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