Average slopeAverage slope

Find the rate of change if it takes 3 hours to drive 210 miles.

What is your average speed or velocity?

( ) (3 0

3

)

0

f f

If it takes 3 hours to drive 210 miles If it takes 3 hours to drive 210 miles

then we averagethen we average

A.A. 1 mile per minute1 mile per minute

B.B. 2 miles per minute2 miles per minute

C.C. 70 miles per hour70 miles per hour

D.D. 55 miles per hour55 miles per hour

Instantaneous slopeInstantaneous slope

What if h went to What if h went to zero?zero?

0'( ) l

( (im

) )h

f x h x

hf x

f

DerivativeDerivative

if the limit exists as one real if the limit exists as one real number. number.

0'( ) l

( (im

) )h

f x h x

hf x

f

DefinitionDefinitionIf f : D -> K is a function then the derivative of f If f : D -> K is a function then the derivative of f

is a new function, is a new function, f ' : D' -> K' as defined above if the limit f ' : D' -> K' as defined above if the limit

exists. exists. Here the limit exists every where except at x = 1Here the limit exists every where except at x = 1

0'( ) l

( (im

) )h

f x h x

hf x

f

Guess at Guess at

0

( ) ( )1lim

1'( )

hf x

f h

h

f

Guess at Guess at

0

( ) ( )1lim

1'( )

hf x

f h

h

f

ThusThus

d.n.e.d.n.e.

0'( ) li

1 1m

( ) ( )h

f fhf

hx

Guess at Guess at

f’(0) – slope of f when x = 0f’(0) – slope of f when x = 0

0'( ) l

( (im

) )h

f x h x

hf x

f

Guess at f ’(3)Guess at f ’(3)

-1.0-1.0

0.490.49

Guess at f ’(-2)Guess at f ’(-2)

-4.0-4.0

2.992.99

Note that the rule is Note that the rule is f '(x) is the slope at the point ( x, f(x) ), f '(x) is the slope at the point ( x, f(x) ), D' is a subset of D, butD' is a subset of D, butK’ has nothing to do with KK’ has nothing to do with K

0'( ) l

( (im

) )h

f x h x

hf x

f

K is the set of distances from homeK is the set of distances from homeK' is the set of speeds K' is the set of speeds K is the set of temperaturesK is the set of temperaturesK' is the set of how fast they rise K' is the set of how fast they rise K is the set of today's profits , K is the set of today's profits , K' tells you how fast they changeK' tells you how fast they changeK is the set of your averages K is the set of your averages K' tells you how fast it is changing. K' tells you how fast it is changing.

0'( ) l

( (im

) )h

f x h x

hf x

f

Theorem If f(x) = c where c Theorem If f(x) = c where c is a real number, then f ' (x) is a real number, then f ' (x) = 0.= 0.

Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =

Lim (c - c)/h = 0.Lim (c - c)/h = 0.

Examples Examples

If f(x) = 34.25 , then f ’ (x) = 0If f(x) = 34.25 , then f ’ (x) = 0

If f(x) = If f(x) = , then f ’ (x) = 0, then f ’ (x) = 0

If f(x) = 1.3 , find f’(x)If f(x) = 1.3 , find f’(x)

0.00.0

0.10.1

Theorem Theorem If f(x) = x, then f ' (x) = 1. If f(x) = x, then f ' (x) = 1.

Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =

Lim (x + h - x)/h = Lim h/h = 1Lim (x + h - x)/h = Lim h/h = 1

What is the derivative of x What is the derivative of x grandson?grandson?

One grandpa, one.One grandpa, one.

Theorem If c is a constant,Theorem If c is a constant,(c g) ' (x) = c g ' (x) (c g) ' (x) = c g ' (x)

Proof : Lim [c g(x+h)-c g(x)]/h =Proof : Lim [c g(x+h)-c g(x)]/h =

c Lim [g(x+h) - g(x)]/h = c g ' (x) c Lim [g(x+h) - g(x)]/h = c g ' (x)

Theorem If c is a constant,Theorem If c is a constant,(cf) ' (x) = cf ' (x) (cf) ' (x) = cf ' (x)

( 3 x)’ = 3 (x)’ = 3 or( 3 x)’ = 3 (x)’ = 3 or

If f(x) = 3 x then If f(x) = 3 x then

f ’(x) = 3 times the derivative of xf ’(x) = 3 times the derivative of x

And the derivative of x is . . And the derivative of x is . .

