Elementary Analysis

Elementary Analysis Review Notes (Math 55 Notes to be added)

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

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

2. Derivatives...........................................................9 2.1.

Slopes and the Derivative......................9

2.2.

Techniques in Differentiation..............10

2.3.

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

4.8.

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

2.4.

Chain Rule ............................................10

4.9.

Volume by Cylindrical Shells ............... 23

2.5.

Implicit Differentiation ........................10

4.10.

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

2.6.

Higher-Order Derivatives ....................10 5. Special Functions and Cases ............................. 25

2.7.

Rectilinear Motion Problems ..............10

2.8.

Rates of Change ...................................11

2.9.

Local Linear Approximation and Differentials .........................................11

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

5.9.

Inverse Hyperbolic Functions ............. 29

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 1

5.10.

Indeterminate Forms and L’Hopital’s 9. Elementary Vector Analysis ............................. 48 Rule.......................................................30 9.1. Vector-valued Functions ..................... 48 6. Integration Techniques .....................................31 9.2. Calculus of Vector-valued Functions .. 48 6.1.

Integration by Parts .............................31

9.3.

Arc Length ........................................... 49

6.2.

Trigonometric Integrals .......................31

9.4.

Arc Length Parametrization ................ 49

6.3.

Trigonometric Substitution .................32

9.5.

6.4.

Integration by Partial Fractions ...........32

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

6.5.

Improper Integrals ...............................33

9.6.

Curvature............................................. 50

6.6.

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

9.7.

Curvilinear Motion .............................. 51

9.8.

Projectile Motion ................................ 51

6.7.

Orthogonal Trajectories ......................34 10. Multivariate Differential Calculus .................... 53 7. Parametric and Polar Curves ............................35 10.1. Multivariate Functions........................ 53 7.1. Review on Conic Sections ....................35 10.2. Limits and Continuity .......................... 53 7.2.

Parametric Equations ..........................36

10.3.

Partial Derivatives ............................... 54

7.3.

Derivatives of Parametric Equations ..36

10.4.

Implicit Partial Differentiation ........... 54

7.4.

Arc Length of Parametric Curves .........37

10.5.

Local Linear Approximation ................ 55

7.5.

Polar Coordinates ................................37

10.6.

Differentiability ................................... 55

7.6.

Graphs of Polar Equations ...................37

10.7.

Differentials......................................... 55

7.7.

Tangent Lines of Polar Curves .............40

10.8.

Multivariate Chain Rule ...................... 55

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 2

Elementary Analysis I

Types of Functions

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

1. Polynomial Functions of degree n ( ) i.

∑

Constant function ( ( )

)

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

ii. Linear function ( ( )

)

Operations on Functions  Let 1. ( 2. (

be functions; then we have: )( ) ( ) ( ) ( )( )

3. . / ( ) 4. (

) * )( )

( ) ( ) ( ) ( )

(

. /

(

+ ( ) ( ( )) ( ) +

(

)

)

)

*

iii. Quadratic function ( ( ) Zeroes: √

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 3

, (

)

.

/)

2. Rational Functions ( ) ( ) ( )

5. Greatest Integer function ( ) ⟦ ⟧   Denotes the greatest integer less than or equal to x, that is,

∑

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

6. Signum function ( )

( )

{

4. Absolute Value function √

( )

7. Piecewise function  Different functions in different intervals

2

1.2. Limits  Consider 3 functions: ( )

( )

( )

2

o

 Properties: a. b. c. d. | | e. f. g. h.

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 is,

) 4

( ) exists, then it is unique, that ( ) ( )



2. ( ) 3. If a function is given as an identity function, ( ) then 4. If ( ) and ( ) exists, then ( ) a. b.

( (

)

)

c. 5.

( ( ))

(

( ))

6.

( ( ))

(

7.

( )

( )) ( ) ( )



Theorem:  

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: is a polynomial or rational function, then ( ) ( ) ( ) ( ) ( )  If the limit of a function exists, then ( ) , ( ) ( ) o Remark: ( ) ( o  Limit theorems also apply to one-sided limits ) If

1.6. Infinite Limits 1.4. Formal Definition of a Limit  Consider ( )  Consider ( ) , a rational function o If , and is infinitesimal, then where ( ) and ( ) , with limits, ( ) means that the as approaches 0, 1 and 0, respectively function is really close to o The values of the function increase o It seems that , such that without bound as assumes values whenever , so that going closer and closer to 0, then ( ) ( )  If a function on some open interval about ,  Let be a function defined on both sides of ( ) then ( ) , then , means that ( ) the values of can be made arbitrarily large as we please by taking values of sufficiently close to 1.5. One-Sided Limits Recall:  

Theorem:

definition of a limit is defined on any open interval ( where ( ) ( ) ( )

 ),

If

, then o o {

5

that are



( )

( ) exists,

If if

( )

o

, and

o

, then

Notes:





 ( )

o

, then



  

( )

If then o

,

o 

-

( ) If exists, then , o ,

o

( ) exists,

and ,

{ and -

-

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

Notes:



 

are not ( )

real numbers, thus does not mean that it

exists The theorems sill fold for one-sided limits 

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

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- Theorem: ( ) infinite interval, then  If two functions are continuous on , then means that the values of the function can be the following are continuous on : made arbitrarily close to by taking o sufficiently large values of o Theorem: o 

If



, then o o

 { 6

All polynomial functions are continuous everywhere A rational function is continuous everywhere in its domain

  



If



, then ( ) √ is continuous Everywhere in

o 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: 



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

If is a function, then the possible points of discontinuity are: Theorem: Intermediate Value Theorem o Values of x that makes ( ) o Endpoints of piecewise intervals  If a function is continuous on , -, and ⟦ ⟧ o ( ) ( ), then If ( ) has neither removal nor essential , ( ) ( )( ) , discontinuity, then the function is simply  The value of is not necessarily unique discontinuous Corollary: Intermediate Value Theorem 

Remarks: 

To remove the discontinuity at is the equivalent of redefining the value of ( ) to form the function ( ), such that 1.9. Limits and ( ) Functions ( ) { ( ) Theorem: 

Continuity on Intervals  



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

 ) if it is A function is continuous on ( continuous on every real number in the  interval A function is continuous from the left[right] Note: of if: ( ) o  ( ) o , ( ) o , - ( ) A function that is continuous at is continuous on both sides of 7

Continuity

of

Trigonometric

The trigonometric functions are continuous on their respective domains

Theorem: Squeeze/Sandwich Theorem 

Let (

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

8



2. Derivatives 2.1. Slopes and the Derivative  Suppose a secant line passes through 2 points, and , on the graph of a function, ( )



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 , ( ), ( )- is given as written as ( )

[



(

)

]

( ) ( ) derivatives exist

( )

if

both

Note:   o

The slope of the secant line is ( )

o o

(

)

( )

 

)

If we let approach , approaches , and approaches 0 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 ,

-

(

)

)

(

(





 

)

(

)

Notations

for ,

)

-

, ( )-

derivatives: , ( )-

Remarks:

)



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 (

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 ( )

o



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

 

( )

( ) 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 9

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

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

( ) ( ) ( ) ( ) ( )

( )

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

( )

( )

2.3. Derivatives of Trigonometric Functions 1.

2.6. Higher-Order Derivatives  The nth derivative of the function , denoted ( ), that by ( ), is the derivative of

2.

is,

3.

