# TANGENT AND NORMAL.pdf

• View
219

0

Embed Size (px)

### Text of TANGENT AND NORMAL.pdf

• 7/30/2019 TANGENT AND NORMAL.pdf

1/29

4.0

APPLICATIONS OF

DIFFERENTIATION

• 7/30/2019 TANGENT AND NORMAL.pdf

2/29

4.1 TANGENT AND NORMAL

EQUATION

• 7/30/2019 TANGENT AND NORMAL.pdf

3/29

At the end of this topic, students

should be able :

a) To find the equations of tangent andnormal to a curve including the parametric

equation

• 7/30/2019 TANGENT AND NORMAL.pdf

4/29

1 . is the gradient to the function ofdxdyxf )('

)(xfy

2. At point ( ) on the curve )(xfy the equation of tangent is

)( 111 xxmyy

11, yx

• 7/30/2019 TANGENT AND NORMAL.pdf

5/29

where

is the gradient of the tangent to

the curve at the point

dxdym 1

),( 11 yx

• 7/30/2019 TANGENT AND NORMAL.pdf

6/29

)(xfy

normaltangent

11, yx

• 7/30/2019 TANGENT AND NORMAL.pdf

7/29

EXAMPLE

Find the tangent and normal equation to

the curve at x= 2322 xy

• 7/30/2019 TANGENT AND NORMAL.pdf

8/29

Solution,

If

Differentiate

so the gradient for tangent at x = 2 is

4(2) = 8

xdx

dy

4

32 2 xy

• 7/30/2019 TANGENT AND NORMAL.pdf

9/29

If x = 2

The tangent equation at point (2, 11) is

113)2(22

y

)2(811 xy

58

16811

xy

xy

• 7/30/2019 TANGENT AND NORMAL.pdf

10/29

tangent is perpendicular to the

normal thus,

121 mm

2m8

1

1)(8 2 m

• 7/30/2019 TANGENT AND NORMAL.pdf

11/29

normal equation at point (2,11)

9082888

8

2

8

111

)2(8

111

xyxy

xy

xy

• 7/30/2019 TANGENT AND NORMAL.pdf

12/29

32 2 xy

908 xynormaltangent

58 xy

• 7/30/2019 TANGENT AND NORMAL.pdf

13/29

EXAMPLE

Find the tangent and normal equation to

the curve at P(1,1)

Solution,

Differentiate by using the implicit derivative

method,

223 xyyx

• 7/30/2019 TANGENT AND NORMAL.pdf

14/29

)3()2(

02)1(3

223

223

yxyxyxdx

dy

dxdyxyyyx

dxdyx

xyxyxy

dxdy

2)3( 3

22

• 7/30/2019 TANGENT AND NORMAL.pdf

15/29

the gradient for tangent at P(1,1) is

_

so tangent equation at P(1,1) is

743

3

4

3

41

)1(3

41

xy

xy

xy

3

4

)1)(1(21

))1()1(31(3

22

• 7/30/2019 TANGENT AND NORMAL.pdf

16/29

gradient for normal at P(1,1) is

_

4

3

13

4

2

2

m

m

• 7/30/2019 TANGENT AND NORMAL.pdf

17/29

normal equation at P(1,1)

134

4

3

4

31

)1(4

31

xy

xy

xy

E i Fi d th t t d

• 7/30/2019 TANGENT AND NORMAL.pdf

18/29

Exercise: Find the tangent and

normal equation to the curve xy = 8

at P(4,2)

Tangent equation 2y = -x + 8Normal equation y = 2x 6

xy

dxdy

• 7/30/2019 TANGENT AND NORMAL.pdf

19/29

EXAMPLE

Find the equation of normal to the curve

given parametrically by

and

at point where t= 1

tx 2 13

2 ty

• 7/30/2019 TANGENT AND NORMAL.pdf

20/29

• 7/30/2019 TANGENT AND NORMAL.pdf

21/29

so

when t=1

the gradient for tangent is -3(1) = -3

32

3)

2

(6 tt

t

dx

dy

3

• 7/30/2019 TANGENT AND NORMAL.pdf

22/29

• 7/30/2019 TANGENT AND NORMAL.pdf

23/29

the normal equation at (2,2) is

43

3

2

3

12

)2(3

12

xy

xy

xy

• 7/30/2019 TANGENT AND NORMAL.pdf

24/29

EXAMPLE

Given parametric equation

and wheret 1

Find the tangent equation which is parallelto line 3y = x

1

3

tx2)1(

2

ty

• 7/30/2019 TANGENT AND NORMAL.pdf

25/29

Solution

2)1(

3

tdtdx

3)1(

4

tdtdy

)1(34

3

1

)1(

4 2

3

t

t

tdx

dt

dt

dy

dx

dy

• 7/30/2019 TANGENT AND NORMAL.pdf

26/29

from 3y = x the gradient for the line is

So

4 = 1+ t

t = 3

3

1

31

)1(34

tdx

dy

• 7/30/2019 TANGENT AND NORMAL.pdf

27/29

when t = 3

tangent equation is

43

133

x81

)31(2

2 y

3824

123

318

)4

3(

3

1

8

1

xy

xy

xy

• 7/30/2019 TANGENT AND NORMAL.pdf

28/29

Exercise:

Given parametric equation x =

and y =

Find the tangent equation when t= 0

23tt

2

42

t

t

• 7/30/2019 TANGENT AND NORMAL.pdf

29/29

3y = 4x - 6 ##### 3.5 Tangent Planes and Linear Approximationsksuweb.kennesaw.edu/~plaval/math2203/funcnD_tgplanes.pdfTo –nd a normal to the tangent plane, we –nd two vectors on the tangent plane
Documents ##### Minutes of the Tangent Club International 2 Annual General ...tangentdanmark.dk/sites/tangent/files/Minutes - Tangent Club... · Minutes of the Tangent Club International 2nd Annual
Documents ##### Derivatives and the Tangent Line Problemprofb.yolasite.com/resources/Derivatives and the Tangent Line Proble… · Derivatives and the Tangent Line Problem Objective: Find the slope
Documents ##### Skid Resistance of Adjacent Tangent and Nontangent Sections …onlinepubs.trb.org/Onlinepubs/trr/1989/1215/1215-015.pdf · Skid Resistance of Adjacent Tangent ... adjacent tangent
Documents ##### 18.1 Tangent Ratio - Mr. Jagpal · 18.1 Tangent Ratio. 18.1 Tangent Ratio *YOU can use a calculator to apprcyimate the sine, cosine, and the tangent. Make sure that goor calculator
Documents Documents