The document discusses the concept of partial derivatives and their importance in optimization problems, focusing on finding maximum and minimum values of functions. It details the definitions of absolute and local extrema, the equations for tangent planes and normal lines, and the procedures to determine stationary points and their nature. Additionally, it touches on measurement errors and their impact on functions, illustrated by an example of calculating volume errors for a cone.

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.