(

( )

)

( )

4.

Remarks:

5.



The derivative of a function is also called the first derivative



Notations:

6. 

2.4. Chain Rule

( )

, ( )( )

, ( )-

( )

.

/

Theorem: 

If the function is differentiable at , and is differentiable at ( ), then 2.7. Rectilinear Motion Problems ( )( )  Derivatives can be used to describe the ( ( )) ( ) behaviour of a moving object, say, a particle, Note: using its position function ( ), where denotes time and must be at least 0.  The Chain Rule can also be stated as  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  The Chain rule can also be extended to a approaches 0, or simply put, the derivative finite number of functions of the position function

10

o

The sign of the instantaneous Note: velocity at time is interpreted as  The average rate of change is the slope of the direction of the moving object the secant line along its position function  The derivative of a function at a point can be ( )  means that the interpreted as the rate of change of per object is moving to the unit change of at the instant positive direction of the system ( )  means that the object is moving to the 2.9. Local Linear Approximation and Differentials negative direction of the system Recall: ( )  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  The equation of the tangent line at point object along its position function ( )) is ( ) ( )( ( ( )  means that the ) moving object is speeding up ( ) ( ) ( )  means that the ( )  moving object is slowing ( ) o down ( ) ( )  means that the ( ) ( ) ( ) o moving objects is travelling ( ) ( ) ( ) at a constant rate ( )  Let ( ) be a function differentiable at a point 2.8. Rates of Change o The differential of the independent  A derivative can be viewed as the rate of variable, denoted as , denotes an change in per unit change in arbitrary increment of  The instantaneous rate of change of with o The differential of the dependent respect to is the limit of the average variable associates , denoted by change of with respect to , as the change ( ) , is given as in x approaches 0  The approximation to the function , given ( ) ( ) , is called the by ( ) 11

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 approximation of near approximation of near A function that is differentiable said to be locally linear at (

 

( If

( ) values of

)

local linear is the best at a point is ( ))

( ) ( ) ( ) , then , thus for sufficiently small

12

3. Behaviour and Analysis of Functions Theorem: 3.1. Related Rates  If has a relative extremum at , then  Let be a quantity that is a function of time, ( ) , then is the rate of change of with  If is continuous from the left[right] of respect to time and [ ] ( ) exists, then  A problem on related rates is a problem , -( ) [ ] ( ) involving rates of changes of several variables where a variable is dependent on Note: another  In particular, if is dependent on , then the  If has a relative maximum[minimum] value rate of change of with respect to time at , then ( ( )) is the relative extremum depends also on the rate of change of with point and ( ) is the relative extremum respect to time value  The converse of the first theorem is not true Note: – 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 )  If the derivative is equal to 0, then is  We call the points on the graph of the constant as time increases function at which the first condition of the theorem is satisfied, as the stationary point 3.2. Analysis of Functions: Relative Extrema  A function is said to have a relative maximum[minimum] value at a point if 3.3. Analysis of Functions: Increasing, Decreasing, and the First Derivative Test ( ) is defined and  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



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

13

Theorem:

Theorem:





Let be a function that is continuous on , - and differentiable on its open interval , ( ) ( ) o , ( ) ( ) o ( ) ( ) means that o the function is constant on

Theorem: First Derivative Test

 

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 continuous on ( and is  Let be a function such that both differentiable on the same interval, except derivatives exist in that contains and possible at ( ) ( ) ( ) ( ) o If o If ( ) , then the function has a ( ), then has a relative relative maximum at maximum at o If ( ) , then the function has a ( ) ( ) ( ) o If relative minimum at ( ), then has a relative o If ( ) , then SDT fails minimum at o If ( ) ( ), then there Note: 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

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

 3.4. 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 ( )) is point with coordinates ( above[below] the tangent line to the graph of at 3.5. Sketching of Functions  The function has a point of inflection (POI) at is is continuous at and if the Recall: function changes concavity at , that is, , ( )  If ( ) there is an open interval ( ) ( ) ( ) o , ( ) ( ) ( ) ( ) ( ) ( ) o ( ) 14

( )

-, then

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 

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

The line with equation is a horizontal asymptote of the graph of the function ( ) Theorem: if at least one of the possible statements is true: ( )  If has an absolute extremum on an open ), then it must occur at any interval, ( Oblique Asymptotes critical number 

The line with the general linear equation is an oblique asymptote of the graph of the function ( ) ( ) if at least one of Theorem: Extreme Value Theorem the possible statements is true: - then  If a function is continuous on , ( ) has both an absolute maximum and minimum on the endpoints or in between

3.6. Rolle’s Theorem and Theorem for Derivatives

the

Mean-Value Note:  

Theorem: Rolle’s Theorem 

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

Theorem: Mean-Value Theorem for Derivatives 

- and If the function is continuous on , 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 ( )

15

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

4. Integration Note: 4.1. The Indefinite Integral  Theorem 4 can be extended to a finite  The function is an antiderivative of the number of functions. If a set of functions ( ) function on an interval if have their antiderivatives on the same ( ) ,∑ ( )interval, then  The process of finding the set of ∑ [ ( ) ] antiderivatives of is called antidifferentiation, or integration Theorem: 

If

4.2. Integration by Substitution

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

Note:  

Let ( ) be an antiderivative of the continuous function o If is a differentiable function with range I, then ( ( )) ( ) ( ( ))

The antiderivative of a function is not unique If ( ) is an antiderivative of ( ) on , then ( ) is also an antiderivative of Note: , ( ) , ( )( ) since , ( ) ( ) ( ) ( ) ( ) ,  If then  The symbol denotes the operation of ( ) ( ) ( )) ( integration o This is called the method of ( )  The expression is called the substitution or the Chain Rule of ( ) indefinite integral [ ( ) ] Antidifferentiation  We now interpret ( ) as the set of all  Objectives when using the method of functions whose derivative is ( ) substitution , ( )( ) ( )  o Simplify the integrand to a form that ( ) can be integrated o Substitution usually involves radicals and repetitive functions Techniques in Integration 1. *

2. 3. 4.

( ) , ( ) ( )

( ) ( )( )

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

+ ( )

( ) ( )

5. 6. 7. 8. 9. 10.

