Transcript
Elementary Analysis
Review Notes
Table of Contents
1. Limits and Continuity ..........................................4 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. Review of Functions ..............................4 Limits ......................................................5 Computing Limits ...................................5 Formal Definition of a Limit ..................6 One-Sided Limits ....................................6 Infinite Limits .........................................6 Limits at Infinity .....................................7 Continuity...............................................7 Limits and Continuity of Trigonometric Functions ................................................8
3.4. 3.5. 3.6. 3.7.
Analysis of Functions: Concavity and the Second Derivative Test ................. 15 Sketching of Functions ........................ 15 Rolle’s Theorem and the Mean-Value Theorem for Derivatives ..................... 16 Absolute Extrema................................ 16
4. Integration ........................................................ 17 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. The Indefinite Integral ........................ 17 Integration by Substitution ................ 17 Separable Differential Equations ....... 17 Area ..................................................... 18 The Definite Integral ........................... 19 Fundamental Theorems of Calculus and the Mean Value Theorem for Integration........................................... 20 Calculation of Area as a Definite Integral ................................................ 21 Volume by Slicing, Disks, and Washers .. ............................................................. 22 Volume by Cylindrical Shells ............... 24 Arc Length of a Plane Curve ................ 24
2. Differentation....................................................10 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. Slopes and the Derivative....................10 Techniques in Differentiation..............11 Derivatives of Trigonometric Functions .. ..............................................................11 Chain Rule ............................................11 Implicit Differentiation ........................11 4.8. 4.9. 4.10. 4.7.
Higher-Order Derivatives ....................11 5. Special Functions and Cases............................. 26 Rectilinear Motion Problems ..............11 Rates of Change ...................................12 Local Linear Approximation and Differentials .........................................12 5.2. 5.3. 5.4. 5.5. 5.7. 5.8. 5.9.
1
5.1.
The Natural Logarithmic Function from the Integral Point-of-View .................. 26 Logarithmic Differentiation ................ 26 Integration of the Natural Logarithmic Function............................................... 26 Inverse Functions ................................ 26 The Natural Exponential Function ...... 27 Derivatives and Antiderivatives of Inverse Trigonometric Functions ........ 29 Hyperbolic Functions .......................... 29 Inverse Hyperbolic Functions ............. 30
3. Behaviour and Analysis of Functions ...............14 3.1. 3.2. 3.3. Related Rates .......................................14 Analysis of Functions: Relative Extrema . ..............................................................14 Analysis of Functions: Increasing, Decreasing, and the First Derivative Test .......................................................14
Indeterminate Forms and L’Hopital’s 9. Elementary Vector Analysis ............................. 49 Rule ......................................................31 9.1. Vector-valued Functions ..................... 49 6. Integration Techniques .....................................32 9.2. Calculus of Vector-valued Functions .. 49 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. Integration by Parts .............................32 Trigonometric Integrals .......................32 Trigonometric Substitution .................33 Integration by Partial Fractions ...........33 Improper Integrals ...............................34 Review on Separable Differential Equations and Applications .................34 9.6. 9.7. 9.8. 9.3. 9.4. 9.5. Arc Length ........................................... 50 Arc Length Parametrization ................ 50 Unit Tangent, Normal, and Binormal Vectors................................................. 51 Curvature............................................. 51 Curvilinear Motion .............................. 52 Projectile Motion ................................ 52
5.10.
