Presentation on Application of Partial

Differentiation

Geometrical Meaning of Partial Derivative

The figure shows a plane y=b intersects the surface z=f(x, y) in the curve APCB. then (¶z/¶x)y denotes the tangent of the angle which the tangent to the curve in which plane y=b meets the surface z=f(x,y) makes with the +ve direction of the x-axis.

TANGENT

The equation of the tangent plane

to the surface F(x,y,z)=0 is

NORMAL

Equation of the normal at (x1,y1,z1) is

Example of tangent and normal

Find the equation of the tangent plane and normal line to the surface xyz=a3 at (x1,y1,z1)

Taylor’s theorem for a function of two variables

Taylor’s theorem for a function of a single variable x, we have

Continue……

Cor.1. putting x=a and y=b in Taylor's

theorem, we have

Continue….

In Cor.1. putting a+h=x and b+y=k so that

h=x-a and y=b-k

Maclaurin’s theorem

In Cor.2. put a=0,b=0,we have

Example…

Errors and Approximations

Errors and Approximations

Example

What error in the common logarithm of a number will be produced by an error of 1% in the number.?

Maximum and Minimum values

Maximum and Minimum values

Local Maximum Value

Local minimum value

Maximum and minimum values

Working rule to find the extreme values of a function z=f(x,y)

Lagrange’s method of undetermined Multipliers

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Advantages of Lagrange’s method

1. The stationary values of f(x,y,z) can be determined directly even without determining x,y,z explicitly

2. This method can be extended to a function of several variables and subject to any number of constraints.

Disadvantages of LaGrange's method

1. This method does not enable us to find whether stationary point in maximum and minimum. Further investigations are needed.

2. The only necessary condition but not sufficient condition is