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
1 Limits: Formal Definition
2 Partial Derivatives
3 Higher-Order Partial Derivatives
4 Implicit Differentiation
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
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
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
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
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
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
1 Limits: Formal Definition
2 Partial Derivatives
3 Higher-Order Partial Derivatives
4 Implicit Differentiation
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
1 Limits: Formal Definition
2 Partial Derivatives
3 Higher-Order Partial Derivatives
4 Implicit Differentiation
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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).
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
1 Limits: Formal Definition
2 Partial Derivatives
3 Higher-Order Partial Derivatives
4 Implicit Differentiation
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 .
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 .
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 .
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
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