Elementary Analysis

Elementary

Analysis

Review Notes (Math 55 Notes to be added)

1

Table of Contents

1. Limits and Continuity ... 3

1.1. Review of Functions ... 3

1.2. Limits ... 4

1.3. Computing Limits ... 4

1.4. Formal Definition of a Limit ... 5

1.5. One-Sided Limits ... 5

1.6. Infinite Limits ... 5

1.7. Limits at Infinity ... 6

1.8. Continuity... 6

1.9. Limits and Continuity of Trigonometric Functions ... 7

2. Derivatives ... 9

2.1. Slopes and the Derivative ... 9

2.2. Techniques in Differentiation ...10

2.3. Derivatives of Trigonometric Functions .. ...10

2.4. Chain Rule ...10

2.5. Implicit Differentiation ...10

2.6. Higher-Order Derivatives ...10

2.7. Rectilinear Motion Problems ...10

2.8. Rates of Change ...11

2.9. Local Linear Approximation and Differentials ...11

3. Behaviour and Analysis of Functions ...13

3.1. Related Rates ...13

3.2. Analysis of Functions: Relative Extrema . ...13

3.3. Analysis of Functions: Increasing, Decreasing, and the First Derivative Test ...13

3.4. Analysis of Functions: Concavity and the Second Derivative Test ... 14

3.5. Sketching of Functions ... 14

3.6. Rolle’s Theorem and the Mean-Value Theorem for Derivatives ... 15

3.7. Absolute Extrema ... 15

4. Integration ... 16

4.1. The Indefinite Integral ... 16

4.2. Integration by Substitution ... 16

4.3. Separable Differential Equations ... 16

4.4. Area ... 17

4.5. The Definite Integral ... 18

4.6. Fundamental Theorems of Calculus and the Mean Value Theorem for Integration ... 19

4.7. Calculation of Area as a Definite Integral ... 20

4.8. Volume by Slicing, Disks, and Washers .. ... 21

4.9. Volume by Cylindrical Shells ... 23

4.10. Arc Length of a Plane Curve ... 23

5. Special Functions and Cases ... 25

5.1. The Natural Logarithmic Function from the Integral Point-of-View ... 25

5.2. Logarithmic Differentiation ... 25

5.3. Integration of the Natural Logarithmic Function ... 25

5.4. Inverse Functions ... 25

5.5. The Natural Exponential Function ... 26

5.7. Derivatives and Antiderivatives of Inverse Trigonometric Functions ... 28

5.8. Hyperbolic Functions ... 28

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5.10. Indeterminate Forms and L’Hopital’s

Rule...30

6. Integration Techniques ...31

6.1. Integration by Parts ...31

6.2. Trigonometric Integrals ...31

6.3. Trigonometric Substitution ...32

6.4. Integration by Partial Fractions ...32

6.5. Improper Integrals ...33

6.6. Review on Separable Differential Equations and Applications ...33

6.7. Orthogonal Trajectories ...34

7. Parametric and Polar Curves ...35

7.1. Review on Conic Sections ...35

7.2. Parametric Equations ...36

7.3. Derivatives of Parametric Equations ..36

7.4. Arc Length of Parametric Curves ...37

7.5. Polar Coordinates ...37

7.6. Graphs of Polar Equations ...37

7.7. Tangent Lines of Polar Curves ...40

7.8. Arc Length of Polar Curves ...40

7.9. Areas in Polar Coordinates ...40

7.10. Conic Sections in Polar Coordinates ....40

8. The Real Space ...42

8.1. Three Dimensional Coordinate System .. ...42

8.2. Surfaces ...42

8.3. Vectors ...43

8.4. Dot Product ...44

8.5. Cross Product ...45

8.6. Parametric and Vector Equations of Lines ...46

8.7. Planes ...47

9. Elementary Vector Analysis ... 48

9.1. Vector-valued Functions ... 48

9.2. Calculus of Vector-valued Functions .. 48

9.3. Arc Length ... 49

9.4. Arc Length Parametrization ... 49

9.5. Unit Tangent, Normal, and Binormal Vectors ... 50

9.6. Curvature ... 50

9.7. Curvilinear Motion ... 51

9.8. Projectile Motion ... 51

10. Multivariate Differential Calculus ... 53

10.1. Multivariate Functions ... 53

10.2. Limits and Continuity ... 53

10.3. Partial Derivatives ... 54

10.4. Implicit Partial Differentiation ... 54

10.5. Local Linear Approximation ... 55

10.6. Differentiability ... 55

10.7. Differentials ... 55

10.8. Multivariate Chain Rule ... 55

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Elementary Analysis I 1. Limits and Continuity 1.1. Review of Functions

 Let and be non-empty sets,  A function from to ( ) is a set of

ordered pairs, ( ) ( )

 A function can be represented numerically, geometrically, algebraically, and verbally

Note:

 If is a function from to ,

o is the domain and is the codomain

o If the ordered pair is in , is the image of , and is the pre- or inverse- image of

o The set of all elements ( ) , is called the range

Operations on Functions

 Let be functions; then we have: 1. ( )( ) ( ) ( ) ( ) 2. ( )( ) ( ) ( ) ( ) 3. . / ( ) ( ) _{( )} . / ( ) * ( ) + 4. ( )( ) ( ( )) ( ) * ( ) +

