Assigned work: pg 74#5cd,6ad,7b,8-11,14,15,19,20

Slope of a tangent to any x value for the curve f(x) is:

This is know as the “Derivative by First Principles”

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0

( ) ( )limh

f x h f x

h

DifferentiabilitySymbols for Derivative:

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1) '( ) " "

2) " "

3) " " ( ' )

4) " ( ' )x

f x f prime of x

dydee y dee x

dxy y prime don t use

D y dee x of y don t use

DifferentiabilityEx 1: Find the equation of the normal to the curve

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2( ) 2f x x x at x

Note: An linear equation needs a slope and a point.

To get the slope take the derivative.To get a point find f(-2).

A normal is a line at a point on f(x) perpendicular to the tangent line at that point.

See solution next slide

Differentiability

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2 2'

0

2 2 2

0

2

0

0

'

:

( ) ( )( ) lim

2 )lim

2 )lim

(2 1)lim

2 1

( 2) 3

1( tan )

3

h

h

h

h

n

SLOPE

x h x h x xf x

h

x xh h x h x x

h

xh h h

h

h x h

h

x

f

m negative reciprocal of gent slope

2( ) 2f x x x at x

:

( 2) 2

( 2,2)

POINT

f

/ :

12 ( 2)3

NORMAL EQUATION POINT SLOPE FORM

y x

Differentiability

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Graphs of Derivatives: The graph of a derivative is the graph of the changing slopes of the tangent.

Ex 2: Given the following graphs sketch the graph of the derivative:

a)

Differentiability

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b)

When does a derivative exist?

For a function, to be Differentiable at x=c:• Left hand limit = Right hand limit of BOTH

•AND the function MUST be continuous at x = c

Differentiability Implies Continuity•If a function is differentiable at x=c then it MUST be continuous at x=c BUT if a function is continuous it does not have to be differentiable.

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( )f x

( ) ( )f c and f c

DifferentiabilityThe following are NOT differentiable at x = a

a a a

CUSP VERTICAL TANGENT DISCONTINUOUS

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DifferentiabilityMr.Function knew that so he took the slopes from the s-t graph to get a v-t graph and then the slopes from the v-t graph to get the a-t graph. Mr. Calculus said “there is a problem with Mr. Function’s graphs”. Explain in terms of differentiability what the problem is.

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Position Time Graph s-t

Velocity Time Graph v-t

Acceleration Time Graph a-t

( ) ( ) ( ) ( )a t v t and v t s t

DifferentiabilityEx. 1: For the following piecewise function determine if the

function is differentiable at a) x=6 b) x= 4 c) x= 2 d) x=0

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2

2

4, 0

1, 0 2

1( ) 4 2 4

21

4 4 62

2340 6

2

x

x x

f x x x

x x

x x x

Differentiability

a) Since for x=6..

Therefore, is differentiable at x = 6.

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6 6

6 6

lim ( ) lim ( ) 7

1lim ( ) lim ( )

2

x x

x x

f x f x

f x f x

( )f x

Differentiability

b) At x = 4..

BUT is NOT continuous at x = 4

Therefore, is NOT differentiable at x = 4.

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4 4

4 4

lim ( ) lim ( ) 6

1lim ( ) lim ( )

2

x x

x x

f x f x

f x f x

( )f x

( )f x

Differentiability

c) At x = 2..

Therefore, is NOT differentiable at x = 2.

d) At x = 0

Therefore, is NOT differentiable at x = 2.

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2 2lim ( ) lim ( )x x

f x f x

( )f x

0 0lim ( ) 1 lim ( ) 4x x

f x f x

( )f x