aplikasi_pembezaan.ppt

7/27/2019 aplikasi_pembezaan.ppt

1/24

The Derivative-Instantaneous rate of change

0

( ) ( )( ) lim

h

dy f a h f af a

dx h

0

( ) ( )( ) limh

dy f x h f xf x

dx h

The derivative of a function,fat a specific

value ofx, say a is a value given by:

The derivative of a function,fas a function of

x, is calledf(x) and is given by:


2/24


3/24

Related problems 2( ) 3 4 7f x x x

1) Find the slope off(x) atx = 3,x = -2

3) Find the point onf(x) for which the slope is 2

4) Find the point for whichf(x) has a

horizontal tangent line

2) Write the equation of the tangent line at x = -2


4/24

Solutions2( ) 3 4 7f x x x

1) Find the slope off(x) atx = 3,x = -2

2) Write the equation of the tangent line at x = -2

( ) 6 4f x x so

( 2) 6( 2) 4 16f

(3) 6(3) 4 14f

1 1( )y y m x x

2( 2) 3( 2) 4( 2) 7 27f

27 16( 2)y x

27 16( 2)y x 16 5y x

use the point-slope formula

Find the value of y


5/24

Solutions 2( ) 3 4 7f x x x

3) Find the point onf(x) for which the slope is 2

4) Find the point for whichf(x) has a horizontal

tangent line

( ) 6 4f x x 6 4 2

1

x

x

2

3x

2

(1) 3(1) 4(1) 7 6f The point is (1, 6)

6 4 0x

22 2 2 17( ) 3( ) 4( ) 7

3 3 3 3

f

The point is (2/3, 17/3)


6/24

Figure 2.7: Derivatives at endpoints are one-sided limits.

Derivatives at Endpoints are one-sided limits.


7/24

How a derivative can fail to exist

A cornerA vertical

tangentA discontinuity

Which of the three examples are the functions

continuous?


8/24

The graph of a function

The graph of the derivative (slope) of the function

Where f(x) is increasing

(slope is positive)

Where f(x) is decreasing

(Slope is negative)

Horizontal tangent

(slope =0)


9/24

3.3 Differentiation formulas

Simple Power rule 1n nd

x nxdx

Sum and difference rule ( ) ( )d d d

u v u vdx dx dx

Constant multiple rule ( )d d

cu c udx dx

Constant ( ) 0d

cdx


10/24

Find the derivative function for:

21

( ) 175 3

xf x

x

2( ) (3 2)f x x

12 2

1 1( ) 17

5 3

f x x x

3

2

2 1( )

5

6

f x x

x

2( ) 9 12 4f x x x

( ) 18 12f x x

rewrite

rewrite

3

21 1 1

( ) 2 * ( )( ) 05 3 2f x x x


11/24

Rules for Finding Derivatives

u and v are functions of x.

Simple Power rule 1n nd x nxdx

Sum and difference rule ( ) ( )d d du v u vdx dx dx

Constant multiple rule ( )d d

cu c udx dx

Product rule ( ) ( )d d d

uv u v v udx dx dx

Quotient rule2

( ) ( )d d

v u u vd u dx dx

dx v v


12/24

Differentiate

2 1 3

(3 2 )(4 5)y x x x

( ) ( )

d d duv u v v u

dx dx dx

2 1 3 3 2 1

(3 2 ) (4 5) (4 5) (3 2 )

dy d d

x x x x x xdx dx dx

2 1 2 3 2(3 2 )(12 ) (4 5)(6 2 )dy

x x x x x xdx

4 4 2

4 2

36 24 24 8 30 10

60 14 10

dyx x x x x x

dx

dyx x x

dx

Product rule


13/24

Differentiate 25 2

1

xy

x

2 2

2 2

( 1) (5 2) (5 2) ( 1)

( 1)

d dx x x x

dy dx dx

dx x

2

( ) ( )d d

v u u vd u

dx dxdx v v

2

2 2

( 1)(5) (5 2)(2 )

( 1)

dy x x x

dx x

2 2

2 2

(5 5) (10 4 )

( 1)

dy x x x

dx x

2

2 2

5 4 5

( 1)

dy x x

dx x

Quotient rule


14/24

Find the derivative function for:

2( ) (3 2 )(5 4 )f x x x x

2

5 2( )

