Functions with Several Variables
Partial Derivatives
Chain Rule
Functions with several variables
Dr. Samir Kumar Bhowmik
Dept. of Mathematics
University of Dhaka
Dhaka, Bangladesh
bhowmiksk@gmail.com
August 16, 2020
Copyright c Calculus by Howard & Anton
Dr. Samir Kumar Bhowmik
University of Dhaka
1 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Chapter contents
1
2
3
2
Functions with Several Variables
Definition of Functions of Several Variables
Limit
Continuity
Partial Derivatives
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Definition
3
Functions of Several Variables
A function of several variables is a function where the domain D is a subset of
R2 or R3 and range is a subset of R.
Example
The function f defined by f (x, y ) = x − y is a function of two variables x and y .
Its domain is R2 and its range is R.
Example
x −y
is a function of three variables x, y
y −z
3
and z. Its domain is in a subset of R and its range is R.
The function g defined by g (x, y , z) =
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Finding the Domain
4
Domain
To find the domain of a function of several variables, we look for zero
denominators and negatives under square roots.
Example
Find the domain of the function
√
f (x, y ) =
x −y
.
x +y
Example
Find and sketch the domain for:
1
2
f (x, y ) = x ln y .
2x
g (x, y ) =
.
y − x2
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Example
Let f (x, y ) =
f.
√
Definition of Functions of Several Variables
Limit
Continuity
y + 1 + ln(x 2 − y ). Find f (e, 0), and sketch the natural domain of
Example
Find and sketch the domain for the following functions, also describe the graph of
the function in an xyz-coordinate system:
1
2
f (x, y ) = 1 − x − 12 y .
p
g (x, y ) = − x 2 + y 2 .
Example
Sketch the contour plot of f (x, y ) = 4x 2 + 9y 2 using level curves of height
k = 0, 1, 2, 3, 4, 5.
Example
Describe the level surfaces f (x, y , z) = x 2 + y 2 + z 2 , g (x, y , z) = z 2 − x 2 − y 2 .
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Limits
6
Example
Find the following limits:
lim
(xy − 2),
lim
(x,y )→(2,3)
(sin xy − x 2 y ) and
(x,y )→(−1,π)
lim
(x,y )→(1,0)
y
.
x +y −1
Formal Definition of Limit
Let f be defined on the interior of a circle centred at (a, b), except possibly at
(a, b) itself.
lim
f (x, y ) = ` if for every ε > 0 there exists δ > 0 such that
(x,y )→(2,3)
p
|f (x, y ) − `| < ε whenever 0 < (x − a)2 + (y − b)2 < δ.
Example : verify that
lim
x =a
and
(x,y )→(a,b)
Dr. Samir Kumar Bhowmik
lim
y = b.
(x,y )→(a,b)
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Limit Theorem
7
Operations on limits
If f (x, y ) and g (x, y ) both have limits as (x, y ) approaches (a, b), then
1
lim
[f (x, y ) ± g (x, y )] =
(x,y )→(a,b)
2
lim
[f (x, y )g (x, y )] = [
(x,y )→(a,b)
3
lim
lim
g (x, y );
lim
g (x, y )];
(x,y )→(a,b)
f (x, y )
(x,y )→(a,b)
lim
lim
(x,y )→(a,b)
f (x, y )][
(x,y )→(a,b)
f (x, y )
=
(x,y )→(a,b) g (x, y )
lim
f (x, y ) ±
lim
(x,y )→(a,b)
g (x, y )
, if
lim
g (x, y ) 6= 0.
(x,y )→(a,b)
(x,y )→(a,b)
A polynomial in two variables x and y is a sum of cx n y m .
The limit of any polynomial is found by substitution.
2x 2 y + 3xy
14
Example:
lim
=
.
13
(x,y )→(2,1) 5xy 2 + 3y
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Limit along a Path
8
Definition
A Path is a curve, like a line, a parabola and so on. Examples: The following are
pathes passing through the point (a, b) : x = a, (vertical line); y = b, (horizontal
line); y = g (x) where b = g (a); and x = g (y ), where a = g (b).
Limit along a Path
To evaluate the limit along a path y = g (x) passing through (a, b), we repalce y
by g (x) and find the limit when x tends to a. When x = g (y ), we repalce x by
g (y ) and find the limit when y tends to b.
Example : Compute the limit along the indicated path:
1
2
y
y
. Path: x = 1
lim
. Path: y = 0
(x,y )→(1,0) x + y − 1
(x,y )→(1,0) x + y − 1
xy
xy
. Path: y = x
lim
. Path: x = y 2 .
lim
(x,y )→(0,0) x 2 + y 2
(x,y )→(0,0) x 2 + y 2
lim
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Limit along a Path
9
Theorem
Let
lim
(x,y )→(a,b)
f (x, y ) = `1 along a path P1 and
path P2 . If `1 6= `2 then
lim
lim
(x,y )→(a,b)
f (x, y ) = `2 along a
f (x, y ) does not exist.
(x,y )→(a,b)
Example
Evaluate
xy 2
.
(x,y )→(0,0) x 2 + y 4
lim
Examples
Do the following limits exist
y
1
lim
.
(x,y )→(1,0) x + y − 1
xy
2
lim
?
(x,y )→(0,0) x 2 + y 2
Dr. Samir Kumar Bhowmik
University of Dhaka
9 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Very important Theorem
10
Theorem
Suppose that |f (x, y ) − `| ≤ g (x, y ) for all (x, y ) in the interior of some circle
centred at (a, b), except possibly at (a, b). If
lim
g (x, y ) = 0, then
(x,y )→(a,b)
lim
f (x, y ) = `.
(x,y )→(a,b)
Example
Evaluate
lim
x 2y
.
+ y2
(x,y )→(0,0) x 2
Example
Does
(x − 1)2 ln x
exist ?
(x,y )→(1,0) (x − 1)2 + y 2
lim
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Continuity
11
Definition
Suppose that f (x, y ) is defined in the interior of a circle centred at the point
(a, b). We say that f is continuous at (a, b) if
lim
f (x, y ) = f (a, b).
(x,y )→(a,b)
If f is not continuous at (a, b) we call (a, b) a discontinuity point of f .
Example: Find the disconttinuity points of
x4
x
,
2
(a) f (x, y ) = 2
and (b) g (x, y ) =
x(x + y 2 )

x −y
0,


if (x, y ) 6= (0, 0)
if (x, y ) = (0, 0)
Example
Let f (x, y ) =
(x − 1)2 ln x
. How to choose f (1, 0) so that f be continuous ?
(x − 1)2 + y 2
Dr. Samir Kumar Bhowmik
University of Dhaka
11 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Definition of Functions of Several Variables
Limit
Continuity
Composition of continuous functions
12
Theorem
Suppose f (x, y ) is continuous at (a, b) and g (x) is continuous at f (a, b). Then
h(x, y ) = (g ◦ f )(x, y ) = g (f (x, y ))
is continuous at (a, b).
Example
Determine where f (x, y ) = e x
2
y
is continuous.
Case of function with Three variables
All what we said about functions with two variables is applies on functions with
three variables: limit, continuity, . . .
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Partial Derivatives
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Intuitive Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Definition of Partial Derivatives
15
Definition
Let f (x, y ) be a function of two variables. Then the partial derivatives of f are:
fx =
∂f
f (x + h, y ) − f (x, y )
∂f
f (x, y + h) − f (x, y )
= lim
and fy =
= lim
,
h→0
h→0
∂x
h
∂y
h
if these limits exist.
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Example
Let f (x, y ) = 2x + 3y . Find
∂f
∂f
and
.
∂x
∂y
Example
For f (x, y ) = 3x 2 + x 3 y + 4y 2 , compute
∂f
∂f
and
, fx (1, 0) and fy (2, −1).
∂x
∂y
Example
Let f (x, y ) = −3x 2 y + 5y 3 .
1
Find the slope of the surface z = f (x, y ) in the x− direction at the point
(2, −3).
2
Find the slope of the surface z = f (x, y ) in the y − direction at the point
(2, −3).
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Examples
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Higher Order Partial Derivatives
18
Definition
We can define second derivatives for functions of two variables. For functions of
two variables, we have four types:
fxx =
∂2f
∂ ∂f
=
( ),
∂2x
∂x ∂x
fyx =
∂2f
∂ ∂f
=
( ),
∂x∂y
∂x ∂y
fxy =
∂2f
∂ ∂f
=
( )
∂y ∂x
∂y ∂x
fyy =
∂2f
∂ ∂f
=
( ).
∂2y
∂y ∂y
Example
Let f (x, y ) = ye x . Find fx , fy , fxx , fxy , fyx and fyy .
Theorem
Let f (x, y ) be a function with continuous second order derivatives, then fxy = fyx .
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Functions of More Than Two Variables
19
Example
Suppose that f (x, y , z) = xy − 2yz is a function of three variables, then we can
define the partial derivatives in the same way as we defined the partial derivatives
for three variables. We have :
fx = y , fy = x − 2z, and fz = −2y .
Example
Compute fx , fxy and fxyz , for: f (x, y , z) = e xy + xyz.
Example
Let f (x, y , z) = xy cos z. Find fz (2, 1, π2 ).
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Example
1
2
3
Find all second-order partial derivatives of f (x, y ) = x 3 y 2 + x 5 y 3 .
Find fxyy , fyxx of and fxxxy of f (x, y , z) = y 2 e −x + y 3 .
Show that the function u(x, t) = sin(x − ct) is a solution of
Dr. Samir Kumar Bhowmik
University of Dhaka
∂2u
∂t 2
2
= c 2 ∂∂xu2 .
20 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Exercises
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Differential and Differentiability
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Increment
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Definition:2D
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Definition:3D
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Theorem
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Total Differential
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Examples
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Examples
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Local Linear Approximation
A function f (x, y ) can be approximated by
L(x, y ) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
and is referred to as the local linear approximation of f at (x0 , y0 ).
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
Exercises
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Definitions and examples
Higher Order Partial Derivatives
Differential and Differentiability
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Chain Rule
Formula:01
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Formula:02
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Formula-Illustration 1
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Formula-Illustration 2
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Exercises
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Exercises
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Definition
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Example
√
Let f (x, y ) = xy . Find and interpret Du f (1, 2) for the unit vector u =
Dr. Samir Kumar Bhowmik
University of Dhaka
3
2 i
+ 21 j.
51 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Theorem
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Example
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Tangent Plane
54
Theorem
Suppose that f (x, y ) has continuous first partial derivatives at (a, b). A normal
vector to the tangent plane to z = f (x, y ) at (a, b) is then hfx (a, b), fy (a, b), −1i.
Further, an equation of the tangent plane is given by
z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b).
Example
Find an equation of the tangent plane to z = 6 − x 2 − y 2 at the point (1, 2, 1).
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Local extrema
55
Definition
We call f (a, b) a local maximum of f if there is an open disk R centered at (a, b),
for which f (a, b) ≥ f (x, y ) for all (x, y ) ∈ R. Similarly, f (a, b) a local mminimum
of f if there is an open disk R centered at (a, b), for which f (a, b) ≤ f (x, y ) for
all (x, y ) ∈ R. In either case, f (a, b) is called a local extremum of f .
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Critical points
The point (a, b) is a critical point of the function f (x, y ) if (a, b) is in the domain
∂f
∂f
∂f
∂f
(a, b) =
(a, b) = 0 or one or both of
and
do not
of f and either
∂x
∂y
∂x
∂y
exist at (a, b).
Theorem
If f (x, y ) has a local extremum at (a, b), then (a, b) must be a critical point of f .
Theorem
If f (x, y ) is continuous on a closed and bounded set R, then f has both an
absolute maximum and an absolute minimum on R.
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Theorem
If f (x, y ) has a relative extremum at a point (x0 , y0 ), and if the first order partial
derivatives of f exist at this point, then
fx (x0 , y0 ) = 0, and fy (x0 , y0 ) = 0.
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Dr. Samir Kumar Bhowmik
Directional Derivatives
Tangent Plane
Extrema of function with several variables
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Local extrema
59
Example
Find all critical points of f (x, y ) = xe −x
graphically.
2
/2−y 3 /3+y
and analyze each critical point
Sadle point
The point P(a, b, f (a, b)) is a saddle point of z = f (x, y ) if (a, b) is a critical
point of f and if every open disk centered at (a, b) contains points (x, y ) in the
domain of f for which f (x, y ) < f (a, b) and points (x, y ) in the domain of f for
which f (x, y ) > f (a, b).
Example
Find all sadle points of the function f (x, y ) = 2x 2 − y 3 − 2xy .
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University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Second Derivatives Test
60
Theorem
Suppose that f (x, y ) has continuous second-order partial derivatives in some open
disk containing the point (a, b) and that fx (a, b) = fy (a, b) = 0. Define the
discriminant D for the point (a, b) by
D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 .
(i) If D(a, b) > 0 and fxx (a, b) > 0, then f has a local minimum at (a, b).
(ii) If D(a, b) > 0 and fxx (a, b) < 0, then f has a local maximum at (a, b).
(iii) If D(a, b) < 0, then f has a saddle point at (a, b).
(iv) If D(a, b) = 0, then no conclusion can be drawn.
Example
Locate and classify all critical points for f (x, y ) = x 3 − 2y 2 − 2y 4 − 3x 2 y .
Dr. Samir Kumar Bhowmik
University of Dhaka
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Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Guidline for finding absolute extrema on a closed and
bounded region R
61
Follow the steps
(i) Find all critical points of f in the region R.
(ii) Find the maximum and minimum values of f on the boundary of R.
(iii) Compare the values of f at the critical points with the maximum and
minimum values of f on the boundary of R.
Example
Find the absolute extrema of f (x, y ) = 5 + 4x − 2x 2 + 3y − y 2 on the region R
bounded by the lines y = 2, y = x and y = −x.
Dr. Samir Kumar Bhowmik
University of Dhaka
61 / 62
Functions with Several Variables
Partial Derivatives
Chain Rule
Directional Derivatives
Tangent Plane
Extrema of function with several variables
Exercises
62
Local exterma
all the examples and In Page 9868, Exercises 9, 10, 31, 32, 33.
Dr. Samir Kumar Bhowmik
University of Dhaka
62 / 62