Basic Types of Functions and their Graphs Recall:

 The graph of a function is a set of points in the Cartesian plane having its coordinate ordered pair belonging to the function  The zeroes of the function are the values of

for which it will make the whole function equal to zero

 The graph must pass the vertical line test

Types of Functions

1. Polynomial Functions of degree n ( ) ∑

i. Constant function ( ) ( )

ii. Linear function ( )

( )

iii. Quadratic function ( )

( ) Zeroes:

√

4 2. Rational Functions ( ) ( ) ( ) ∑

3. Functions involving Radical Expressions i.e. ( ) √

4. Absolute Value function

( ) √ 2 Properties: a. b. c. d. | | e. f. g. h. ( ) ( )

5. Greatest Integer function  ( ) ⟦ ⟧

 Denotes the greatest integer less than or equal to x, that is,

6. Signum function ( ) ( ) { 7. Piecewise function

 Different functions in different intervals

1.2. Limits  Consider 3 functions: ( ) ( ) ( ) 2

o For the 3 functions, as approaches 1, the functions will be approaching the value 3; or, we can take the values of the 3 functions as close as we like to 3 by taking values of sufficiently, but no reaching, 1

 The limit of a function, as approaches to is , written as ( ) , means that

the values of the function get closer and closer to as assumes values going close and closer, but not reaching,

1.3. Computing Limits

1. If ( ) exists, then it is unique, that

is, ( ) ( )

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2. ( )

3. If a function is given as an identity function, then ( )

4. If ( ) and ( ) exists, then

a. ( ) b. ( ) ( ) c. 5. ( ( )) ( ( )) 6. ( ( )) ( ( )) ( ) 7. ( ) ( ) Theorem:

 If is a polynomial or rational function, then ( ) ( ) ( )

 If the limit of a function exists, then o ( ) , ( )

-

o ( ) ( )

1.4. Formal Definition of a Limit

 Consider ( )

o If , and is infinitesimal, then ( ) means that the function is really close to

o It seems that , such that whenever , so that ( )

 If a function on some open interval about , then ( )

( )

1.5. One-Sided Limits

Recall: definition of a limit

 is defined on any open interval ( ), where ( )  ( ) ( )  Consider ( ) √ o √ √ √

is false since there is no open interval about 1 such that the function is defined on such interval o It can be said that the limit of the

function as approaches 1 from the right is 0

 The limit of a function as approaches to from the left [right] is , that is _{, -} ( ) , if we can make values of the function arbitrarily close to by taking to be sufficiently close to from values of that is less [greater] than

Theorem:

 ( ) ( ) ( )

Remark:

 Limit theorems also apply to one-sided limits

1.6. Infinite Limits

 Consider ( ) , a rational function where ( ) and ( ) , with limits, as approaches 0, 1 and 0, respectively

o The values of the function increase without bound as assumes values going closer and closer to 0, then ( )

 Let be a function defined on both sides of , then ( ) , means that

the values of can be made arbitrarily large as we please by taking values of that are sufficiently close to Theorem:  If , then o o {

6  If ( ) exists, ( ) , and if o ( ) , then   o ( ) , then    If ( ) and ( ) exists, then o , - o , - {  If ( ) and ( ) exists, then o , - o , - { Notes:

 are not real numbers, thus ( ) does not mean that it exists

 The theorems sill fold for one-sided limits

1.7. Limits at Infinity

 Consider ( )

o The values of the function eventually get closer to zero as x increases

without bound, thus

( )

 Let be a function defined on some half-infinite interval, then ( )

means that the values of the function can be made arbitrarily close to by taking sufficiently large values of

Theorem:  If , then o o { o Notes:

 When evaluating limits at infinity of rational functions, the numerator and denominator is divided by the highest power of

 Limit theorems 2, 4, infinite, and one-sided hold for limits at infinity

1.8. Continuity

 A function is continuous at a point, , if all the conditions are satisfied:

o ( )

o ( ) o ( ) ( )

Types of Discontinuity

 If a function is discontinuous at a point , then the discontinuity is said to be

o Removable if ( ) exists; or

o Essential if ( ) does not

exists

 If a function is essentially discontinuous at a point , and

o ( ) and ( )

both exists, then the discontinuity is called jump discontinuity

o ( ) and ( ) is , then the discontinuity is called infinite discontinuity

Theorem:

 If two functions are continuous on , then the following are continuous on :

o o o

 All polynomial functions are continuous everywhere

 A rational function is continuous everywhere in its domain

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 If , then ( ) √ is continuous o Everywhere in o

 Absolute Value functions are continuous everywhere

 If ( ) and the function is

continuous at , then ( )( )

( ( )) ( ( )) ( ) o This theorem holds for one-sided

limits

 If the function is continuous at and if the function is continuous at ( ), then is continuous at

Note:

 If is a function, then the possible points of discontinuity are:

o Values of x that makes ( ) o Endpoints of piecewise intervals o ⟦ ⟧

 If ( ) has neither removal nor essential discontinuity, then the function is simply discontinuous

Remarks:

 To remove the discontinuity at is the equivalent of redefining the value of ( ) to form the function ( ), such that ( ) { ( )

( )

Continuity on Intervals

 A function is continuous on ( ) if it is continuous on every real number in the interval

 A function is continuous from the left[right] of if:

o ( )

o _{, -} ( ) o ( ) _{, -} ( )

 A function that is continuous at is continuous on both sides of

 A function is continuous on

o , - if it is continuous on its open interval, from the right of , and from the left of

o , ) if it is continuous on its open interval and from the right of o ( - if it is continuous on its open

interval and from the left of

o ( ) if it is continuous ( )

o ( ) if it is continuous ( ) o ( - if it is continuous on its open interval and from the left of o , ) if it is continuous on its open

interval and from the right of

Theorem: Intermediate Value Theorem

 If a function is continuous on , -, and

( ) ( ), then

, ( ) ( )- , - ( )  The value of is not necessarily unique

Corollary: Intermediate Value Theorem

 If a function is continuous on , - and if ( ) ( ) , then ( ) ( )

1.9. Limits and Continuity of Trigonometric Functions

Theorem:

 The trigonometric functions are continuous on their respective domains





Note:

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Theorem: Squeeze/Sandwich Theorem

 Let be defined on some open interval ( ) about, except possibly at, c, and * + ( ) ( ) ( ), then ( ) ( )

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2. Derivatives

2.1. Slopes and the Derivative

 Suppose a secant line passes through 2 points, and , on the graph of a function, ( )

o The slope of the secant line is

( ) ( ) ( ) ( )

o If we let approach , approaches , and approaches 0 o The slope of the tangent line will be

( ) ( )

 If the function is defined at , then the tangent line to the graph of the function at point is the line

o Passing through and where the slope is ( ) ( )

o

_, - ( ) ( )

o Otherwise, there does not exists a tangent line to the graph of the function at point

 The line normal to the given graph of the function at point is the line perpendicular to the tangent line at the same point

 The derivative of a function , denoted by ,

is the function

( ) ( ) ( )

and the limit exists

 If the derivative of the function at exists, then the function is said to be differentiable at that point

 The function is differentiable on an open interval if its differentiable for all real numbers in that interval

 The function is differentiable everywhere if it is differentiable on all real numbers

 If the function is defined at , then the derivative from the left[right] of , written as ( ), ( )- is given as _{[ ]} ( ) ( )

 ( ) ( ) ( ) if both derivatives exist

Note:

 may refer to the steepness or flatness of the tangent line

 The line to the graph of a function may intersect the graph at points other than the point of tangency



 The slope of the tangent line to the graph of the function at point P is ( ). If the limit in the definition of the derivative does not exist, then the slope of the graph of the function is undefined at point P

 The alternate definition for ( ) is ( ) ( )

 Notations for derivatives:

, - , ( , ( )- Remarks:

 If a function is discontinuous at a point, then it is not differentiable at such point

 A function may be continuous at a point, but fail to be differentiable at such point

 A function is not differentiable at a point if o The function is discontinuous at such

point

o The graph has a vertical tangent line at such point

o The graph has no well-defined tangent line at such point

10 2.2. Techniques in Differentiation 1. Constant Rule ( ) ( ) 2. Power Rule ( ) ( ) 3. ( ) ( ) ( ) ( ) 4. Sum Rule ( ) ( ) ( ) ( ) ( ) ( ) 5. Product Rule ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6. Quotient Rule ( ) ( ) _{( )} ( ) ( ) ( ) ( ) ( ) ( ) ( )

2.3. Derivatives of Trigonometric Functions

1. 2. 3. 4. 5. 6. 2.4. Chain Rule Theorem:

 If the function is differentiable at , and is differentiable at ( ), then ( )( ) ( ( )) ( )

Note:

 The Chain Rule can also be stated as . ( )

/ . ( ) /

. / ( , ( )- ( ) )  The Chain rule can also be extended to a

finite number of functions

2.5. Implicit Differentiation

 Given an equation in and , we assume that is a differentiable function of

 To obtain a derivative without explicitly solving for in terms of , implicit differentiation is used

Assumptions in Implicit Differentiation

1. Look at the variable as a differentiable function of

2. Since equal functions have the same derivative on both sides, differentiate both sides of the equation using the Chain Rule when necessary

3. Solve for

2.6. Higher-Order Derivatives

 The nth derivative of the function , denoted by ( ), is the derivative of ( ), that is, ( )

_{( ) ( )}

Remarks:

 The derivative of a function is also called the first derivative

 Notations: , ( )- , ( )-  ( ) ( ) _{( )} . /

2.7. Rectilinear Motion Problems

 Derivatives can be used to describe the behaviour of a moving object, say, a particle, using its position function ( ), where denotes time and must be at least 0.