Orthogonal Trajectories ......................35 10. Multivariate Differential Calculus .................... 54 7. Parametric and Polar Curves ............................36 10.1. Multivariate Functions........................ 54 7.1. Review on Conic Sections ....................36 10.2. Limits and Continuity .......................... 54 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. Parametric Equations ..........................37 Derivatives of Parametric Equations ..37 Arc Length of Parametric Curves .........38 Polar Coordinates ................................38 Graphs of Polar Equations ...................38 Tangent Lines of Polar Curves .............41 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. Partial Derivatives ............................... 55 Implicit Partial Differentiation ........... 55 Local Linear Approximation ................ 56 Differentiability ................................... 56 Differentials......................................... 56 Multivariate Chain Rule ...................... 56
Arc Length of Polar Curves ..................41 11. Behaviour and Analysis of Multivariate Areas in Polar Coordinates ..................41 Functions .......................................................... 57 Conic Sections in Polar Coordinates....41 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. Directional Derivatives ....................... 57 Gradient............................................... 57 Tangent Planes to Surfaces................. 58 Extreme Values ................................... 58 Absolute Extreme Values on a Closed and Bounded Region ........................... 59 Lagrange Multipliers ........................... 59
8. The Real Space ..................................................43 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. Three Dimensional Coordinate System .. ..............................................................43 Surfaces ................................................43 Vectors .................................................44 Dot Product ..........................................45
Cross Product .......................................46 12. Multivariate Integral Calculus .......................... 60 Parametric and Vector Equations of Lines .....................................................47 Planes ...................................................48
2
12.1.
Parametric Surfaces and Multivariate Vector-valued Functions ..................... 60
12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. 12.9. 12.10.
Calculus of Multivariate Vector-valued Functions ..............................................60 Double Integrals...................................60 Double Integrals over Polar Regions ...62 Surface Area .........................................62 Mass and Moments of Mass of a Lamina ..................................................63 Triple Integrals .....................................64 Triple Integrals in Cylindrical Coordinates ..........................................65 Triple Integrals in Spherical Coordinates ..............................................................65 Mass and Moments of Mass of a Solid ..................................................66
13. Analysis of Vector Fields ...................................67 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. Vector Fields ........................................67 Line Integrals of Scalar Fields ..............67 Line Integrals of Vector Fields .............69 Conservative Vector Fields and Independence of Path .........................69 Green’s Theorem .................................70 Surface Integrals ..................................70 Flux .......................................................71
14. Analysis of Sequences and Series .....................72 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. Sequences ............................................72 Series ....................................................72 Convergence Tests ...............................72 Alternating Series and Conditional Convergence ........................................73 Power Series ........................................74 Taylor and Maclaurin Series ................74 Taylor Polynomials and Approximations ..............................................................75
3
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 codomain o If the ordered pair image of , and inverse- image of o The set of * | ( the range , and is the
Types of Functions 1. Polynomial Functions of degree n ( ) i. ∑ )
Constant function ( ( ) |
is in , is the is the pre- or all elements +, is called )
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
4
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 ( ) | | √ 2 7. Piecewise function Different functions in different intervals ( ) {
1.2. Limits Consider 3 functions: ( ) ( ) o 2
( )
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,
|
| | (
| | ) ( )
5
1.3. Computing Limits 1. If is, ( ) 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: If ( ( )) ( ( )) ( ) ( ( ( )) ( )) ( ) ( ) ( ( ) )
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 )
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: definition of a limit is defined on any open interval ( where ( ) ( ) | | | | ( ) | ), Theorem: If o o {
6
that are
, then
If if o o
( ) exists, ( ) , then
( )
, and Notes:
o
( )
, then
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
If then o o
( ) , , -
and { and {
( ) exists, 1.8. Continuity A function is continuous at a point, , if all the conditions are satisfied: ( ) o ( ) ( ) o ( ) o ( )
( ) If exists, then , 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 o o {
7
, then
All polynomial functions are continuous everywhere A rational function is continuous everywhere in its domain
If
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
, then ( ) √ is continuous Everywhere in
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:
If a function is continuous on , - and if ( ) ( ) ( )| ( ) , then
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
of
Trigonometric
Continuity on Intervals
The trigonometric functions are continuous on their respective domains
) 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
8
Theorem: Squeeze/Sandwich Theorem Let ( be defined on some open interval ) about, except possibly at, c, and * + ( ) ( ) ( ), then ( ) ( ) ( )
9
2. Differentation 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
( ) ( ) [ ]
Note: o o o The slope of the secant line is
( ) ( ) ( ) ( )
( ) ( ) derivatives exist
( )
if
both
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
( ) ( )
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: 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
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
10
( ) ( )
o
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. 3. 4. 5. 6.