16

    

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

 





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 ,

Tips: ( ) 1. To solve ( ) sides of the equation

, integrate both

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

17

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 , -

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:    

∑ ∑ ∑ ∑



∑



∑



∑



 () ∑ , () ( )() ∑ (

() )

∑



()

) (

)( (



)

)

(

∑

() ∑ (

)(

)

 

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

-, then it is If a function is continuous on , integrable on the same interval Let be a function defined on , o If has finitely many discontinuities - and ( ) on , , 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



o

The length of each partition is

o

The largest subinterval is called the norm, ‖ ‖ Choose an arbitrary on every subinterval

o

( )

If ( ) exists, then

Theorem: )(

Note:  

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 ( ) ( ) o

18

If

, ( )

( )-

-

( )

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

( )

( ) o is odd, then ( ) is the velocity function of an If ( ) object moving along a line, then ( ) is the displacement of the o

Note:   

 The lower and upper limits of integration are a and b, respectively The variable in the integral is a dummy particle during the time between the variable limits of integration Let be a function that is continuous on , ( ) o is the distance of the o If the function is non-negative on the moving object during the time interval, then the integral is equal to between the limits of integration the area of the region under the -,  If functions and are integrable on , graph of the function and the and if interval o is non-negative on the same o The integral is the area of the region, ( ) interval, then that is, above the interval and below , ( ) ( ) o the graph minus the area of the ( ) ( ) region that is below the interval and above the graph of the function, ( ) o , ( ) 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 Theorem: Fundamental Theorem of Calculus, Part 1 on the same interval 

Let the function be continuous on , , and let o If is the function defined by ( ) ( ) , then ( )

4.6. Fundamental Theorems of Calculus and the Mean Value Theorem for Integration  Let be function that is continuous on , , - or ( ) ( ) , ( ) ( )  ( ) ( )  If is non-negative on the same closed interval, then the function is the area of Theorem: Fundamental Theorem of Calculus, Part 2 the region under the graph of over the - and  If the function is continuous on , closed interval is the antiderivative of on the same Theorem: ( ) ( ) ( ) interval, then 



-, and if If is continuous on , is Theorem: Mean-Value Theorem for Integrals continuous on the range of on the same interval, then ( ( )) ( ) -, then  If the function is continuous on , ( ) , ( ) ( )( ) ( ) ( )  If the function is continuous on , -, then -, and If the function is integrable on , the average/mean value of the function on if ( ) ( ) o is even, then such interval is ̅ ( )

19

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



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  



-, 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 and are continuous functions -, and , on , Let be the region bounded by the graphs of and , and the closed interval

( )

o o o o

, ‖ ‖ , ( ) ∑ ‖ ‖

, ( )

20

( )∑

( )-

Note:   

Method of Slicing 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:    



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

o o

Pass planes at each endpoint of the subintervals of , thus slicing the thin slabs Let ( ) be the area of the cross , region where The volume of the ith slab is approximately equal to the volume of the right cylinder of height and base area ( ) ∑ ( ) ∑ ( ) ‖ ‖

A right cylinder with a quadrilateral for a base is called a parallelepiped o A right cylinder with a circle for a base is o called a right circular cylinder ( ) Each cross section of a right cylinder is congruent to the base of the right cylinder 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

21

Volume by Disks

o o

( )

( ( ))

∑

o

‖ ‖

∑

( ( ))

, ( )-

Volume by Washers



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:

A washer is a circular disk with a hole in the middle ( ) Its volume is



The cross sections of a solid of revolution are Method of Washers perpendicular to its axis of revolution and  Suppose and are functions which are are circles , continuous on ,  Let be the region bounded by the two Method of Disks functions and the closed interval, and let  Suppose the function is continuous and be the solid of revolution obtained when is - and non-negative on , is the region revolved about the -axis 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 22

o o o o

( )

( )

0( ( ))

( ( )) 1

∑ 0( ( ))

( ( )) 1

Note: 

4.9. Volume by Cylindrical Shells  A cylindrical shell is a solid contained between two cylinders having the same center and axis

4.10. 



Arc Length of a Plane Curve Suppose is continuous on ,

o

̅̅̅̅̅̅̅̅ √(

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

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 Define each arc length of each ̅̅̅̅̅̅̅̅̅ segment as √(

o





-

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

Method of Cylindrical Shells 

The moving strips should be parallel to the axis of revolution

Let be the solid of revolution obtained when is revolved around the y-axis ( ) o ( ) o

  

23

)

)

( (

)

(

))

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 Suppose is continuous on ,

o

It implies that is defined and continuous on the same interval √

o o o

(

)

(

)

/

is continuous and differentiable on , , - ( ) 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

24

5. Special Functions and Cases 5.1. The Natural Logarithmic Function from the 5.2. Logarithmic Differentiation Integral Point-of-View Steps in applying Logarithmic Differentiation

Recall: 

 Consider  

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 Function

of

the

Natural

Logarithmic

Theorem:

is continuous on (

o

    

)

o (

o

Remarks:

)

o 



Properties: (

a. b. If  

5.4. Inverse Functions  One-to-one Functions o A function is said to be one-to-one ( ) if ( ), then

)

( ) . /

o

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



Theorem: 

{

The natural logarithmic function is defined as

 ,

-

0

1

25

Another verification contrapositive

is

its

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 increasing/decreasing on an interval

, then the function is one-to-one on Theorem: 



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

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

( )

(

( ))

Properties: a.

5.5. The Natural Exponential Function  The natural exponential function, denoted b. The graphs of and are symmetric with by , is the inverse of the natural respect to the identity function logarithmic function, that is, if , then 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 

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

Suppose is a function differentiable at f. point P1 and ( ) Theorem:    

( ) ( ) , ,

-

( )



5.6. Exponential and Logarithmic Functions and their Derivatives and Antiderivatives o o o o o

( ) Exponential Functions ( )( ) Equation of first tangent line:  If , then the function ( )( ) ( ) is called the exponential function Equation of second tangent line: with base ( ) ( ) (

)( )

( )

(

( ))

26

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:   Properties:

  

a.

( ) b. It is continuous on its domain c. It is increasing on its domain if ( ) decreasing if d. Its graph is concave up on all points



, and



  ( (



. /

) )



 

Theorem: 

,

-



Logarithmic Functions 

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

Properties: a.

(

-

Summary: Types of Functions with Exponents

Laws of Exponents

 

,

) 27

Positive Variable Base and Constant Exponent ( ) , ( ), ( ), ( )Positive Constant Base with Variable Exponent ( ) ( ) ( ) , ( )( ) Positive Variable Base with Variable Exponent ( ) ( ) ( ) , ( )- ( ) *Use Logarithmic Differentiation

5.7. Derivatives and Antiderivatives of Inverse Theorem: Trigonometric Functions ,  Restrictions for Trigonometric Functions ,   Sine ,  o 0 1 ,  , o  Cosine ,  , o , o ,   Tangent .

o  

o Cotangent o o Secant

(

)

o

20

/

{, 

/

o Cosecant



/3

0



-

,

-

(

(

√

{

,

√

-

(

) - 2 3

) 1 * +

0

√

/

0

-

,

√ √



)



. / . /

√

. /

/3

1 * +}

{0 o

-



- 2 3} (

√

-

√

{ 

√

-

√

20

o

-

5.8. Hyperbolic Functions  Hyperbolic Sine Function

)



o o Hyperbolic Cosine Function



o , ) o Hyperbolic Tangent Function



o ( ) o Hyperbolic Cotangent Function

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





28

o o * + , Hyperbolic Secant Function o ( o Hyperbolic Cosecant Function o o * + * +

-

Properties:

5.9. Inverse Hyperbolic Functions

Restrictions of Hyperbolic Cosine and Secant a. b. Functions c. The hyperbolic sine, tangent, cotangent, and  Hyperbolic Cosine cosecant functions are odd and one-to-one, , ) o while the hyperbolic cosine and secant functions , ) o are even  Hyperbolic Secant d. Hyperbolic functions are not periodic , ) o Identities: ( o      

Inverse Hyperbolic Functions  ( (



) )



 

 

Theorem:               

, , , , , ,

Identities:  

(

)

( (

√ √



.