 Velocity is the ratio of the difference in speed and the difference in time, while instantaneous velocity can be viewed as the limit of the velocity as the change in time approaches 0, or simply put, the derivative of the position function

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o The sign of the instantaneous velocity at time is interpreted as the direction of the moving object along its position function

 ( ) means that the object is moving to the positive direction of the system

 ( ) means that the object is moving to the negative direction of the system

 ( ) means that the moving object is changing direction

 The instantaneous speed of the moving object can be viewed as the absolute value of its velocity at a certain time,

 Acceleration can be viewed as the ratio of the difference in velocity and the difference in time, while instantaneous acceleration can be viewed as the limit of the ratio as the change in time approaches 0, or simply put, the derivative of the velocity function

o The sign of the instantaneous acceleration at time is interpreted as the behaviour of the moving object along its position function

 ( ) means that the moving object is speeding up  ( ) means that the moving object is slowing down

 ( ) means that the moving objects is travelling at a constant rate

2.8. Rates of Change

 A derivative can be viewed as the rate of change in per unit change in

 The instantaneous rate of change of with respect to is the limit of the average change of with respect to , as the change in x approaches 0

Note:

 The average rate of change is the slope of the secant line

 The derivative of a function at a point can be interpreted as the rate of change of per unit change of at the instant

2.9. Local Linear Approximation and Differentials Recall:

 The equation of the tangent line at point ( ( )) is ( ) ( )( )  ( ) ( ) ( ) o ( ) ( ) o ( ) ( ) ( ) ( ) ( ) ( ) ( )

 Let ( ) be a function differentiable at a point

o The differential of the independent variable, denoted as , denotes an arbitrary increment of

o The differential of the dependent variable associates , denoted by , is given as ( )

 The approximation to the function , given by ( ) ( ) ( ) , is called the

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local linear approximation of the function at such point, that is, the tangent line of the graph of the function at such point approximates the graph of the function when is near that point

Note:

 ( ) ( )

 The symbol can be seen either as the derivative of with respect to , or the quotient of the differentials, that is geometrically, the rise and run of the tangent line at a point

Remarks:

 It can be shown that the local linear approximation of near is the best approximation of near

 A function that is differentiable at a point is said to be locally linear at ( ( )) 

( ) ( ) ( )

 If , then ( ) ( ) , thus for sufficiently small values of

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3. Behaviour and Analysis of Functions 3.1. Related Rates

 Let be a quantity that is a function of time, , then is the rate of change of with respect to time

 A problem on related rates is a problem involving rates of changes of several variables where a variable is dependent on another

 In particular, if is dependent on , then the rate of change of with respect to time depends also on the rate of change of with respect to time

Note:





 If the derivative is equal to 0, then is constant as time increases

3.2. Analysis of Functions: Relative Extrema

 A function is said to have a relative maximum[minimum] value at a point if ( ) is defined and ( ) ( ), ( ) ( )-

 A function is said to have a relative extremum value at if it has either a relative maximum or minimum value at

Theorem:

 If has a relative extremum at , then ( )

 If is continuous from the left[right] of and [ ] ( ) exists, then

, -( ) [ ] ( )

Note:

 If has a relative maximum[minimum] value at , then ( ( )) is the relative extremum point and ( ) is the relative extremum value

 The converse of the first theorem is not true – a function may be defined on its interval and the first theorem is satisfied, but will not have a relative extremum at a point

 The number c is a critical number of the function if ( ) ( )  We call the points on the graph of the

function at which the first condition of the theorem is satisfied, as the stationary point

3.3. Analysis of Functions: Increasing, Decreasing, and the First Derivative Test

 If the function is defined on an interval , then the function is said to be increasing [decreasing] on if ( ) ( ), ( ) ( )-

 A function is said to be monotonic on if it is either increasing or decreasing

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Theorem:

 Let be a function that is continuous on , - and differentiable on its open interval

o ( ) ( ) , - o ( ) ( ) , - o ( ) ( ) means that

the function is constant on

Theorem: First Derivative Test

 Let be continuous on ( ) and is differentiable on the same interval, except possible at

o If ( ) ( ) ( ) ( ), then has a relative maximum at

o If ( ) ( ) ( ) ( ), then has a relative minimum at

o If ( ) ( ), then there is no relative extremum at

Note:

 To find all relative extrema of a function, o Determine its critical numbers o Apply the First Derivative Test

3.4. Analysis of Functions: Concavity and the Second Derivative Test

 The function is said to be concave up[down] at point ( ( )) if its derivative at exits and if , the point with coordinates ( ( )) is above[below] the tangent line to the graph of at

 The function has a point of inflection (POI) at is is continuous at and if the function changes concavity at , that is, ( ) o ( ) ( ) ( ) ( ) o ( ) ( ) _{( )} ( ) Theorem:

 Let be a function such that its first two derivatives exist in an open interval ( )

o ( ) ( ) o ( ) ( )  If P is a POI, then ( ) ( )  is the critical number of

Theorem: Second Derivative Test

 Let be a function such that both derivatives exist in that contains and

( )

o If ( ) , then the function has a relative maximum at

o If ( ) , then the function has a relative minimum at o If ( ) , then SDT fails Note:  _{( ) ( ) ( ) ( )}  _{( ) ( ) ( ) ( )} Remarks:

 The graph of a function may not have a POI at even if the conditions in the second theorem are satisfied

 To find all the POI of a function, o Find all critical numbers of

 The critical numbers should be in the domain of

o Check the concavity of the intervals containing the critical numbers

3.5. Sketching of Functions Recall:

 If ( ) , ( ) -, then

there is an open interval ( ) , ( ) -

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Vertical Asymptotes

 The line with equation is a vertical asymptote of the graph of the function ( ) if at least one of the possible statements is true: ( )

