Application of Partial Derivatives with Two Variables Maxima And Minima Values. Maximum And Minimum Values. Tangent and Normal. Error And Approximation.

1. By:Patel Dipen
Patel Sagar
Patel Kirtan
Vaghela Nayan
Patel Darpan
Patel Akshay

2. 

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 fᵪ y.
for

3. 
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.

4. 

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.

5. 
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.

6. 
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.

7. 
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.

8. 
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.

9. 


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

10. 
Extreme value is useful for
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?
1.

Determination of Extreme Value
Consider the function u= f(x , y). Obtain the
first and second order derivatives such as p=
fᵪ q= fᵪr= fᵪᵪ fᵪᵪ fᵪᵪ
,
,
, s= , t= .

11. 

a.
I.
II.
Take p=0 and q=0 and solve. Simultaneously
obtain the Stationary Points.
(xᵪ , yᵪ),(x₁ , y₁),…. Be simultaneously points.
Consider the stationary points (xᵪ , yᵪ) and
obtain the value of r, s, t.
If rt-s²>0 then the extreme value exists.
If r<0, then value is Maximum.
If r>0, then value is Minimum.

12. b.
c.

If rt-s²<0, then the extreme value does not
exist.
If rt-s²=0, we cannot state about extreme
value & further investigation is required.
Follow the Same procedure for the other
stationary point.
Saddle Point
If rt-s²=0, then the point (xᵪ , yᵪ) is called a
Saddle point.

13. Z = f(x , y) be a continuous function of x and y
where fᵪ fᵪ the errors occurring in the
& be
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).


14. 
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.

15. 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
1.

16. 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
=0
So,
The Error in the measurement in the Volume is
Zero.