/



.

/



(



(

√

) )

) √

)

Theorem: (





-

The inverse hyperbolic sine function defined as The inverse hyperbolic cosine function defined as The inverse hyperbolic tangent function defined as The inverse hyperbolic cotangent function defined as The inverse hyperbolic secant function defined as The inverse hyperbolic cosecant function defined as

) |

|

29



,

-



,

-



,

-



,

-



,

-



,

-

√ √

√ √

(

)

is is is is is is

 √ 

except possibly at a point a in I and ( ) * + ( )  If is an ( )

(

√

) (

√

√

indeterminate form of type

)



|

̅ , then

{ 

. /

√

√ 

. /

√

(

) |

(

)

(

( ) ( )

(

( )



|

,

which

will

,

which

will

( )

become of type 

o Type o

if ) ( ) if ( )

Apply the theorem for type Combine the two function to obtain the indeterminate forms of type

o Apply the theorem for type Types ( ) o Write

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 ( ) ( ) Note: ( ) ( ) is of type ( ) o if ( )  If o

( ) as

become of type

)

Indeterminate Forms and L’Hopital’s Rule If ( ) ( ) ,

o

( ) ( )

. /

then



( )

Write

. /

{



Type o









(

)

√

5.10. 

( ) ( ) ( ) ( )

and

|

( )

( )

( )

o

( )

( ) ( )

( )

( )

( )

Solve for

is of type , and the derivatives

( ) ( ) ( ) ( )



In

( )

( )

( )

are continuous at a with

( )

( )

( ) ( )

( )

( )

( )

( )

, then

( ) ( )

general,

Theorem: L’Hopital’s Rule 

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

30



are indeterminate forms

not

Elementary Analysis II 6. Integration Techniques

Integration by Parts of Definite Integrals

6.1. Integration by Parts  Suppose we want to evaluate an integral of ( ) ( ) , assuming that is the form differentiable and ( ) is an antiderivative of ( ) o o

, ( ) ( )-

o

Integrating ( ) ( ) ( ) ( )

The definite integral can be solved with integration by parts, provided that the functions satisfy its conditions



( ) ( )

( ) ( ) Deriving the equation, , ( ) ( )-



6.2. Trigonometric Integrals 1. Integrating powers of sine and cosine

( ) ( )

( ) ( ) both ( ) ( )

sides,





Theorem: 2. Integrating products of sine and cosine  o If is odd,  Split off a factor of  Use the Pythagorean identity Note:  Let  As a rule of thumb, the order of choosing the o If is odd, term is: Logarithmic, Inverse  Split off a factor of trigonometric, Algebraic, Trigonometric, and  Use the Pythagorean identity Exponential  Let o If both and are even, 

By letting ( )

( )

( ) and ( ), then , which is integration by parts

Tabular Integration by Parts 





( ) ( ) , tabular integration by Given parts can be used if one of the functions is finitely differentiable and the other function is integrable ( ) ( ) ∑ ( ) ( ) ( ) ( )





Use



Use

( ) ( o

) 

and Use ,

o

31

(

)

(

)-

(

)

(

)-

, 

Use ,

The consequence of tabular integration by parts is that it cannot be used when the first function is infinitely differentiable

,

o

, 

o

√

Use ,

(

)

(

)-

3. Integrating powers of tangent and secant

Steps in Integration Substitution



using

Trigonometric

1. Substitute the values for and 2. Integrate 3. Return the variables to its original form



4. Integrating products of tangent and secant  6.4. Integration by Partial Fractions o If is odd,  Linear Factor Rule  Split off a factor of o Factors of the form (

o

o

If

If

 Use the Pythagorean identity  Let is even,  Split off a factor of  Use the Pythagorean identity  Let is even and is odd,  Use the Pythagorean identity  Use the reduction formula for powers of

Note:   

) 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, (

)

∑ ( ) For powers of sine and cosine, should be a positive integer Note: For powers of tangent and secant, should be greater than 1  If the degree of the numerator is greater To evaluate integrals of cosecant and than or equal to the degree of the cotangent, use the formulae for tangent and denominator, then long division must first be secant, and substitute the corresponding carried out before advanced to partial cofunctions fraction decomposition  Partial fraction decomposition gives way to the easier use of simple integration

6.3. Trigonometric Substitution  Substitutions are used if the following expressions are found in the integrand: o

√

o

√ 32

6.5. Improper Integrals Improper Integrals with Infinite Discontinuity  Improper integrals are definite integrals  Consider the same function whose limit of integration reaches infinity, is o It has an infinite discontinuity at a value of which makes the graph of the function infinitely discontinuous, or a o By first inverting the interval such combination of both ( -, the new area of the that region bounded by the function and Improper Integrals with Infinite Integration Intervals the interval is 

o

Consider ( )

Theorem: 

-, except at and If is continuous on , infinite discontinuity at , then the improper integral of

over ,

- is

( )

( ) 

-, except at and If is continuous on , infinite discontinuity at , then the improper integral of

o o

( ) The area of the region bounded by , ( ) and is

over ,

- is

( )

( ) 

If is continuous on , infinite discontinuity at improper integral of

-, except at an ( ), then the - is over ,

( ) o

( )

Theorem: 

( )

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 ( )

33

( )

Some Applications of SODEs 

Malthusian Population Model ( ) o Let be time, population Recall: at given time, birth rate, and death rate, respectively (

o

)





Integrating both ( ) *( ) + o The initial population will population at time zero, ( ) ( ) ) +  ( ) ( )  Verhulstian Population Model o

( )

Let

.

sides, ( ) be the that is, *(

/ be time,

population at given time, carrying capacity, and per capita income increase, respectively .

o 

/

Integrating | *

The

sides,

( )

|

*

o

both

+ +

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 34

member of one family is orthogonal to each member of the other

7. Parametric and Polar Curves 7.1. Review on Conic Sections 1. Parabola  A parabola is a set of points on the plane equidistant from a fixed line, called a directrix, and a fixed point, called the focus, not on the line  Assuming that the vertex is located at the origin, let ( ) be the focus, be the equation of the directrix, and is the distance of the focus to the vertex



Assuming that the center of the ellipse is at the origin, let ( ) and ( ) be any point on the ellipse

o

o

(

√

√(

)

̅̅̅̅



√(

)

√(

) √(



)

√(

√(

)



√(

)





) –

the

̅̅̅̅̅

) 

̅̅̅̅

(

√ coordinates of the foci ̅̅̅̅̅ ) √( √(

o

)

.

