lim)('0

factor out h

)236(lim)('0

hxxfh

26 x


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


7

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)

x – c

x c

Is there a limit as x approaches c? YES

x c

f x f cf c

x c( ) ( )'( ) lim

alternative form of derivative


Differentiability and Continuity3.5

3

2.5

2

1.5

1

0.5

-0.5

-1

-1 1 2 3 4

f x x( ) [[ ]]

Does this step function, the greatest integer function, have a limit at 1?

NO: f(x) approaches a different number from the right side of 1 than it does from the left side.

By definition the derivative is a limit. If there is no limit at x = c, then the function is not differentiable at x = c.

Is this step function differentiable at x = 1?


Differentiability and Continuity

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

Is the Converse true?

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

NO


Graphs with Sharp Turns – Differentiable?2.5

2

1.5

1

0.5

-0.5

1 2 3 4 5

f(x) = |x – 2|

Is this function continuous at 2?

m = 1m = -1

xx

2lim | 2 | ?

xx

2lim | 2 | ?

YES

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

2 2

2 0( ) (2)lim lim 12 2x x

xf x fx x

x c

f x f cf c

x c( ) ( )'( ) lim

alternative form of derivative

2 2

2 0( ) (2)lim lim 12 2x x

xf x fx x

Is this function differentiable at 2?


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 fx0

( ) (0)lim0

x

xx

13

0

0lim

x x

20 3

1lim

UND

xx

13

0lim ?

Does a limit exist at 0?

NO

f is not differentiable at 0; slope of vertical line is undefined.


Differentiability Implies Continuity

a b c d

f is not continuous at a therefore not differentiable

f is continuous at b & c, but not differentiable

cornervertical tangent

f is continuous at d and

differentiable


Summary

Documents

Aim: What do slope, tangent and the derivative have to do with each other?