M38 Lec 020614

MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

Outline

1 Limits: Formal Definition

2 Partial Derivatives

3 Higher-Order Partial Derivatives

4 Implicit Differentiation

Figures taken from: J. Stewart, The Calculus: Early Transcendentals,Brooks/Cole, 6th Edition, 2008.

For functions of two variables:

lim(x,y)(x0,y0)

f (x, y)= L

The limit L is the unique value that f (x, y) approaches as (x, y) approaches(x0, y0) along all paths containing (x0, y0).

DefinitionThe function f (x, y) is continuous at P = (x0, y0) if and only if

lim(x,y)(x0,y0)

f (x, y)= f (x0, y0).

What is the graphical interpretation of continuity?

DefinitionA function of 2 variables that is continuous at every point (x, y) is said tobe continuous everywhere, or simply continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.

2 A rational function is continuous at every point in its domain.3 A sum, difference, product, quotient (except where the denominator is

0) or composition of continuous functions are continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.2 A rational function is continuous at every point in its domain.

3 A sum, difference, product, quotient (except where the denominator is0) or composition of continuous functions are continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.2 A rational function is continuous at every point in its domain.3 A sum, difference, product, quotient (except where the denominator is

0) or composition of continuous functions are continuous.

Outline





DistanceLet P and Q be points in 2- or 3-space. We denote the distance d betweenP and Q by d = ||P Q||.

1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space:

d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space:

d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2

DefinitionIf P R2 or R3 and r > 0, then the open ball centered at P and of radiusr is

B(P,r )= {Q R2 | ||P Q|| < r }

DefinitionLet P = (x0, y0) R2 and let f be a function in 2 variables defined on someopen ball B(P,r ) centered at P , except possibly at P . Then

lim(x,y)(x0,y0)

f (x, y)= L

if the following condition is satisfied:For every > 0, there exists > 0 such that if

(xx0)2+ (y y0)2 < ,

then | f (x, y)L| < .

For every > 0, there exists > 0 such that if(xx0)2+ (y y0)2 < ,

then | f (x, y)L| <

Outline





Recall:For functions of a single variable y = f (x):

derivative of the dependent variable with respect to the independentvariable.

f (x),d f

dx, Dx [ f (x)].

geometric interpretation? slope of the tangent line at x.physical interpretation? instantaneous rate of change in y per unitchange in x.

Functions of 2 variables: two independent variables.PARTIAL DERIVATIVES with respect to the independent variables.



f (x),d f

dx, Dx [ f (x)].

geometric interpretation?

slope of the tangent line at x.physical interpretation? instantaneous rate of change in y per unitchange in x.




f (x),d f

dx, Dx [ f (x)].

geometric interpretation? slope of the tangent line at x.

physical interpretation? instantaneous rate of change in y per unitchange in x.




f (x),d f

dx, Dx [ f (x)].

geometric interpretation? slope of the tangent line at x.physical interpretation?

instantaneous rate of change in y per unitchange in x.




f (x),d f

dx, Dx [ f (x)].





f (x),d f

dx, Dx [ f (x)].


Functions of 2 variables: two independent variables.

PARTIAL DERIVATIVES with respect to the independent variables.



f (x),d f

dx, Dx [ f (x)].



DefinitionLet f (x, y) be a function of 2 variables.

1 The partial derivative of f with respect to x is

fx(x, y)= limh0

f (x+h, y) f (x, y)h

2 The partial derivative of f with respect to y is

fy (x, y)= limh0

f (x, y +h) f (x, y)h

Other notations: f

x, f

y

If z = f (x, y): zx

,z

yf1(x, y), f2(x, y)

ExampleLet f (x, y)= 4x23xy .

1 Compute fx and fy for the given function.2 Find fx(1,2) and fy (1,2).

DefinitionLet f (x, y) be a function of 2 variables.