Horizontal Asymptotes

 The line with equation is a horizontal asymptote of the graph of the function ( ) if at least one of the possible statements is true: ( )

Oblique Asymptotes

 The line with the general linear equation is an oblique asymptote of the graph of the function ( ) ( ) if at least one of the possible statements is true: ( )

3.6. Rolle’s Theorem and the Mean-Value Theorem for Derivatives

Theorem: Rolle’s Theorem

 If the function is continuous on , - and differentiable on its open interval, ( ) ( ) ( ) ( )

Theorem: Mean-Value Theorem for Derivatives

 If the function is continuous on , - and differentiable on its open interval, then ( ) ( )

Note:

 The value of is not unique

 The function need not be differentiable at the endpoints of the interval

 The condition of the closed interval’s continuity is necessary

 Rolle’s Theorem is a special case of MVTD, such that ( )

3.7. Absolute Extrema

 A function is said to have an absolute maximum[minimum] on an interval at point if ( ) ( ), ( ) ( )-

 If function has either and absolute maximum or minimum on at the same point, then it is said to have and absolute extremum on at the same point

Theorem:

 If has an absolute extremum on an open interval, ( ), then it must occur at any critical number

Theorem: Extreme Value Theorem

 If a function is continuous on , - then has both an absolute maximum and minimum on the endpoints or in between

Note:

 The converse is not true

 In order to find the absolute extremum of a function on a closed interval, the closed interval method is used

o Find the critical numbers

o Evaluate the function at the critical numbers that are inside the interval, including its endpoints

o The largest[smallest] of the values obtained from the previous step is

the absolute

maximum[minimum]value of the function on such interval

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4. Integration

4.1. The Indefinite Integral

 The function is an antiderivative of the function on an interval if ( ) ( )

 The process of finding the set of antiderivatives of is called antidifferentiation, or integration

Theorem:

 If ( ) and ( ) are antiderivatives of ( ) on an interval , then ( ) ( )

Note:

 The antiderivative of a function is not unique  If ( ) is an antiderivative of ( ) on , then

( ) is also an antiderivative of ( ) since , ( ) - , ( )-

, - ( ) ( ) ( )

 The symbol denotes the operation of integration

 The expression ( ) is called the indefinite integral [ ( ) ( ) ]  We now interpret ( ) as the set of all

functions whose derivative is ( )  ( ) ( ) , ( )- ( ) Techniques in Integration 1. 2. * + 3. ( ) ( ) ( ) 4. , ( ) ( )- ( ) ( ) ( ) ( ) 5. 6. 7. 8. 9. 10. Note:

 Theorem 4 can be extended to a finite number of functions. If a set of functions have their antiderivatives on the same interval, then ,∑ ( )-

∑ [ ( ) ]

4.2. Integration by Substitution Theorem:

 Let ( ) be an antiderivative of the continuous function

o If is a differentiable function with range I, then ( ( )) ( ) ( ( ))

Note:

 If ( ) ( ) , then ( ) ( ) ( ( ))

o This is called the method of substitution or the Chain Rule of Antidifferentiation

 Objectives when using the method of substitution

o Simplify the integrand to a form that can be integrated

o Substitution usually involves radicals and repetitive functions

4.3. Separable Differential Equations

 An ordinary differential equation (ODE) is an equation where the unknown is a function and which involves derivatives and differentials of the unknown

 The order of a differential equation is the order of the highest derivative of the equation

 The function ( ) is a solution of an ODE if the equation is satisfied when and its derivative or differentials are substituted into the equation

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 The set of all solutions of an ODE is called the complete/general solution

 A particular solution of an ODE is a solution of the equation where the parameter assumes a constant value

 The graph of a solution of an ODE is called an integral curve

 The graph of the general solution of an ODE is called the family of integral curves

 A differential equation that can be written in the form ( ) ( ) is said to be separable

Initial Value Problem

 The condition that when where is called the initial condition

 A problem of solving for a particular solution of a differential equation that is subject to an initial condition is called an initial value problem

Tips:

1. To solve ( ) ( ) , integrate both sides of the equation

2. To solve ( ), get the antiderivative of ( ) n times

3. When differentiating, make sure all variables in the integrand is the same as the variable of integration

Note:

 To solve an ODE means to find the general solution

 The general solution of the nth order of the ODE usually involves the same number of arbitrary parameters 4.4. Area  Sigma Notation o If , then ∑ ( ) ( ) ( ) ( ) ( )

 This is called the sigma/summation notation  The numbers and are

called the lower and upper limits of the summation, respectively

 The dummy variable is called the index of the summation

 It has terms  Area as a Limit

o The area is a unique positive real number associated to a polygonal region, and is, intuitively, the size of the region

o Let the function be continuous and non-negative on , -

o To determine the area of the region

R bounded by the graph of

( )

 Divide the interval into subintervals with equal length by inserting equally spaced points between and

 , -

 , -  Choose in each of the

subintervals, where , -

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 The area of the ith

rectangle with height ( ) and width is ( )