2.6. Higher-Order Derivatives The nth derivative of the function , denoted ( ), that by ( ), is the derivative of is, Remarks: The derivative of a function is also called the first derivative Notations: ( ) , ( )( )
, ( ). /
( )
(
)
( )
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
11
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 ( )
12
o
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: If It can be shown that the approximation of near approximation of near A function that is differentiable said to be locally linear at ( ( ( ) values of ) local linear is the best at a point is ( ))
( ) ( ) ( ) , then , thus for sufficiently small
13
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
14
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: 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: First Derivative Test
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 ( )
15
( ) |
-, 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
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
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 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 the Mean-Value 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
3.6. Rolle’s Theorem and Theorem for Derivatives 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 ( )
16
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 ( ) Let ( ) be an antiderivative of the continuous function o If is a differentiable function with range I, then ( ( )) ( ) ( ( ))
Note:
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 ( ) as the set of all We now interpret 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. 5. 6. 7. 8. 9. 10. * ( ) , ( ) ( ) ( ) ( )( ) ( ) ( ) + ( )
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
17
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
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 ,
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
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
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 ,
4.4. Area Sigma Notation o If ∑ () (
( ) ( ) ( )
,
)
then
, Choose in each of the subintervals, where , -
18
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: . / ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑
(
() ∑ , () ( )() ∑
( ( ( ) )( ) )(
() ∑ (
)
() )
∑
()
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 If ( ) exists, then ( )
Theorem:
)( )
-, 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 , ( ) ( )( ) -
4.5. The Definite Integral Suppose a function is defined on , o Divide the interval into subintervals by choosing arbitrary points with o o o The length of each partition is The largest subinterval is called the norm, ‖ ‖ Choose an arbitrary on every subinterval
19
If
( ) is integrable on a closed interval , then ( ) ( ) ( )
Note:
The lower and upper limits of integration are a and b, respectively The variable in the integral is a dummy the particle during the time between variable the 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 ( )
( ) o is odd, then ( ) is the velocity function of an If ( ) object moving along a line, then ( ) o is the displacement of
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 ̅ ( )
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
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 ( )
4.7. Calculation of Area as a Definite Integral
Area between Two Curves Suppose and are continuous functions -, and , on , Let be the region bounded by the graphs of and , and the closed interval
-, 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 ( )
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
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 ,
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:
o
o o
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
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 ( ) ∑ ( ) ∑ ( ) ‖ ‖
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
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
23
o o o o
( ) 0( ( )) ∑
( ) ( ( )) 1 ( ( )) 1
0( ( ))
Note: 4.9. Volume by Cylindrical Shells A cylindrical shell is a solid contained between two cylinders having the same center and axis The moving strips should be parallel to the axis of revolution
4.10.
Arc Length of a Plane Curve Suppose is continuous on ,
-
The volume of the cylindrical shell is ( ) ) . /( 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 ) ( ( ) ( )) )
Method of Cylindrical Shells 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
Let be the solid of revolution obtained when is revolved around the y-axis ( ) o ( ) 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 Suppose is continuous on ,
24
o o o o o o
It implies that is defined and continuous on the same interval √ .