1 The partial derivative of f with respect to x is

fx(x0, y0)= limxx0

f (x, y0) f (x0, y0)xx0

2 The partial derivative of f with respect to y is

fy (x0, y0)= limh0

f (x0, y) f (x0, y0)h

RemarkUse definition 3.4.1 to get the partial derivatives of f (functions)Use definition 3.4.2 if to get the partial derivative at particular point(value)

ExampleLet f (x, y)= 4x23xy . Find fx(1,2) and fy (1,2).

RemarkComputation of Partial Derivatives:

1 To get fx(x, y), treat y as a constant and get the derivative of f wrt x.2 To get fy (x, y), treat x as a constant and get the derivative of f wrt y .

ExampleCompute the partial derivatives of the following functions:

1 f (x, y)= 4x23xy2 f (x, y)= cos(x2y)3 f (x, y)= 4x33x2y2+ ln(x2+ y2)4 f (x, y)= exy cos(2x y)5 f (x, y)= xy

x2+ y26 f (x, y)=

xye sin t d t + tan1(2+ y2)

Extension to functions of three variables:

ExampleCompute the partial derivatives of the following functions:

1 f (x, y,z)= x2y4z3+xy + z2+12 f (x, y,z)= y3/2 sec

(xz

y

)

RemarkGeometric Interpretation of Partial Derivatives:

fx(x0, y0): slope of the surface at (x0, y0) in the direction of thepositive x-axis.fy (x0, y0): slope of the surface at (x0, y0) in the direction of thepositive y-axis.

fx(x0, y0): Slope at (x0, y0) of the trace of the surface on the plane y = y0

fy (x0, y0): Slope at (x0, y0) of the trace of the surface on the plane x = x0

RemarkPartial Derivatives as Rate of Change:

f

x: instantaneous rate of change in f per unit change in x as y is

held constant f

y: instantaneous rate of change in f per unit change in y as x is

held constant

Outline





Partial derivatives are also functions of two variables.

Four types of 2nd-order partial derivatives:

12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx

Note: Take note of the order of integration for types 3 and 4 using the twodifferent notations!



12 f

x2

= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)

= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x

= fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2

= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)

= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y

= fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx

= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)

= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y

= fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy

= x

( f

y

)= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)

= ( fy )x = fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x

= fyx




12 f

x2= x

( f

x

)= ( fx)x = fxx

22 f

y2= y

( f

y

)= ( fy )y = fy y

32 f

yx= y

( f

x

)= ( fx)y = fxy

42 f

xy= x

( f

y

)= ( fy )x = fyx


ExampleFind the 2nd-order partial derivatives of the given function:

1 f (x, y)= 2x34x2y2Previous result: fx(x, y)= 6x2y 8xy2, fy (x, y)= 2x38xy2

2 f (x, y)= cos(x2y) (Use different notation)Previous result: fx(x, y)=2xy sin(x2y), fy (x, y)=x2 sin(x2y)

RemarkLikewise, we can get the higher order partial derivatives of order 3, 4 andso on.

ExampleGiven f (x, y)= 2x3y 4x2y2,

Find3 f

xy2

Find fxyxx(x, y).

Outline





Let z be a function in x and y defined implicitly by the equationF (x, y,z)=C , where C R.

For example:

x2+ y2+ z2 = 1 defines two functions implicitly:z =1x2 y2 and z =

1x2 y2

How can we get the partial derivatives of z without isolating z?

Implicit differentiation allows us to get the partial derivatives of z withrespect to x and y .

Suppose z is a function of x and y . Then

x[ f (z)]= d f

dz zx

ExampleLet z be a function in x and y defined implicitly by the given equation.

1 If x2+ y2+ z2 = 1, find zx

,z

yand

2z

x2

2 If ln(x+ z)= xy , find zx

.

Announcements

Special Make-up Midterm Exam on Monday, 9-11 AM, MB 209(Present excuse slip, only those on the official list will be given a make-up

exam. Please inform me or your recitation instructor.)

Limits: Formal DefinitionPartial DerivativesHigher-Order Partial DerivativesImplicit Differentiation

M38 Lec 020614

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