reviewer - elementary analysis ii
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Elementary Analysis II
6. Integration Techniques
6.1. Integration by Parts
Suppose we want to evaluate an integral of
the form ( ) ( ) , assuming that is
differentiable and ( ) is an antiderivative
of ( )
o
, ( ) ( )- ( ) ( )
( ) ( )
o Deriving the equation, ( ) ( )
, ( ) ( )- ( ) ( )
o Integrating both sides,
( ) ( ) ( ) ( )
( ) ( )
Theorem:
By letting ( ) ( ) and
( ) ( ), then
, which is integration by parts
Note:
As a rule of thumb, the order of choosing the
term is: Logarithmic, Inverse
trigonometric, Algebraic, Trigonometric, and
Exponential
Tabular Integration by Parts
Given ( ) ( ) , tabular integration by
parts can be used if one of the functions is
finitely differentiable and the other function
is integrable
( ) ( )
∑ ( ) ( ) ( ) | ( )
The consequence of tabular integration by
parts is that it cannot be used when the first
function is infinitely differentiable
Integration by Parts of Definite Integrals
The definite integral can be solved with
integration by parts, provided that the
functions satisfy its conditions
|
6.2. Trigonometric Integrals
1. Integrating powers of sine and cosine
2. Integrating products of sine and cosine
o If is odd,
Split off a factor of
Use the Pythagorean identity
Let
o If is odd,
Split off a factor of
Use the Pythagorean identity
Let
o If both and are even,
Use
Use
( ) ( )
o and ,
Use
, ( ) ( )-
o ,
Use
, ( ) ( )-
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o ,
Use
, ( ) ( )-
3. Integrating powers of tangent and secant
4. Integrating products of tangent and secant
o If is odd,
Split off a factor of
Use the Pythagorean identity
Let
o If is even,
Split off a factor of
Use the Pythagorean identity
Let
o If is even and is odd,
Use the Pythagorean identity
Use the reduction formula
for powers of
Note:
For powers of sine and cosine, should be a
positive integer
For powers of tangent and secant, should
be greater than 1
To evaluate integrals of cosecant and
cotangent, use the formulae for tangent and
secant, and substitute the corresponding
cofunctions
6.3. Trigonometric Substitution
Substitutions are used if the following
expressions are found in the integrand:
o √
o √
o √
Steps in Integration using Trigonometric
Substitution
1. Substitute the values for and
2. Integrate
3. Return the variables to its original form
6.4. Integration by Partial Fractions
Linear Factor Rule
o Factors of the form ( ) in the
denominator of a proper rational
functions will contribute to terms
of partial fractions; that is,
( )
∑
( ) * +
Quadratic Factor Rule
o For each factor of the form
( ) , the partial
fraction decomposition contributes
to terms of partial fractions that is,
( )
∑
( )
Note:
If the degree of the numerator is greater
than or equal to the degree of the
denominator, then long division must first be
carried out before advanced to partial
fraction decomposition
Partial fraction decomposition gives way to
the easier use of simple integration
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6.5. Improper Integrals
Improper integrals are definite integrals
whose limit of integration reaches infinity, is
a value of which makes the graph of the
function infinitely discontinuous, or a
combination of both
Improper Integrals with Infinite Integration
Intervals
Consider ( )
o ( )
o The area of the region bounded by
( ) and , - is
o
Theorem:
( )
( )
( )
( )
( )
( )
( )
Improper Integrals with Infinite Discontinuity
Consider the same function
o It has an infinite discontinuity at
o By first inverting the interval such
that ( -, the new area of the
region bounded by the function and
the interval is
o
Theorem:
If is continuous on , -, except at and
infinite discontinuity at , then the improper
integral of over , - is ( )
( )
If is continuous on , -, except at and
infinite discontinuity at , then the improper
integral of over , - is ( )
( )
If is continuous on , -, except at an
infinite discontinuity at ( ), then the
improper integral of over , - is
( )
( )
( )
6.6. Review on Separable Differential Equations
and Applications
A differential equation is an equation
involving the derivative/s of an unknown
function
A first order separable differential equation
is an equation of the form ( )
( )
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Some Applications of SODEs
Malthusian Population Model
o Let ( ) be time, population
at given time, birth rate, and death
rate, respectively
o
( )
Integrating both sides,
( ) ( )
*( ) +
o The initial population will be the
population at time zero, that is,
( ) ( ) *(
) +
( )
( )
Verhulstian Population Model
o Let ( ) .
/ be time,
population at given time, carrying
capacity, and per capita income
increase, respectively
o
.