)

/ – the equation

of the ellipse



– the equation of Orientation of Ellipses the parabola  is an oblate ellipse Orientation of Parabolas  is a prolate ellipse  opens to the right if and Note: left if 

opens upward if

and

 downward if  The names of the axes where the two 2. Ellipse parameters lie are the major and minor axes,  An ellipse is a set of points on the plane respectively whose sum of distances from two fixed 3. Hyperbola points, called the foci, is a given positive  A hyperbola is a set of points on the plane constant that is greater than the distance whose difference of distances from the foci between the foci 35



is a given positive constant that is less than Translation of Conic Sections the distance between the foci  A translated conic section is a conic section Assuming that the center of the hyperbola is whose center (vertex, in the case of at the origin, let ( ) and ( ) be any ) parabolas) is located at ( point on the hyperbola  To translate a conic section, let and , and simply substitute them in the equation of the conic section Note:  

The orientation and shape is preserved in translation of conic sections The values of parameters and will become distances from the center (vertex, in the case of parabolas)

– the coordinates of √ the foci o The proof of the equation of the hyperbola is almost the same with 7.2. Parametric Equations  Let ( ) and ( ) be functions of a that of the ellipse, except for the parameter, signs and the output:  The pair of equations are called parametric o The hyperbola is asymptotic to lines equations  The graph of a pair of parametric equations is called a parametric curve Orientation of Hyperbolae  The direction at which a parametric curve is traced, as the parameter increases, is called  opens horizontally the orientation of the curve  opens vertically o

Note: 



7.3. Derivatives of Parametric Equations  Let ( ) and ( ) be differentiable The names of the axes where the two ( ) functions of , then and parameters lie are the transverse and conjugate axes, respectively ( ) The foci always lie on the transverse axis  If is the parametric curve defined by ( ) and ( ), then the slope of the tangent line to

36

is

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 Conversion of Polar and Rectangular Coordinates 

parametric equations, then 

The second order derivative,

, can be



(

expressed as 

In general, the nth derivative of

Polar to ( ) ( Rectangular )

.√

)

( )

Rectangular ) to Polar

-

/

is Remarks:  

( (

(

) )

7.4. Arc Length of Parametric Curves 7.6. Graphs of Polar Equations  Let be the parametric curve defined by  A polar equation is an equation of the form ( ) ( ) ( )  If no segment of is traced more than once from to , then the arc length of Theorem: from to is  A polar curve is symmetric about the x-axis if , , given that the pair √, ( ) ( ) of parametric equations is differentiable  A polar curve is symmetric about the y-axis if over , ( ) ( )  A polar curve is symmetric about the origin if ( ) ( ) or if negating the equation will still produce an equivalent equation 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 37

1. Lines and Rays 



2. Circles 

By strictly letting

* +





o o o

38

The graphs can be completed from to ( ) Negating would flip the graph to the non-symmetric axis



3. Roses 

* +

The length of the inner loop is

o

Cardioid -

o

Dimpled Limaçon -

(

)

o { o o

The length of each petal is The period of the graph is , , and , 4. Limaçons  o Limaçon with an inner loop (

 o

The distance from the pole to the dimple is

Convex Limaçon -

)



39

The nearest distance is

Remarks:  

The length of the bulb is The graph intersects the vertical axis at



The graph bulbs at the polar axis when , and at the vertical axis at

7.7. Tangent Lines of Polar Curves ( )  Let  In rectangular coordinates, ( ) , and ( ) ( )



( ) ( )(

 ( ) )

To find the area of the region bounded by ( ) o Divide the region into sectors o , with ‖ ‖ , o Choose

Theorem: 

If the polar curve, defined by passes throught the pole at | ( )

|

,

o

The approximation for the area of ∑ the region is ∑

o

(

)

The area of the region is ∑ ‖ ‖ ‖ ‖

is tangent to the curve at the pole

))

Define

( ), then

( (

o

∑

(

)

, ( )7.8. Arc Length of Polar Curves ( ) be the polar equation of the 7.10.  Let curve,   If no segment of is traced more than once from to , then the arc length  of

from

to

is

√

Conic Sections in Polar Coordinates Let be a point of a conic section with a focus at and directrix The eccentricity of a conic, , is defined as ̅̅̅̅ ̅̅̅̅

. / 

7.9. Areas in Polar Coordinates ( )be either non-negative or non Let positive from to , where

40

Let ( ) be a point of a conic section with the pole as the focus, and be the distance of the directrix to the pole, assuming that it is to the right of the pole

(

o ( o

)

) √

√

Note: 

̅̅̅̅ ̅̅̅̅

o o

– the equation of the conic section

Theorem:   

The eccentricity of a parabola is ( The eccentricity of an ellipse is The eccentricity of a hyperbola is

)

Orientation of Polar Form Conic Sections 

– the directrix is to the right of the pole



– the directrix is to the left of the pole



– the directrix is below the pole



– the directrix is above the pole

Special Results for Polar Form Ellipses and Hyperbolae 

Let distance of a point to the nearest focus, and distance of a point to the farthest focus o (

)

41

The same proof goes for the case of hyperbola, except that and are interchanged, and is still constant

8. The Real Space Quadric Surfaces 8.1. Three Dimensional Coordinate System  Ellipsoid  A point in is defined by an ordered o triple, ( ) )  To locate in , find first the point (  Hyperboloid in the -plane then move the point units o One sheet up if , or down if  Distance Between Two Points  

Let in In

(

) and

 (

) be points

 o

the

-plane,



√ 



( ) √( ) Suppose another plane exists where lies,



and 

√ √(

)

)

(

)

o o

The midpoint of the points ( (

Elliptic Paraboloid o

(

Midpoint Formula in 

Two sheets

) is ( ̅ ̅ ̅) ̅

) and *



+

Elliptic Cone o o o

8.2. Surfaces



Cylindrical Surfaces 



Hyperbolic Paraboloid o

o An equation that contains only two of the variables represents a cylindrical o surface in The system can be obtained by the equation Note: in the corrdinate plane of the two variables  that appear in the equation and then  If (if occurs in the quadric), then translating that graph parallel to the axis of a circle will occur in at least one crossthe missing variable section plane

42

8.3. Vectors  A vector is a physical quantity that has a magnitude and direction  A vector, denoted by ⃗, can be represented as a ray with initial and terminal points  The direction where the arrow head points is the direction of ⃗  Its length is called the magnitude ) and ( ), the  Given two points ( ( ) vector ⃗ with initial point and ) is ⃗ ⟨ terminal point ( ⟩  In , the vector ⃗ with initial point ( ) and terminal point ( ) is ⟨ ⟩

2. Vector Subtraction ⟩  If ⃗ ⟨ and ⟩, ⃗⃗⃗ ⟨ then ⃗ ⃗⃗⃗ ⟨ ⟩

3. Scalar Multiplication  If * + and ⃗ is ‖ ⃗‖ units long, then ⃗ is ‖ ⃗‖ units long and points o To the direction of ⃗ o Opposite to ⃗ 4. Norm of a vector ⟩ and ⃗⃗⃗ ⟨ ⟩ be vectors in  Let ⃗ ⟨ and , respectively  The norm of the vectors ⃗ and ⃗⃗⃗ are

Remarks: 

In general, a vector ⃗ in ( ) and (

 

) is ⃗

⟨

with initial point terminal point ⟩

‖ ⃗‖ √ and ‖ ⃗⃗⃗‖ √ 5. Vector normalization  Let ⃗ be a vector with ‖ ⃗‖  Define ⃗⃗ to be the normalized vector of ,

Two vectors are equal if the nth components are equal to each other Vectors are poisitionless, that is, given a ⟩ without knowledge of the vector ⃗ ⟨ initial and terminal points, there are infinitely many vectors in , which can also be extended to

then ⃗⃗

6. Vectors defined by angle and length  Let ⃗ be a vector with angle , then it may be defined as ⃗ ⟨‖ ⃗‖ ‖ ⃗‖ ⟩

Vector Operations 1. Vector Addition ⟩ and  If ⃗ ⟨ ⟩, ⃗⃗⃗ ⟨ then ⟨ ⃗ ⃗⃗⃗ ⟩

⃗⃗ ‖ ⃗⃗‖

Note:    43

The vector with zero length is called the zero vector, a particularly directionless vector The vector with unit length is caled the unit vector Normalized vectors are unit vectors

o Properties of Vector Arithmetic a. Commutativity – ⃗⃗ ⃗ ⃗ ⃗⃗ b. Associativity – ⃗⃗ ( ⃗ ⃗⃗⃗) ( ⃗⃗ ⃗) ⃗⃗⃗ c. Existence of Vector Additive Identitiy ⃗⃗ ⃗⃗ ⃗⃗ d. Existence

of ⃗⃗

Vector

Additive

Inverse

Note: –

⃗⃗ ⃗ 

–

⃗⃗ ( ⃗⃗) ( ⃗) ( ⃗) e. Scalar Associativity – ( f. Scalar Distributivity – ⃗⃗ ⃗) ⃗⃗ ⃗ ( ) ⃗⃗ g. Vector Distributivity – ⃗⃗ ⃗⃗ h. Existence of Scalar Multiplicative Identity – ⃗ ⃗ i. Vector Nullification – ⃗ ⃗⃗ j.

Scalar Nullification –

⃗⃗

By equating the two values of ‖ ⃗⃗ ⃗‖ and further simplification, we get ⃗⃗ ⃗ ‖ ⃗⃗‖‖ ⃗‖

   

In

,

0

/

⃗⃗ ⃗

. 1 { ⃗⃗ ⃗ The dot product can be extended to ⃗⃗ ⃗ ∑ Two vectors are said to be orthogonal if and only if their dot product is zero The zero vector is orthogonal to any nonzero vector The dot product gives an idea on how two vectors are positioned with each other

⃗⃗ Properties of a Dot Product

8.4. Dot Product ⟩ and ⃗ ⟨ ⟩ be vectors  Let ⃗⃗ ⟨ in o The dot product is defined as ⃗⃗ ⃗  The dot product can also be expressed trigonometrically

a. Commutativity - ⃗⃗ ⃗ ⃗ ⃗⃗ b. Distributivity - ⃗⃗ ( ⃗ ⃗⃗⃗) ⃗⃗ ⃗ ⃗⃗ ⃗⃗⃗ c. ⃗⃗ ⃗⃗ ‖ ⃗⃗‖ ( ⃗⃗ ⃗) d. ⃗⃗ ⃗ ⃗⃗ ⃗ e. ⃗⃗ ⃗

Direction Angles  

o o

‖ ⃗⃗‖ By cosine law ‖ ⃗⃗ ⃗‖ ‖ ⃗‖ ‖ ⃗⃗‖‖ ⃗‖ By property of the dot product, ‖ ⃗⃗ ⃗‖ ( ⃗⃗ ⃗) ( ⃗⃗ ⃗) ‖ ⃗⃗‖ ‖ ⃗‖ ⃗⃗ ⃗

44

⟩ be a nonzero vector in Let ⃗ ⟨ Let be the angles between ⃗ and the unit axis vectors ̂ ̂ ̂ called the direction angles of ⃗



The direction cosines of ‖ ⃗⃗‖

*

are defined as



+

In general, ⃗⃗ ‖ ⃗⃗‖

⃗⃗

⃗⃗ ⃗⃗

⃗

‖ ⃗⃗‖

⃗⃗

Note: 8.5. Cross Product  Let o Define the determinant of the matrix



0

Orthogonal Components and Projections 

Let ⃗ ⃗⃗⃗ ⃗⃗⃗ be nonzero vectors, ⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗ ⃗⃗ ‖ ⃗⃗ ‖ * + * + assuming that ⃗ ⃗⃗⃗ ⃗⃗⃗



1

as

|

|

+ If * is the set of constants, then define the determinant of the matrix [

] |

|

|

as |



By basket rule, |



Let ⃗⃗ o

|

| |

|

|

⟨ ⟩ and ⃗ ⟨ ⟩ Define the cross product of and ̂ ̂ ̂ as ⃗⃗ ⃗ | | ⟨

⟩ To find the components of ⃗, the Properties of the Cross Product surrounding vectors may be used  ⃗ ⃗⃗ (∑ ⃗⃗⃗ ) ⃗⃗ ( ⃗ ⃗⃗) a. ⃗⃗ ⃗ ⃗⃗ ) ⃗⃗ (∑ b. Left Distributivity – ⃗⃗ ( ⃗ ⃗⃗⃗) ⃗⃗ * + ⃗⃗⃗ ( ⃗ ⃗⃗ ) ⃗⃗  ⃗ ( ⃗ ⃗⃗ ) ⃗⃗ c. Right Distributivity – ( ⃗⃗ ⃗) ⃗⃗⃗ The vector components of ⃗ will be ⃗ ⃗⃗⃗ * + ( ⃗ ⃗⃗ ) ⃗⃗ d. ( ⃗⃗ ⃗) ( ⃗⃗) ⃗⃗⃗ ⃗⃗ ( ⃗⃗⃗) In trigonometric form, ⃗ ‖ ⃗‖ ⃗⃗ e. ⃗⃗ ⃗⃗ ⃗⃗ ‖ ⃗‖ ⃗⃗ f. ⃗⃗ ⃗⃗ ⃗⃗ However, if ⃗⃗⃗ and ⃗⃗⃗ are those vectors whose components are unknown, vector Theorem: may be projected onto ⃗⃗ ⃗⃗  Let ⃗⃗ ⃗ be vectors in ⃗⃗ ⟨ * + ( ⃗ ⃗⃗ ) ⃗⃗ o ⃗⃗ ⃗ ⟨ ⟩ o

  

45

⃗

⃗⃗

⃗⃗

⃗⃗⃗

⟩ ⃗

o o

⃗⃗ ( ⃗⃗ ⃗) ⃗ ( ⃗⃗ ⃗) ⃗⃗ ⃗ ⃗⃗ ⃗ ‖ ⃗⃗ ⃗‖ ‖ ⃗⃗‖‖ ⃗‖  If ⃗⃗ ⃗ are the adjacent sides of a parallelogram, then ‖ ⃗⃗ ⃗‖ is its area  If ⃗⃗ ⃗ are nonzero, then ‖ ⃗⃗ ⃗‖ ⃗⃗ * +



 

Consider the vectors ̂ ̂ ̂ o ̂ ̂ ̂ o ̂ ̂ ̂ ̂ o ̂ ̂

o



 

o

⃗⃗ ⃗⃗⃗ ⃗⃗



Since a scalar length as

is of same ⃗⃗ ⃗⃗⃗ ⃗⃗, ⃗⃗ ( ⃗⃗ ⃗⃗⃗) ‖ ⃗⃗ ⃗⃗⃗‖

The volume of a cylinder, in general, is ; with this, the volume of the parallelepiped is ⃗⃗ ( ⃗ ⃗⃗⃗)

If the scalar triple product is 0, then the vectors are coplanar The above remark proves that ⃗⃗ ( ⃗⃗ ⃗) ⃗ ( ⃗⃗ ⃗)

8.6. Parametric and Vector Equations of Lines ⟩ be a vector parralel to a line  Let ⃗ ⟨ ( ) and  The parametric equations of are

⟩ ⃗ ⟨ ⟩ Let ⃗⃗ ⟨ ⟨ ⟩ ⃗⃗⃗ be vectors in o The number ⃗⃗ ( ⃗ ⃗⃗⃗) is called the scalar triple product, defined as |

, ⃗⃗ ( ⃗⃗ ⃗⃗⃗)-( ⃗⃗ ⃗⃗⃗) ‖ ⃗⃗ ⃗⃗⃗‖



Remarks:

Vectors ungrouped in a string are not associative The cross product is only defined in

Theorem: Scalar Triple Product

be the base, therefore, ‖ ⃗ ⃗⃗⃗‖

⃗⃗ ⃗⃗⃗ ⃗⃗

Note: 

Let

{ o

|

If the vectors are nonzero and the adjacent edges of a parallelepiped, then the volume of the parallelepiped is ⃗⃗ ( ⃗ ⃗⃗⃗)



In ⟨ ⟨ Extending (

vector

⟩

⟩

⟨

⃗

to

⟩

{

In

vector

⟨

⟩

⟨

form, ⟩

46

⟨

⟩

) o

form, ⟩ ⟨

⟩ ⟨

⟩

⟨

⟩

o

Assume that a vector ⃗⃗

o

⟩ Form ⃗⃗⃗⃗⃗⃗ ⟨  ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗⃗⃗⃗ ( ) ( ) ⃗⃗ ⃗⃗⃗⃗⃗⃗ ( ) – the equation of the plane

o

Theorem: 



as 



⟩ and In any space, by letting ⃗ ⟨ ⃗ ⟨ ⟩, ⃗ ⃗ ⃗ o As is being exhausted, the terminal points of ⃗ ⃗ generate the line The parametric equations can be interpreted

Let be constants, where the first three are nonzero, then the general form of the equation of a plane normal to the vector ⟨ ⟩ is

Distance between a Point and a Plane

{

The symmetric equations of a line are defined as

8.7. Planes  The graph of are all planes  Let be a plane containing ( ( ) be any point in

) and



(

Let ( o o

⟨

⟩

) Define ⃗⃗⃗⃗⃗⃗ ⟩ ⟨ Suppose

⟨ ⟩ ̅̅̅̅ | ⃗⃗ ⃗⃗⃗⃗⃗⃗ | ‖ ⃗⃗‖

⃗⃗ ⃗⃗⃗⃗⃗⃗

‖‖ ⃗⃗‖ ⃗⃗‖ o

) ⃗⃗

√

Simplifying, and letting ,

√

,

which incorporates the general form of the plane equation

47

9. Elementary Vector Analysis 9.1. Vector-valued Functions ( ) ( )  Let ( ) be functions of t o Define ⃗( ) ⟨ ( ) ( ) ( )⟩, called a vector-valued function o The domain of ⃗ is , while its counterdomain is the set of vectors  The domain of ⃗ is the intersection of the domains of ( ) ( ) ( )  The graph of ⃗ is a line in real space, whose points are traced by ⃗, as is exhausted

o

In the case in the figure above,

o

⃗( )

o

By dividing

⃗( ) is pointing opposite to ⃗( ) ⃗( )

is

now the new vector that points to 9.2. Calculus of Vector-valued Functions Limits  

If the limit of ⃗( ), as written as Let ⃗( ) ⃗( ) ⟨ ( )

approaches , is ⃗⃗, ⃗⃗

⃗( ) ⟨ ( ) ( ) ( )⟩, ( )

then ( )⟩

Continuity 

⃗( )

o

By dividing

⃗( ) is said to be continuous at and only if  ⃗( )  ⃗( )  ⃗( ) ⃗( )

The derivative of ⃗( ) at ⃗( )

⃗( ) is pointing to ⃗( ) ⃗( )

is

the parralel vector that points to

if

 

⃗ ( ) is called the tangent vector to the curve at ⃗( )



always points to the direction of

Theorem:

Derivatives

Let

o

⃗( ) ⃗( )





In this case in the next figure,

be the curve defined by ⃗( ), and

Let o



o

is defined as

⃗( ) ⃗( )

Let ⃗( )



, ⃗⃗( ) ⃗( )-



, ⃗⃗( ) ⃗( )-

be the curve defined by ⃗ 48

⃗( ) ⟨ ( ) ( ) ( )⟩, ⟨ ( ) ( ) ( )⟩ ⃗( )-

⃗⃗( ) ⃗ ( ) , ⃗⃗( )

then

⃗⃗ ( ) ⃗( )

⃗ ( )-, ⃗⃗ ( )

Note: 

The cross product rule, unlike the real-valued Integrals product rule, is not commutative nor  Let ⃗( ) ⟨ ( ) ( ) ( )⟩, then simultaneously associative ⟨ ( ) ( ) ( ) ⟩ ⃗( ) ⟩ is the constant ⃗, where ⃗ ⟨ vector of integration Tangent Lines  The definite integral is defined as 

⃗( ) ) be a point on the curve Let ( defined by ⃗( ), and ⃗( ) be the vector ( ) ( ) ( ) ⟩ ⟨ whose intial and terminal points are the origin and , respectively Note: ⃗⃗ ( ) ( ) o If ⃗ , then ⃗ is parallel  Integration of vector-valued functions is also to the tangent line to the curve at defined in



The tangent line ⃗ ( ) is defined as ( ) ( ) ( ) ( ) { 9.3. Arc Length ( ) ( )  A vector-valued function ⃗( ) is said to be smooth if ⃗( ) ⃗( ) Recall: Dot Product Theorems 

The derivative of the dot product between ,⃗ ⃗ -

two vector-valued functions is ⃗

⃗

o 

Recall: Arc Length of Parametric Equations

⃗



⃗( )

⃗ If ⃗

⃗

⃗,

, ⃗ ⃗-

⃗

If ‖ ⃗‖ ⃗

,

, ⃗ ⃗-

‖ ⃗‖

o 

⃗

⟨ ( ) ( ) ( )⟩

⟨ ⃗

‖ ‖

⟩ √. /

. /

. /

The arc length of a vector-valued function form

Theorem: 

⃗

o

⃗

The dot product of the same vector is ⃗ ⃗ ‖ ⃗‖ o

⃗( ) ⃗( )

Let

to

is

⃗

‖ ‖

9.4. Arc Length Parametrization  Let be the curve defined by ⃗( ) o Select a point , called the reference point o The vector with the terminal point as its reference point is denoted by ⃗( )

If ⃗( ) is a vector-valued function and if ‖ ⃗( )‖ is a constant, then ⃗( ) ⃗ ( ) 49

9.5. Unit Tangent, Normal, and Binormal Vectors  Let ⃗( ) be a smooth function o

⃗ ( ) , ‖ ⃗ ( )‖

Define ⃗⃗( )

called the unit

tangent vector o

o

o o 

⃗⃗ ( ) ‖ ⃗⃗ ( )‖

⃗ ( ) , ‖ ⃗ ( )‖

called the unit normal vector Define ⃗⃗( ) ⃗⃗( ) ⃗⃗( ), called the unit binormal vector

o From the reference point, define an arbitrary direction along the curve as the positive direction; and the Remark: oppositve direction as the negative  The unit tangent, normal, and binormal direction vectors make up of what is known as the All points in the positive direction moving trihedral are said to have positive arc lengths  Let be an A.L.P. All points in the negative direction are said to have negative arc lengths o ⃗⃗( ) ⃗ ( ) ⃗

as the arc length

o

⃗⃗( )

parametrization of ⃗ with reference point

o

⃗⃗

Define

⃗⃗( )

Define

‖ ‖



Properties: ⃗

⃗⃗

⃗⃗ ( ) ‖ ⃗⃗ ( )‖

⃗⃗

⃗ ( ) ‖ ⃗ ( )‖

⃗⃗

⃗⃗

⃗⃗

‖ ⃗⃗‖ The unit normal vector points to the concavity of ⃗

a. ‖ ‖ ⃗

b. ‖ ‖ 9.6. Curvature Note:  Let be a smooth curve o The sharpness of bend of is ⃗, where ⃗ is a  If ⃗ is an A.L.P., then ⃗ measured by its curvature vector-valued function of o The curvature of a curve is  The A.L.P. is a function dependent on arc ⃗⃗ defined by ⃗ as ( ) ‖ ‖ length , which finds the vector ⃗( ) that is ‖ ⃗ ( )‖ units along from the reference point o Other forms for are ( ) ‖ ⃗⃗ ( )‖ ‖ ⃗ ( )‖ { ) ( )‖ ⃗ ‖ ⃗( ‖ ⃗ ( )‖

50

Radius of Curvature  

o

be a circle with radius , then ( )

Let

o

⃗( )

⃗( )

Since

⃗⃗( )

is defined to be the radius of curvature ‖

Note: ̅



( )

(

⃗⃗( )

‖, then

implies ⃗⃗ ( )

)

o

With . /

the ⃗⃗( )

.

⃗⃗( )

⃗⃗( )

/

⃗⃗⃗( )

‖

and

⃗⃗⃗( )

‖

⃗⃗

⃗⃗( ), which ⃗⃗( )

above,

⃗( )

⃗⃗( )

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

o

, the rate of change of the arc length with respect to time, is the speed of the particle The velocity vector is defined as ⃗⃗( ) ⃗( ) 

Distance and Displacement 



Define

as the tangential scalar

component of acceleration, and

Define ⃗ as the displacement vector of a . / as the normal scalar component of particle travelling from to , then acceleration ⃗ ⃗( ) ⃗( ) Let be the distance travelled of a particle Remarks: from to  Other formulas for the scalar components ⃗ ‖ ⃗( )‖ o ‖ ‖ ‖ ⃗⃗ ⃗⃗‖ ‖ ⃗⃗ ⃗⃗‖ ⃗⃗ ⃗⃗ include , , ‖ ⃗⃗‖ ‖ ⃗⃗‖ ‖ ⃗⃗‖

Normal and Acceleration

Tangential

Componenents

of

9.8. Projectile Motion ⟩, where  Let ⃗( ) ⟨ is the acceleration due to gravity ⃗⃗( ) Recall: ⃗( ) ⟨ ⟩ ⃗ o ⃗( ) ⃗( ) ) and ⃗ be the initial position and  The acceleration vector may be derived from  Let ( the velocity vector velocity of the particle, respectively 51

o

⃗( ) ⃗ ⟨ ⟩ ⃗

o

⃗( )

o

⃗ ⃗ ) ⃗( ⃗

⃗ ,

then

⃗( )

⟨

⟨

⟩,

⟨

⟩

⃗( ) ⟩

then

⃗( )

⃗

Parametric Equations of Projectile Motion



⃗

Let

⟨

‖⃗ ‖ 

⃗( ) ‖⃗ ‖ {

⟩; ‖⃗ ‖

then

⟨‖ ⃗ ‖ ⟩ ‖⃗ ‖ ‖⃗ ‖

52

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 *

(

) ( )





( ), then the projection of the Let trace of on the plane , onto the -plane is called the level curve of at



Let ( at

10.2. 

)

, if ( ), (

+

Level Curves and Surfaces

(

In

), then the graph of , is called a surface of



) (

is defined by ( and if ( ) )

( ) and ), then

( ( ) ( )) o Unlike the limits in , infinitely many points are being approached ) in infinitely many paths, to ( in which, the well defined curve C is required o To get the limit, the points of are projected onto the surface ) is said to be continuous at A function ( ( ) if and only if ( ) o ) o ( ) ( ) ( ( ) ) o ( ) ( ) (

Theorem: Limits and Continuity Let be a smooth parametric curve defined  If ( ) is continuous at , and if ( ) is ( ) ( ) ( ), by and ( ) ( ) continuous at , then ( ) ( ) suppose that at ) is continuous at ( ( ) ( ) ) is continuous at ( ) and if  If ( ( ), o Let then ( ) is continuous at ( ), then ( ) ( ) ( ) )) is continuous at ( ) ( ( ( ( ) ( ) ( ))

53

Remarks: 

If

( (

) (

 (



)

(

)

)

| (



) (

(

)

,

) (

(

) )

(

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

then

| ) (



) ), then the limit

derivatives

does not exist The sum/difference/product of two continuous functions is also continuous Higher-Order Partial Derivatives The quotient of two continuous functions is ( );  Let then continuous, except at those points where the denominator is zero . /, . /,

. /, and

. / 10.3. 

Partial Derivatives Theorem: Clairaut’s Theorem ( ) be a continuous function Let o The partial derivative of with  If the partial derivatives and ) is defined as respect to at ( continuous and defined on ( ( ) ( ) ( ) ( ) o

are both ), then

Similarly, the partial derivative of ) is with respect to at ( defined as

(

)

(

)

10.4. 

Implicit Partial Differentiation ( ) is expressed Suppose a function ( ) 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 ( ), implicit partial simply as 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 54

10.5. 

 

Local Linear Approximation 10.8. ( )( ) Let  o Define the local linear approximation ) as of at point ( ( ) ( ) ( )  ) that are very close to For those points ( ( ), then ( ) In general, if ( ), then ∑

Differentiability A function differentiable (



In (

) (

general,

point

)

( at (

√(

)

) is said to be ( ) if ) (

)

( ) a function ) is said to be differentiable at

10.7. 



( )

if

√∑

(



)

 ( ) is the error in the approximation if the local linear approximation is used

Differentials ( ) Let o Define the total differential of ( ) as ( ) ( ) Define o o If

then define

∑

)

( ),

, provided that

the end-function is univariate

(

)

(

( and )

at

)

(

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:

Note: 

(

In general, if

Remark: 

10.6. 

Multivariate Chain Rule ( ) ( ) ( ) Let o If the end-function of the general function is univariate, then define

, then )

55



( ) and ( ) can be substituted directly ( ) and proceed with after 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