One grandpa, one !!One grandpa, one !!

If f(x) = -2 x then f ’(x) If f(x) = -2 x then f ’(x) = =

-2.0-2.0

0.10.1

TheoremsTheorems

1. (f + g) ' (x) = f ' (x) + g ' (x), and 1. (f + g) ' (x) = f ' (x) + g ' (x), and

2. (f - g) ' (x) = f ' (x) - g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x)

1. (f + g) ' (x) = f ' (x) + g ' 1. (f + g) ' (x) = f ' (x) + g ' (x) (x) 2. (f - g) ' (x) = f ' (x) - g ' 2. (f - g) ' (x) = f ' (x) - g ' (x) (x)

If f(x) = 3If f(x) = 322 x + 7, find f ’ x + 7, find f ’ (x)(x)

f ’ (x) = 9 + 0 = 9f ’ (x) = 9 + 0 = 9

If f(x) = x - 7, find f ’ (x)If f(x) = x - 7, find f ’ (x)

f ’ (x) = - 0 = f ’ (x) = - 0 =

55 5

If f(x) = -2 x + 7, find f ’ If f(x) = -2 x + 7, find f ’ (x)(x)

-2.0-2.0

0.10.1

If f(x) = thenIf f(x) = then f’(x) = f’(x) =

Proof : f’(x) = Lim [f(x+h)-f(x)]/h = Proof : f’(x) = Lim [f(x+h)-f(x)]/h =

x1

2 x

If f(x) = then f’(x) = If f(x) = then f’(x) =

A.A. ..

B.B. ..

C.C. ..

D.D. ..

x

0limh

x h x

h

0limx

x h x

h

0limh

x x

h

x h x

h

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-

1)1)

If f(x) = xIf f(x) = x44 then f ' (x) = 4 xthen f ' (x) = 4 x33

If If 2

3( )g x

x 23x

2 2 3'( ) (3 ) ' 3( ) ' 3( 2 )g x x x x 3

3

66x

x

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n xn-1 n-1

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4 xf ' (x) = 4 x3 3 + . . . .+ . . . .

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3

f ’ (1) = 4 + 9 – 4 – 3 = 6f ’ (1) = 4 + 9 – 4 – 3 = 6

3y

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-

1)1)If f(x) = If f(x) = xx44 then f ' (x) = 4then f ' (x) = 4 x x33

If f(x) = If f(x) = 44 then f ' (x) = 0then f ' (x) = 0 If If ( ) 3g x x

1

23x1 1 1

2 2 21

'( ) (3 ) ' 3( ) ' 3( )2

g x x x x

1

23 3

2 2x

x

If f(x) = then f ‘(x) =If f(x) = then f ‘(x) =x

1 1

2 21

'( ) ( ) '2

f x x x

1

2 x

Find the equation of the line Find the equation of the line tangent to g when x = 1. tangent to g when x = 1.

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

g ' (x) = 3 xg ' (x) = 3 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

g (1) =g (1) =

g ' (1) =g ' (1) =

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g (1)find g (1)

0.00.0

0.10.1

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g’ (1)find g’ (1)

-4.0-4.0

0.10.1

Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.

g(1) = 0g(1) = 0

g ' (1) = – 4g ' (1) = – 4

14

0

x

y

4(0 1)xy

( 1)4y x

Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f (1) = 1 + 3 – 2 – 3 + 4 = 3f (1) = 1 + 3 – 2 – 3 + 4 = 3

f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6

Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.

f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3

f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6

61

3Y

X

Write the equation of the Write the equation of the tangent line to f when x = 0. tangent line to f when x = 0.

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f (0) = write downf (0) = write down

f '(0) = for last questionf '(0) = for last question

Write the equation of the Write the equation of the line tangent to f(x) when x line tangent to f(x) when x = 0.= 0.A.A. y - 4 = -3xy - 4 = -3x

B.B. y - 4 = 3xy - 4 = 3x

C.C. y - 3 = -4xy - 3 = -4x

D.D. y - 4 = -3x + 2y - 4 = -3x + 2

http://www.youtube.com/watch?v=P9dpTTpjymE Derive Derive

http://www.9news.com/video/player.aspx?aid=52138&bw= Kids= Kids

http://math.georgiasouthern.edu/~bmclean/java/p6.html Secant Lines Secant Lines