( ) ( )
/
is continuous and differentiable on , , -| ( ) By MVTD,
( ) ( )
, ( )√ 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 5.2. Logarithmic Differentiation Integral Point-of-View Recall: Consider 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 Function Theorem: Remarks:
of
the
Natural
Logarithmic
| | | | | | | | | |
o o o o (
is continuous on ( )
)
{
The natural logarithmic function is defined as
| |
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 Another verification contrapositive is its
. /
c. It is continuous and increasing on its domain d. It is concave down at all points e. Theorem: , 0 1
26
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 ( ) 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
Differentiation of Inverse Functions
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 ( ) ( ) ( )( )
( ) ( ( ))
27
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 Laws of Exponents Theorem: , ( ( . / ) ) , and , -
Summary: Types of Functions with Exponents Positive Variable Base and Constant Exponent ( ) , ( ), ( ), ( )Positive Constant Base with Variable Exponent ( ) ( ) ( ) , ( )( ) Positive Variable Base with Variable Exponent ( ) ( ) ( ) , ( )- ( ) *Use Logarithmic Differentiation
Logarithmic Functions If , then the logarithmic function with base a is the inverse of the exponential function with base , that is,
Properties: a. ( )
28
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 {, o Cosecant o {0 o ( . / 0 / 1 * +} , ) , 0 /3 ) /3
√ √ √
√ | |√
√ √
{
√ | |√
( , ( ) 1 * +
) - 2 3
{
√
0
(
)
20 ( 20
/
. / . / . /
- 2 3}
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 * + * + -
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
29
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 Theorem: , , , , , , Inverse Hyperbolic Functions ( ( ) ) 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 is is is is is is
Identities: ( | ) | Theorem:
30
( ( . . ( (
√
√ √ / / )
√ | |
) )
)
(
) | |
, , , , , ,
-
√ √
| | | |
√ | |√
(
)
√ √ { √ √
√
( ) ( ) | | | | | | . / ) . / ) |
( )
√
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 Type o Apply the theorem for type Combine the two function to obtain the indeterminate forms of type
{
5.10.
Indeterminate Forms and L’Hopital’s Rule If ( ) ( ) , then
( ) ( )
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
(
is said to be indeterminate
o Apply the theorem for type Types ( ) o Write
( ) ( ) ( ) ( )
( )
o
Solve for ( )
( )
( )
( )
o
if ) ( ) if ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
is of type , and the derivatives ( ) , then
( ) ( ) ( ) ( )
are continuous at a with
( ) ( )
( ) ( )
Theorem: L’Hopital’s Rule Type and o Let and be differentiable functions on some open interval I,
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 ( ) ( ) , assuming that is the form differentiable and ( ) is an antiderivative of ( ) o o o , ( ) ( )( ) ( ) ( ) ( ) sides, 6.2. Trigonometric Integrals 1. Integrating powers of sine and cosine ( ) ( ) Deriving the equation, , ( ) ( )Integrating ( ) ( ) ( ) ( ) Integration by Parts of Definite Integrals The definite integral can be solved with integration by parts, provided that the functions satisfy its conditions |
( ) ( ) both ( ) ( )
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 ( ) ( ) ∑ ( ) ( ) ( ) | ( ) The consequence of tabular integration by parts is that it cannot be used when the first function is infinitely differentiable
32
Use Use ) and Use , ( , , ) ( )-
( ) ( o o
Use , ( ) ( )-
o Use ,
, ( ) ( )-
o
√ using Trigonometric
3. Integrating powers of tangent and secant
Steps in Integration Substitution
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 ( 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
o
If
) in the denominator of a proper rational functions will contribute to terms of partial fractions; that is,
( ) ( )
∑
* +
o
If
Quadratic Factor Rule o For each factor of the form ( ) , the partial fraction decomposition contributes to terms of partial fractions that is,
( )
Note:
∑ ( ) 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 √ √
33
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 Consider ( ) o
Theorem: -, except at and If is continuous on , infinite discontinuity at , then the improper integral of over , ( ) -, 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 , ( ) If is continuous on , infinite discontinuity at improper integral of ( ) o ( ) ( ) -, except at an ( ), then the - is over , - is ( ) - is ( )
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 ( ) ( )
34
Some Applications of SODEs Malthusian Population Model ( ) o Let be time, population Recall: at given time, birth rate, and death rate, respectively o ( ) sides, ( ) be the that is, *( Integrating both ( ) *( ) + o The initial population will population at time zero, ( ) ( ) ) + ( ) ( ) Verhulstian Population Model o Let ( ) .