 An approximation of the area of the region is

∑

 The area of the region R is

given by ∑ ( ) Theorem:  ∑  ∑ ( ) ∑ ( )  ∑ , ( ) ( )- ∑ ( ) ∑ ( )  ∑ ( ) ∑ ( )  ∑ ( )  ∑ ( )( )  ∑ ( )  ∑ ( )( )( ) Note:

 The area doesn’t depend on the choice of  Some common choices for :

o Left Endpoint:

( )

o Right Endpoint: o Midpoint: ( )

. /

4.5. The Definite Integral

 Suppose a function is defined on , - o Divide the interval into

subintervals by choosing arbitrary points with

o The length of each partition is

o The largest subinterval is called the norm, ‖ ‖

o Choose an arbitrary on every subinterval

o The area of the ith

rectangle with height ( ) and width is ( )

o The approximation of the area of the region is ∑

o The area of the region is ‖ ‖ ∑ ( )

 is said to be integrable on the closed interval if the limit exists and does not depend on the choice of

 The definite integral, or Riemann integral, of from to is given as ( ) _{‖ ‖} ∑ ( )

 If is less than , and the function is integrable on the closed interval, then ( ) ( )

 If ( ) exists, then ( )

Theorem:

 If a function is continuous on , -, then it is integrable on the same interval

 Let be a function defined on , -

o If has finitely many discontinuities on , - and ( ) , then the function is integrable on the same interval  The Riemann sum is equivalent to the area

with equal partitions, that is, (‖ ‖ ) ( ) ( )

‖ ‖ ∑ ( )

∑ ( )

 ( )

 Let and be integrable functions on , - o ( ) ( ) o , ( ) ( )- ( )

( )

 If is integrable on a closed interval , then ( )

( ) ( )

19

Note:

 The lower and upper limits of integration are

a and b, respectively

 The variable in the integral is a dummy variable

 Let be a function that is continuous on , -

o If the function is non-negative on the interval, then the integral is equal to the area of the region under the graph of the function and the interval

o The integral is the area of the region, that is, above the interval and below the graph minus the area of the region that is below the interval and above the graph of the function, called the net signed area between the graph of the function and the interval

o Theorem 5b can be extended to a finite number of integrable functions on the same interval

4.6. Fundamental Theorems of Calculus and the Mean Value Theorem for Integration

 Let be function that is continuous on , -  , - ( ) ( ) ( )  If is non-negative on the same closed

interval, then the function is the area of the region under the graph of over the closed interval

Theorem:

 If is continuous on , -, and if is continuous on the range of on the same interval, then ( ( )) ( )

( )

( ) ( )

 If the function is integrable on , -, and if

o is even, then ( ) ( )

o is odd, then ( )  If ( ) ( ) is the velocity function of an

object moving along a line, then

o ( ) is the displacement of the particle during the time between the limits of integration

o ( ) is the distance of the moving object during the time between the limits of integration  If functions and are integrable on , -,

and if

o is non-negative on the same interval, then ( ) o ( ) ( ) , - ( ) ( ) o ( ) , - ( ) ( ) ( )

Theorem: Fundamental Theorem of Calculus, Part 1

 Let the function be continuous on , - and let , -

o If is the function defined by ( ) ( ) , then ( ) ( ) , - or ( ) ( )

Theorem: Fundamental Theorem of Calculus, Part 2

 If the function is continuous on , - and is the antiderivative of on the same interval, then ( ) ( ) ( )

Theorem: Mean-Value Theorem for Integrals

 If the function is continuous on , -, then , - ( ) ( )( )  If the function is continuous on , -, then

the average/mean value of the function on such interval is ̅ ( )

20

Note:

 FOTC-I shows that all continuous functions on I has an antiderivative on

 FOTC-II states that any derivative of may be chosen since if is an antiderivative of and c is a constant term, then ( ) ( ) ( ) ( ( ) ) ( ) ( )

 FOTC tells us that differentiation and integration are inverse processes

 If the indefinite integral is the set of all antiderivatives of a function, then the definite integral is the limit of the Riemann sum, which is a real number

4.7. Calculation of Area as a Definite Integral

 If is continuous and non-negative on , -, and be the region bounded by the graph of , the closed interval, and the -axis, then

( )

 If is non-positive, then ( )

 Suppose the function ( ) is continuous and non-negative on , - and is the region bounded by the graph of , the closed interval, and the -axis, then ( )

 If is non-positive, then ( )

Area between Two Curves

 Suppose and are continuous functions on , -, and , -

 Let be the region bounded by the graphs of and , and the closed interval

o ‖ ‖ , - o , ( ) ( )- o ∑

o _{‖ ‖} ∑ , ( ) ( )-

21

Note:

 The same principle applies to functions ( ) and ( ), only if  If and are non-negative, we have

( ) ( )  Useful tips in finding

o Sketch the region and identify the boundaries of

o Slice into vertical[horizontal] strips[rectangles] of area and express its length as a function of [ ]

o Determine the limits of integration from the figure and integrate with respect to [ ] to obtain the area of

4.8. Volume by Slicing, Disks, and Washers

 A right cylinder is a solid generated by moving a plane region (which is the base of the right cylinder) along a line or axis that is perpendicular to the region through a distance (height of the right cylinder)

Note:

 A right cylinder with a quadrilateral for a base is called a parallelepiped

 A right cylinder with a circle for a base is called a right circular cylinder

 Each cross section of a right cylinder is congruent to the base of the right cylinder 

Method of Slicing

 Suppose is a solid whose cross sections are perpendicular to the -axis and is bounded to the left and right by the planes that are perpendicular to the -axis and , -

o Pass planes at each endpoint of the subintervals of , thus slicing the thin slabs

o Let ( ) be the area of the cross region where , -

o The volume of the ith

slab is approximately equal to the volume of the right cylinder of height and base area ( )

o ∑ ( )

o _{‖ ‖} ∑ ( ) ( )

Useful tips in finding the Volume of a Solid with known Cross-Section by Slicing

1. Partition the axis that is perpendicular to the known cross-section

2. Slice into thin slabs by drawing planes perpendicular to the axis

3. Approximate the volume of the thin slab by treating it as a right cylinder and express the volume of as a definite integral

22

Volume by Disks

 A solid of revolution is the solid generated when a plane region is revolved about a line that lies in the plane of the region/axis of revolution

Note:

 The cross sections of a solid of revolution are perpendicular to its axis of revolution and are circles

Method of Disks

 Suppose the function is continuous and non-negative on , - and is the region bounded by the graphs of , the closed interval, and the x-axis

 Let be the solid of revolution obtained when is revolved about the -axis

o ( ) ( ( )) o ∑

o _{‖ ‖} ∑ ( ( )) , ( )-

Volume by Washers

 A washer is a circular disk with a hole in the middle

 Its volume is ( )

Method of Washers

 Suppose and are functions which are continuous on , - , -  Let be the region bounded by the two

functions and the closed interval, and let be the solid of revolution obtained when is revolved about the -axis

o ( ) ( )

o 0( ( )) ( ( )) 1 o ∑

23

4.9. Volume by Cylindrical Shells

 A cylindrical shell is a solid contained between two cylinders having the same center and axis

 The volume of the cylindrical shell is ( ) . / ( )

Method of Cylindrical Shells

 Suppose the function is continuous and non-negative on , - and is the region bounded by the graphs of , the closed interval, and the -axis

 Let be the solid of revolution obtained when is revolved around the y-axis

o ( ) o ( )

Note:

 The moving strips should be parallel to the axis of revolution

4.10. Arc Length of a Plane Curve

 Suppose is continuous on , -

o ̅̅̅̅̅̅̅̅

√( ) ( ( ) ( ))

o The line segment ̅̅̅̅̅̅̅̅

* + form a polygonal path from to that approximates the length of arc of the graph of over the closed interval

o Define each arc length of each segment as ̅̅̅̅̅̅̅̅̅

√( ) ( ( ) ( ))

o The approximate length of arc of over the closed interval is given by ∑

 Let the function be continuous on , - and the arc length of the graph of the function over the closed interval is ‖ ‖ ∑ , if the limit exists

 If exists, then the function is said to be rectifiable

 A function is said to be smooth on an interval if its derivative is continuous on the same interval

24

o It implies that is defined and continuous on the same interval o √ . ( ) ( )

/

o is continuous and differentiable on , -

o By MVTD, , - ( )

( ) ( )

o √ , ( )-

o The arc length of the graph of on the closed interval is given as √ , ( )-

Theorem:

 If the same conditions are satisfied for the function ( ) smooth on , -, then the arc length template can be used on

25

5. Special Functions and Cases

5.1. The Natural Logarithmic Function from the Integral Point-of-View Recall:  Consider o is continuous on ( ) o o ( ) o

 The natural logarithmic function is defined as Properties: a. ( ) b. If  ( )  . / 

c. It is continuous and increasing on its domain d. It is concave down at all points

e.

Theorem:

 , - 0 1

5.2. Logarithmic Differentiation

Steps in applying Logarithmic Differentiation

 Take the absolute value of both sides of the equation

 Take the natural logarithm of both sides of the equation

 Differentiate both sides of the equation implicitly with respect to

5.3. Integration of the Natural Logarithmic Function Theorem:      Remarks:  { 5.4. Inverse Functions  One-to-one Functions

o A function is said to be one-to-one if ( ) ( ), then

o Another verification is its contrapositive

 Horizontal Line Test

o A function is one-to-one if every horizontal line intersects the graph of the function in at most 1 point  First Derivative Test

o If a function is

26

, then the function is one-to-one on

 If the function is one-to-one, then the inverse function of , denoted by , is the set of all ordered pairs ( ) defined by _{( ) if ( )}

Properties:

a.

b. The graphs of and are symmetric with respect to the identity function

Theorem:

 If the function is continuous and increasing/decreasing on , then has an inverse function defined on the interval ( ) * ( ) +, that is continuous and increasing/decreasing on ( )

Differentiation of Inverse Functions

 Suppose is a function differentiable at point P1 and ( )

o ( )

o ( _{)( )}

o Equation of first tangent line: ( )( )

o Equation of second tangent line: _{( )}( )

o ( _{)( )}

( ) ( _{( ))}

Theorem:

 If the function is one-to-one and differentiable on an open interval , then ( _{)( )}