1

xf x

x

2 2( ) (3 2 ) (5 4 ) (5 4 ) (3 2 )d df x x x x x x xdx dx

2(3 2 )4 (5 4 )(3 4 )x x x x 224 4 15x x

2 2

2 2

( 1) (5 2) (5 2) ( 1)( )

( 1)

d dx x x x

dx dxf xx

2 2 2

2 2 2 2

2

2 2

( 1)5 (5 2)2 (5 5) (10 4 )

( 1) ( 1)

5 4 5

( 1)

x x x x x x

x

x x

xx


15/24

Velocity. The particle is moving forward for the

first 3 seconds and backwards the next 2 sec,

stands still for a second and then moves forward.

forward

motion

means

velocity is

positive

backward

motion

means

velocity is

negative

If velocity = 0, object is standing still.


16/24

The graphs ofs and v as functions of time;s islargest when v = ds/dt= 0. The graph ofs isnotthe path of the rock: It is a plot of heightversus time. The slope of the plot is the rocks

velocity graphed here as a straight line.

a) How high does the rock go?b) What is the velocity when the rock is 256 ft.

above the ground on the way up? On the way

down?

c) What is the acceleration of the rock at any time?

d) When does the rock hit the ground? At what velocity?

3.4 applications

A dynamite blast blows a heavy rock straight up with a launch

velocity of 160 ft/sec. Its height is given by s = -16t2 +160t.


17/24



a) How high does the rock go?Maximum height occurs when v =0.

-32t+ 160 = 0v = s= -32t+ 160

t= 5 sec.

s = -16t2 +160t

At t = 5,

s = -16(5)2 +160(5)

= 400 feet.


18/24



b) What is the velocity when the rock is 256 ft. above theground on the way up? On the way down?

v =-32t+ 160

at t= 2v=-32(2)+160 = 96 ft/sec.

at t = 8

v=-32(8)+160 = -96 ft/sec

-16t2 +160t = 256

-16t2 +160t256=0

-16(t2 - 10t + 16)=0-16(t2) (t- 8) = 0

t = 2 or t = 8

Find the times

Substitute the timesinto the velocity

function

Set position = 256


19/24



c) What is the acceleration of the rock at any time?

d) When does the rock hit the ground? At what velocity?

s = -16t2 +160t

v = s= -32t+ 160

a = v = s= -32ft/sec2

s = -16t2 +160t = 0

t = 0 and t = 10

v =-32t+ 160

v = -32(10)+ 160 = -160 ft/sec.

Set position = 0


20/24

3.5 Derivatives of trig functions-formulas needed

sin(x+h) = sin x*cos h+cos x*sin h

0

sinhlim 1h h

cos(x+h) = cos x*cos h- sin x*sin h

0

cos 1lim 0h

h

h


21/24

Derivative of y = sin x0

( ) ( )( ) limh

dy f x h f xf x

dx h

0

sin( ) sin( )

( ) limh

x h x

f x h

0

sin( )cos( ) cos( )sin( ) sin( )limh

x h x h x

h

0

sin( )cos( ) sin( ) cos( )sin( )limh

x h x x h

h

0

sin( )(cos( ) 1) cos( )sin( )

limhx h x h

h h

0

sin( )(cos( ) 1) cos( )sin( )

limhx h x h

h h

0 +cos(x)*1 = cos (x)


22/24

3.5 Derivatives of Trigonometric Functions

sin cosd

x xdx

cos sind

x xdx

2tan secd x xdx

2cot cscd x xdx

sec sec tand

x x xdx csc csc cotd

x x xdx


23/24

Figure 25: The curvey = sinx as the graph of the slopes of

the tangents to the curvey = cosx.

Slope of y = cos x

Fi d h d i i


24/24

Find the derivatives

21( ) 5sin sec tan 7 32

f x x x x x x

21( ) 5 cos sec tan sec tan (1) 142f x x x x x x x x

1 sin( )

cos

xf x

x x

2

( cos ) (1 sin ) (1 sin ) ( cos )

( )

( cos )

d dx x x x x x

dx dxf x

x x

2

( cos )(cos ) (1 sin )(1 sin )( )

( cos )

x x x x xf x

x x

2 2 2 2

2 2

( cos cos ) (1 sin ) cos cos 1 sin( )

( cos ) ( cos )

x x x x x x x xf x

x x x x

2

cos( )

( cos )

x xf x

x x

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