/
Integrating both sides,
|
| ( )
* +
* +
o The initial population will be
( )
* +
( )( * + )
( )
( )
6.7. Orthogonal Trajectories
Two curves are said to be orthogonal if their
tangent lines are perpendicular at every
point of intersection
Two families of curves are said to be
orthogonal trajectories of each other if each
member of one family is orthogonal to each
member of the other
Recall:
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Note:
If
and
, then has a horizontal
tangent line
If
and
, then has a vertical
tangent line
If
, then has a singular point
Higher Derivatives
Let ( ) and ( ) be a pair of
parametric equations, then
The second order derivative,
, can be
expressed as
In general, the nth derivative of is
7.4. Arc Length of Parametric Curves
Let be the parametric curve defined by
( ) ( )
If no segment of is traced more than once
from to , then the arc length of
from to is
√, - , -
, given that the pair
of parametric equations is differentiable
over , -
7.5. Polar Coordinates
A point ( ) on the polar coordinate system
can be determined by its distance from the
pole , and the angle of the radial line with
respect to the polar axis
Conversion of Polar and Rectangular Coordinates
Polar to Rectangular -
( ) ( )
Rectangular to Polar -
( ) .√
/
Remarks:
( ) ( )
( ) ( )
7.6. Graphs of Polar Equations
A polar equation is an equation of the form
( )
Theorem:
A polar curve is symmetric about the x-axis if
( ) ( )
A polar curve is symmetric about the y-axis if
( ) ( )
A polar curve is symmetric about the origin if
( ) ( ) or if negating the equation
will still produce an equivalent equation
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8. The Real Space
8.1. Three Dimensional Coordinate System
A point in is defined by an ordered
triple, ( )
To locate in , find first the point ( )
in the -plane then move the point units
up if , or down if
Distance Between Two Points
Let ( ) and ( ) be points
in
In the -plane,
√| | | |
√( ) ( )
Suppose another plane exists where and
lies,
√ | |
√( ) ( ) ( )
Midpoint Formula in
The midpoint of the points ( ) and
( ) is ( )|
* +
8.2. Surfaces
Cylindrical Surfaces
An equation that contains only two of the
variables represents a cylindrical
surface in
The system can be obtained by the equation
in the corrdinate plane of the two variables
that appear in the equation and then
translating that graph parallel to the axis of
the missing variable
Quadric Surfaces
Ellipsoid
o
Hyperboloid
o One sheet
o Two sheets
Elliptic Paraboloid
o
o
o
Elliptic Cone
o
o
o
Hyperbolic Paraboloid
o
o
o
Note:
If (if occurs in the quadric), then
a circle will occur in at least one cross-
section plane
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o From the reference point, define an
arbitrary direction along the curve as
the positive direction; and the
oppositve direction as the negative
direction
o All points in the positive direction
are said to have positive arc lengths
o All points in the negative direction
are said to have negative arc lengths
Define ‖
‖
as the arc length
parametrization of with reference point
Properties:
a. ‖
‖
b. ‖
‖
Note:
If is an A.L.P., then , where is a
vector-valued function of
The A.L.P. is a function dependent on arc
length , which finds the vector ( ) that is
units along from the reference point
9.5. Unit Tangent, Normal, and Binormal Vectors
Let ( ) be a smooth function
o Define ( ) ( )
‖ ( )‖, called the unit
tangent vector
o Define ( ) ( )
‖ ( )‖
( )
‖ ( )‖,
called the unit normal vector
o Define ( ) ( ) ( ), called
the unit binormal vector
Remark:
The unit tangent, normal, and binormal
vectors make up of what is known as the
moving trihedral
Let be an A.L.P.
o ( ) ( )
o ( ) ( )
‖ ( )‖
( )
‖ ( )‖
o
‖ ‖
The unit normal vector points to the
concavity of
9.6. Curvature
Let be a smooth curve
o The sharpness of bend of is
measured by its curvature
o The curvature of a curve is
defined by as ( ) ‖
‖
‖ ( )‖
o Other forms for are ( )
{
‖ ( )‖
‖ ( )‖
‖ ( ) ( )‖
‖ ( )‖
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Radius of Curvature
Let be a circle with radius , then ( )
is defined to be the radius of curvature
Note:
( ) ( )
9.7. Curvilinear Motion
Let ( ), a smooth vector-valued function,
be the position function of partical moving in
space
o The unit tangent vector points to
the direction of motion of a particle
o
, the rate of change of the arc
length with respect to time, is the
speed of the particle
o The velocity vector is defined as
( )
( )
Distance and Displacement
Define as the displacement vector of a
particle travelling from to , then
( ) ( )
Let be the distance travelled of a particle
from to
o ‖
‖
‖ ( )‖
Normal and Tangential Componenents of
Acceleration
Recall: ( )
( )
The acceleration vector may be derived from
the velocity vector
o ( ) ( )
. ( )
/
( )
o Since ( ) ( )
‖ ( )
‖
and
‖ ( )
‖, then
( ), which
implies ( )
( )
o With the above, ( )
.
/ ( )
( )
Define
as the tangential scalar
component of acceleration, and
.