( _{( ))}

( ) ( _{( ))}

5.5. The Natural Exponential Function

 The natural exponential function, denoted by , is the inverse of the natural logarithmic function, that is, if , then

Properties:

a. ( )

b. It is continuous and increasing on its domain c. ( ) ( )

( )

d. The graph is concave up on all points

e. The graph is symmetric with respect to the identity function, and is the reflection of f. Theorem:  ( )  ( )  , -  , - ( ) 

5.6. Exponential and Logarithmic Functions and their Derivatives and Antiderivatives

Exponential Functions

 If , then the function ( ) is called the exponential function with base

27

Properties:

a. ( )

b. It is continuous on its domain

c. It is increasing on its domain if , and decreasing if ( )

d. Its graph is concave up on all points

Laws of Exponents    ( )  ( )  . /  Theorem:  , -  Logarithmic Functions

 If , then the logarithmic function with base a is the inverse of the exponential function with base , that is,

Properties:

a. ( )

b. It is continuous on its domain

c. It is increasing and concave up at all points when , and decreasing and concave down at all points when ( )

d. ( )

e. The graph of is symmetric to the graph of on the identity function

Theorem:       , -

Summary: Types of Functions with Exponents

 Positive Variable Base and Constant Exponent

( ) , ( )- , ( )- , ( )-

 Positive Constant Base with Variable Exponent

( ) ( )

, ( )- ( ) _{( )}

 Positive Variable Base with Variable Exponent

( ) , ( )- ( ) ( ) ( )

28

5.7. Derivatives and Antiderivatives of Inverse Trigonometric Functions

Restrictions for Trigonometric Functions

 Sine o 0 1 o , -  Cosine o , - o , -  Tangent o . / o  Cotangent o ( ) o  Secant o 20 / 0 /3 {, - 2 3} o ( - , )  Cosecant o 20 / 0 /3 {0 1 * +} o ( - , )

Inverse Trigonometric Functions

 The inverse sine function is defined as

 The inverse cosine function is defined as

 The inverse tangent function is defined as

 The inverse cotangent function is defined as

 The inverse secant function is defined as

 The inverse cosecant function is defined as Theorem:  , _- √  , _- √  , _-  , _-  , _{- {} √ ( ) √ , - 2 3  , _- { √ ( ) √ 0 1 * +  √   √  √ _{. /}  _{. /}  √ _{. /} 5.8. Hyperbolic Functions

 Hyperbolic Sine Function o o  Hyperbolic Cosine Function

o o , )  Hyperbolic Tangent Function

o o ( )  Hyperbolic Cotangent Function

o

o * + , -  Hyperbolic Secant Function

o

o ( -  Hyperbolic Cosecant Function

o o * + * +

29

Properties:

a. b.

c. The hyperbolic sine, tangent, cotangent, and cosecant functions are odd and one-to-one, while the hyperbolic cosine and secant functions are even

d. Hyperbolic functions are not periodic

Identities:      ( )  ( )   Theorem:  , -  , -  , -  , -  , -  , -        ( )   _{( )}  | |

5.9. Inverse Hyperbolic Functions

Restrictions of Hyperbolic Cosine and Secant Functions  Hyperbolic Cosine o , ) o , )  Hyperbolic Secant o , ) o ( -

Inverse Hyperbolic Functions

 The inverse hyperbolic sine function is defined as  The inverse hyperbolic cosine function is

defined as  The inverse hyperbolic tangent function is

defined as  The inverse hyperbolic cotangent function is

defined as  The inverse hyperbolic secant function is

defined as  The inverse hyperbolic cosecant function is

defined as Identities:  _{( √ )}  _{( √ )}  _. /  _. /  ₍ √ ₎  ₍ √ ) Theorem:  , _- √  , _- √  , _-  , _-  , _- √ ( )  , _- √

30  √ ₍ √ )  √ ₍ √ )  | | {  √ _{. / (} √ )  √ _{. / (} √ ) ( )  | ( ) | { _{. / . /}

5.10. Indeterminate Forms and L’Hopital’s Rule

 If ( ) ( ) ,

then _{( )} ( ) is said to be indeterminate

of the form or , and may exist even if it does not exist nor

 ( ) ( ) is of type if ( ) ( )  , ( ) ( )- is of type if ( ) ( )  ( ) ( ) is of type o if ( ) ( ) o if ( ) _{( )} ( ) o if ( ) ( )

Theorem: L’Hopital’s Rule

 Type and

o Let and be differentiable functions on some open interval I,

except possibly at a point a in I and ( ) * +

 If _{( )} ( ) is an

indeterminate form of type and ( )_{( )} ̅ , then _{( )} ( )  Type o Write ( ) ( ) as  ( ) ( ) , which will become of type  ( ) ( ) , which will become of type

o Apply the theorem for type  Type

o Combine the two function to obtain the indeterminate forms of type

o Apply the theorem for type  Types o Write ( ) ( ) ( ) ( ) ( ) ( ) o Solve for ( ) ( ) ( ) ( ) Note:

 If ( ) _{( )} is of type , and the derivatives

are continuous at a with ( ) , then ( )_{( )} ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  In general,  are not indeterminate forms

31

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 , ( ) ( )-

32

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

33

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 ( ) ( )

34

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