Application of Partial Derivatives with Two

Variables

By:-Patel DipenPatel SagarPatel KirtanVaghela NayanPatel DarpanPatel Akshay

Definition

Let Z= f(x,y) the derivative of Z with respect to x is, if it is, when x alone varies & y remains constant is called partial derivative of Z w.r.t x.

It is denoted by ¶Z/¶x or fᵪ And fᵧ for y.

OPTIMIZATION PROBLEMS

Some of the most important applications of differential calculus

are optimization problems.

In these, we are required to find the optimal (best) way of doing something.

These problems can be reduced to finding the maximum or minimum values of a function.

MAXIMUM & MINIMUM VALUES

A function f has an absolute

maximum (or global maximum) at c if f(c) ≥ f(x) for all x in D, where D is the domain of f.

The number f(c) is called the maximum value of f on D.

MAXIMUM & MINIMUM VALUES

Similarly, f has an absolute minimum

at c if f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D.

The maximum and minimum values of f are called the extreme values of f.

LOCAL MAXIMUM VALUE

If we consider only values of x near b—for instance, if we restrict our attention to the interval (a, c)—then f(b) is the largest of those values of f(x).

It is called a local maximum value of f.

LOCAL MINIMUM VALUE

Likewise, f(c) is called a local minimum value of f because f(c) ≤ f(x) for x near c—for instance, in the interval (b, d).

The function f also has a local minimum at e.

MAXIMUM & MINIMUM VALUES

In general, we have the following definition.

A function f has a local maximum (or relative maximum) at c if f(c) ≥ f(x) when x is near c.

This means that f(c) ≥ f(x) for all x in some open interval containing c.

Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c.

Tangent Plane and Normal Line Equation of the Tangent plane and

Normal line can be made with the help of partial derivation.

Equation of Tangent Plane to any surface at P is given by,

(X – x)¶f/¶x + (Y – y)¶f/¶y = 0

Equation of Normal Line is given by,(X – x)/¶f/¶x = (Y – y)/¶f/¶y

Extreme value

Extreme value is useful for 1. What is the shape of a can that minimizes

manufacturing costs?2. What is the Maximum Area or Volume which can be

obtained for particular measurements of height, length and width?

Determination of Extreme Value Consider the function u= f(x , y).

Obtain the first and second order derivatives such as p= fᵪ , q= fᵧ, r= fᵪᵪ, s= fᵪᵧ, t= fᵧᵧ.

Extreme Value

Take p=0 and q=0 and solve. Simultaneously obtain the Stationary Points.(xₒ , yₒ),(x₁ , y₁),…. Be simultaneously points.

Consider the stationary points (xₒ , yₒ) and obtain the value of r, s, t.

a. If rt-s²>0 then the extreme value exists.

I. If r<0, then value is Maximum.II. If r>0, then value is Minimum.

Extreme Value

b. If rt-s²<0, then the extreme value does not exist.

c. If rt-s²=0, we cannot state about extreme value & further investigation is required.

Follow the Same procedure for the other stationary point.

Saddle PointIf rt-s²=0, then the point (xₒ , yₒ) is

called a Saddle point.

Error And Approximation

Z = f(x , y) be a continuous function of x and y where fᵪ & fᵧ be the errors occurring in the measurement of the value of x & y. Then the corresponding error ¶Z occurs in the estimation of the value of Z.i.e. Z+¶Z = f(x+¶x , y+¶y)

Therefore, ¶Z = f(x+¶x , y+¶y) – f(x , y).

Error And Approximation

Expanding by using Taylor’s Series and neglecting the higher order terms of ¶x & ¶y, we get,

¶Z = ¶x.¶f/¶x + ¶y.¶f/¶y

¶x is known as Absolute Error in x.¶x/x is known as Relative Error in x¶x/x*100 is known as Percentage

Error in x.

Example

1. In measurement of radius of base and height of a rigid circular cone are incorrect by -1% and 2%. Calculate Error in the Volume.

Solution,Let r be the radius and h be the height

of the circular cone and V be the volume of the cone.

V = π/3*r^2*h

Thus,¶V = ¶r.¶V/¶r + ¶h.¶V/¶h

Now,¶r/r*100 = -1 ¶h/h*100 = 2

Again,¶V = π/3(2rh)(r/100) +

π/3(r*r)2h/100 = 0So,The Error in the measurement in the

Volume is Zero.

Thank You