member of one family is orthogonal to each member of the other
/ be time,
population at given time, carrying capacity, and per capita income increase, respectively o . |
* * +
/ both ( )
+
Integrating |
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
35
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
√
(
)
( –
) the
o
√ coordinates of the foci |̅̅̅̅̅| √( ) √( ) √( √( of the ellipse . ) ) √( /
|̅̅̅̅̅|
o
|̅̅̅̅ | √(
√( ) √( √(
) |̅̅̅̅ | ) )
)
– the equation
– 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 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
36
opens upward if
and
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 is
37
Note: If If If and and , then , then , then has a horizontal has a vertical
tangent line tangent line has a singular point
Higher Derivatives Let ( ) and ( ) be a pair of Conversion of Polar and Rectangular Coordinates , can be Polar to ( ) ( Rectangular ( is Remarks: ( ( ) ( ) ( ) ) ) .√ Rectangular ) to Polar / -
parametric equations, then The second order derivative, expressed as In general, the nth derivative of
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
38
1. Lines and Rays
2. Circles
* +
By strictly letting
o o o
The graphs can be completed from to ( ) Negating would flip the graph to the non-symmetric axis
39
3. Roses
* + o
The length of the inner loop is
Cardioid -
o
Dimpled Limaçon -
(
)
o | The length of each petal is The period of the graph is , , and , 4. Limaçons o Limaçon with an inner loop o o ( ) { | o The distance from the pole to the dimple is
Convex Limaçon -
The nearest distance is
40
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 o Define ( ( )) o The approximation for the area of ∑ the region is ∑ o ( ) The area of the region is ∑ ‖ ‖
‖ ‖
Theorem: If the polar curve, defined by passes throught the pole at | ( ) | , then ( ),
is tangent to the curve at the pole
∑
(
)
, ( )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
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
41
o ( o Note: √ )
(
)
√
The same proof goes for the case of hyperbola, except that and are interchanged, and is still constant
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 pole the pole – the directrix is below the pole – the directrix is above the pole – the directrix is to the left of – the directrix is to the right of
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 ( )
42
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 the √| | | | and o ) o o ) and * + Elliptic Cone o o 8.2. Surfaces Cylindrical Surfaces o Hyperbolic Paraboloid o ( ) be points o -plane, Two sheets Elliptic Paraboloid
( ) √( ) Suppose another plane exists where lies, √ √( ) ( | ) | (
Midpoint Formula in The midpoint of the points ( ( ) is ( ̅ ̅ ̅)| ̅
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
43
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 ⟨ ⟩ Remarks: In general, a vector ⃗ in ( ) and ( ) is ⃗ ⟨ with initial point terminal point ⟩
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 ‖ ⃗‖ √ and ‖ ⃗⃗⃗‖ √ 5. Vector normalization Let ⃗ be a vector with ‖ ⃗‖ Define ⃗⃗ to be the normalized vector of , then ⃗⃗
⃗⃗ ‖ ⃗⃗‖
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
Vector Operations 1. Vector Addition ⟩ and If ⃗ ⟨ ⟩, ⃗⃗⃗ ⟨ then ⟨ ⃗ ⃗⃗⃗ ⟩
6. Vectors defined by angle and length Let ⃗ be a vector with angle , then it may be defined as ⃗ ⟨‖ ⃗‖ ‖ ⃗‖ ⟩
Note:
44
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: – – In ,
By equating the two values of ‖ ⃗⃗ ⃗‖ and further simplification, we get ⃗⃗ ⃗ ‖ ⃗⃗‖‖ ⃗‖
⃗⃗ ⃗ ⃗⃗ ⃗
0
/
⃗⃗ ( ⃗⃗) ( ⃗) ( ⃗) e. Scalar Associativity – ( f. Scalar Distributivity – ⃗⃗ ⃗) ⃗⃗ ⃗ ( ) ⃗⃗ g. Vector Distributivity – ⃗⃗ ⃗⃗ h. Existence of Scalar Multiplicative Identity – ⃗ ⃗ i. Vector Nullification – ⃗ ⃗⃗ j. Scalar Nullification – ⃗⃗ ⃗⃗
. 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 ⟩ be a nonzero vector in Let ⃗ ