DIFFERENTIAL CALCULUS OF
FUNCTIONS OF MORE THAN ONE
VARIABLE
Chapter 1
1
Functions of Several Variables
OBJECTIVES
Consider functions of two or more variables.
At the end of the chapter, you should be able to
2
find its domain and range,
perform operation

Section 1.6
PARTIAL
DERIVATIVES
1
RECALL
Given: y f x
f x h f x
dy
Derivative:
lim
dx h 0
h
What it is?: Change in y with respect to a change in x.
Interpretation: Slope of the tangent line to the graph of
f at the point x, y .
2
PARTIAL DERIVATIVES
Giv

1.3
GRAPHS of
FUNCTIONS
Definition
If f
is a function of n variables, then the graph of f is the
set of all points (x1, x2 , , xn, w) in for which (x1, x2,
, xn) Df and which w = f (x1, x2, , xn).
Note:
y = f(x)
graph is in
2
z = f(x, y)
graph is in
3

1.9
CHAIN RULE
for
Partial Differentiation
1
Chain rule
Given u, a differentiable function
of x and y where x F r ,s and
y G r ,s .
u u x u y
r x r y r
u u x u y
s x s y s
2
Chain rule
u
r
u
u
x
u
y
x
x
r
y
y
r
r
s
r
s
u u x u y
r x r y r
3
Chain rule
u
s

MATH 38
MATHEMATICAL
ANALYSIS III
SECTION T
1st SEM 2016-2017
LECTURER
Prof. Lauro L. Fontanil
Assistant Professor 3
MB 215
(Tentative) CONSULTATION HOURS:
MW: 1-2 PM
F: 1-3 PM
TTh: 1-4 PM
COURSE DESCRIPTION
At the end of the term, a student
should be abl

1.2
OPERATIONS on
FUNCTIONS
1
2.2 Operations on Functions
Operations on Function:
Sum
f P g P
Difference
f P g P
Product
f P g P
2
Quotient
f P
g P
, g P 0
2.2 Operations on Functions
Operations on Function:
Domain
of f + g, f g and f g:
Dom f Dom g
f
D

Section 1.8
HIGHER-ORDER
PARTIAL
DERIVATIVES
1
HIGHER-ORDER PARTIAL DERIVATIVES
If z f x, y and f1 x, y exists, the second-order partial
derivative of f , obtained by first partially differentiating
f with respect to x and then partially differentiating t

Continuity
Section 1.5
1
DEFINITION: Continuous Function
Suppose that f is a function of n variables and A is a point
in n . Then f is said to be continuous at the poinnt A if
and only if the following three conditions are satisfied:
i. f A exists
ii.
lim

Section 1.7
IMPLICIT
DIFFERENTIATION
1
RECALL: Implicit differentiation in MATH 36
dy
Let y be a differentiable function of x. Find
.
dx
x 2 y 2 3xy 0 F x, y
dy
dy
2x 2 y 3 x y 0
dx
dx
dy
dy
2 y 3x
2 x 3 y
dx
dx
Fx x, y
dy 2 x 3 y
2x 3y
dx 2 y 3x
2 y

Chapter 11.2
Definition:
Let cfw_ be a sequence of a real numbers and
= 1 + 2 + 3 + +
Then the sequence cfw_ is called an infinite series.
Notation:
cfw_ ,
=1
Definition.
In an infinites series
,
=1
1 , 2 , , , are called the terms of the infinite s

Alternating Series
Recall:
An infinite series of the form
(1) = 1 + 2 3 +
=1
Or
(1)+1 = 1 2 + 3
=1
Where > 0, is called an alternating series.
The following series are examples of alternating series.
1.
(1)+2
=1
7
20
2.
=1
(1)+1
(2 1)(2 + 1)
3.
(1)
2
=

Chapter 8 Applications of Integration
8.1 Arc Length
Consider a curve = () that is continuous differentiable on[, ].
Then, consider a portion of the given line and lets call it . We will formulate a way to solve for its
distance using Pythagorean Theorem.

Chapter 7.8 Improper Integrals
If is continuous over [, ]and () = () + , then () = () ().
When is an integral improper?
() = () ()
Case 1.Unbounded interval of integration
Case 2. has an infinite discontinuity over the interval of integration.
Infinite d

Limit of a Multivariate Function
Section 1.4
1
RECALL: Limit of a Function (MATH 36)
Let f be a function defined at every number in some
open interval containing a, except possibly at a. The
limit of f x as x approaches a is L, written as
lim f x L,
xa
if