mathematics ii dr. kadhum muttar shabeeb first … · kadhum muttar shabeeb first semester...

UNIVERSITY OF TECHNOLOGYMATERIAL ENGINEERING

DEPARTMENT

MATHEMATICS II

DR. KADHUM MUTTAR SHABEEB

First Semester 2011-2012

Partial Derivatives

Functions of Several Variables

Domains and Ranges

Limits and Continuity

Partial Derivatives of a Function of Two Variables

PARTIAL DERIVATIVES

Notation For Partial Derivative

The partial derivative with respect to y is denoted the same way as the partial derivative with respect to x:

Example 1

Example 2

Example 3

Solution We treat ƒ as a quotient. With y held constant, we get

Example 4

defines z as a function of the two independent variables x and y and the partial derivative exists.

Second-Order Partial Derivatives

When we differentiate a function ƒ(x, y) twice, we produce its second-order derivatives. These derivatives are usuallydenoted by

Example 5

The Mixed Derivative Theorem

Partial Derivatives of Still Higher Order

Example 6

EXERCISES 14.3

CHAIN RULEChain Rule for Functions of Two Independent Variables

Example 1

Chain Rule for Functions of Three Independent Variables

Example 2

Chain Rule for Two Independent Variables and ThreeIntermediate Variables

Example 3

Implicit Differentiation

Example 4: Find dy/dx if

If

Total Differentiation• 1- If w = f(x,y,z), then the total differentiation is• dw = fx dx + fy dy + fz dz

• 2- If w = f(x,y,z), and x=x(t), y=y(t), z=z(t) then, the total differentiation is

•

dtdt

dz

z

wdt

dt

dy

y

wdt

dt

dx

x

wdw

3- If w = f(x,y,z), and x=x(r,s), y=y(r,s), z=z(r,s) then, the total differentiation is

dw = fx dx + fy dy + fz dzwhere

dss

xdr

r

xdx

ds

s

ydr

r

ydy

ds

s

zdr

r

zdz

Exercises 14.4

Extreme Values and Saddle Points

First Derivative Test for Local Extreme Values

Second Derivative Test for Local Extreme Values

Example 2

Example 3

EXERCISES 14.7