/
as the normal scalar component of
acceleration
Remarks:
Other formulas for the scalar components
include
‖ ‖,
‖ ‖
‖ ‖,
‖ ‖
‖ ‖
9.8. Projectile Motion
Let ( ) ⟨ ⟩, where is the
acceleration due to gravity
o ( ) ( ) ⟨ ⟩
Let ( ) and be the initial position and
velocity of the particle, respectively
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o ( ) , then ( )
⟨ ⟩
o ( ) ( ) ⟨
⟩
o ( ) ⟨ ⟩, then ( )
⟨
⟩
Parametric Equations of Projectile Motion
Let ⟨ ⟩; then
‖ ‖ ‖ ‖
( ) ⟨‖ ‖
‖ ‖ ⟩
{ ‖ ‖
‖ ‖
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10. Multivariate Differential Calculus
10.1. Multivariate Functions
A function of 2 variables, and , is a rule
that assigns a unique real number for each
ordered pair ( )
A function of 3 variables, , , and , is a rule
that assigns a unique real number for each
ordered triple ( )
In general, a function of variables,
, is a rule that assigns a
unique real number for each ordered -tuple
( )
The domain of is defined, and strictly
follwed, as
* ( )| ( ) +
Level Curves and Surfaces
Let ( ), then the projection of the
trace of on the plane , onto
the -plane is called the level curve of at
Let ( ), then the graph of
( ) , is called a surface of
at
10.2. Limits and Continuity
Let be a smooth parametric curve defined
by ( ) ( ) ( ), and
suppose that at ( )
( ) ( )
o Let ( ), then
( ) ( ) ( )
( ( ) ( ) ( ))
In , if is defined by ( ) and
( ), and if ( ), then
( ) ( ) ( )
( ( ) ( ))
o Unlike the limits in , infinitely
many points are being approached
to ( ) in infinitely many paths,
in which, the well defined curve C is
required
o To get the limit, the points of are
projected onto the surface
A function ( ) is said to be continuous at
( ) if and only if
o ( )
o ( ) ( ) ( )
o ( ) ( ) ( ) ( )
Theorem:
If ( ) is continuous at , and if ( ) is
continuous at , then ( ) ( ) ( )
is continuous at ( )
If ( ) is continuous at ( ) and if
( ) is continuous at ( ), then
( ( )) is continuous at ( )
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Remarks:
If ( ) ( ) ( ) , then
( ) ( ) ( )
| |
( ) ( ) ( )
( ) ( ) ( ), then the limit
does not exist
The sum/difference/product of two
continuous functions is also continuous
The quotient of two continuous functions is
continuous, except at those points where
the denominator is zero
10.3. Partial Derivatives
Let ( ) be a continuous function
o The partial derivative of with
respect to at ( ) is defined as
( ) ( )
o Similarly, the partial derivative of
with respect to at ( ) is
defined as
( ) ( )
A partial derivative can be interpreted as the
slope of the tangent line at the cross section
of the surface at ( )
Notations include
for -partial
derivatives, and
for -partial
derivatives
Higher-Order Partial Derivatives
Let ( ); then
.
/,
.
/,
.
/, and
.
/
Theorem: Clairaut’s Theorem
If the partial derivatives and are both
continuous and defined on ( ), then
( ) ( )
10.4. Implicit Partial Differentiation
Suppose a function ( ) is expressed
in a general form ( )
o The equation may be solved by
explicitly solving for the partial
derivatives of
o If the equation cannot be expressed
simply as ( ), implicit partial
differentiation is used
Assumptions in Implicit Partial Differentiation
1. Treat the variable as a partially
differentiable function of and
2. Since equal functions have the same
derivative on both sides, partially
differentiate both sides of the equation
3. Solve for the partial derivative
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10.5. Local Linear Approximation
Let ( ) ( )
o Define the local linear approximation
of at point ( ) as
( ) ( )
( )
For those points ( ) that are very close to
( ), then ( )
In general, if ( ), then
∑
10.6. Differentiability
A function ( ) is said to be
differentiable at ( ) if
( ) ( ) ( )
√( ) ( )
In general, a function ( )
( ) is said to be differentiable at
point if
( )
√∑ ( )
Note:
( ) is the error in the approximation if
the local linear approximation is used
10.7. Differentials
Let ( )
o Define the total differential of at
( ) as ( )
( )
Define ( ) ( )
o
o If and , then
( ) ( )
10.8. Multivariate Chain Rule
Let ( ) ( ) ( )
o If the end-function of the general
function is univariate, then define
In general, if ( ) ( ),
then define
∑
, provided that
the end-function is univariate
Remark:
To aid in MCR, a tree diagram may be used
o The use of the tree diagram exhausts
all possible paths from the most
general function to the most specific
function with the variable of interest
Note:
( ) and ( ) can be substituted directly
after ( ) and proceed with
univariate differentiation
If at least one of the functions that define
is multivariate, then MCR produces a partial
derivative
A combination of univariate and multivariate
end-functions is possible
o If such happens, for as long as the
end-function is univariate, a
univariate derivative is multiplied;
else, a partial derivative is multiplied
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