tangent plane

*

VADODARA INSTITUTE OF ENGINEERINGPRESENTED BY:KUSHWAHA AKHILESH (16COMP021)PATEL TINKLE (16COMP022)RAMI JAY (16COMP023)BHATT DHURTI (16COMP024)GUIDED BY:PROF.CHIRAG TRIVEDI

TOPIC:

1.TANGENT PLANE2.NORMAL LINE3.LINERAZATION

TANGENT

PLANE

DEFINITIONTHE PLANE THROUGH A POINT OF A SURFACE THAT CONTAIN THE TANGENT LINES TO ALL THE CURVES ON THE SURFACE THROUGH THE SAME POINT.

FORMULA

NORMAL LINE

DEFINATIONTHE NORMAL LINE IS DEFINED AS THE LINE THAT IS PERPENDICULAR TO THE TANGENT LINE AT THE POINT OF TANGENCY.

FORMULA

= =

EXAMPLE-FIND THE EQUATION OF THE TANGENT PLANE AND NORMAL TO THE SURFACE

Z= AT THE POINT (1,-1,2)Here f(x,y,z) = z - = 0= -2x

= -2y =1

At (1,-1,2), =-2, =2, =1 Therefore equation of the tangent plane at (1,-1,2)(x-1)(-2) + (y+1)(2) + (z-2)(1) = 0Or -2x + 2 + 2y + 2 +z -2 = 0Or 2x – 2y – z = 2 Equation of the normal are = =

LINEARIZATION

DEFINITION

IF F IDIFFERENTIABLE AT X = A,THEN APPROXIMATE FUNCTION L(X) = F(A) + F’(A)(X-A)IS THE LINEARIZATION OF F AT A.THE APPROXIMATE F(X)≈L(X)OF F BY L IS THE STANDARD LINEARAPPROXIMATE OF F AT A.THE POINT X = A IS THE CENTRE OF THE APPROXIMATIONS.

Example- Find the linearization of f(x)=cosx at x=π/2Since f(π/2) = cos(π/2) = 0 f’(x) = -sinx f’(π/2) = -sin(π/2) = -1

L(x) = f(a) + f’(a) (x-a) = 0 + (-1) (x -π/2 ) = -x + π/2

cosx ≈ -x +π/2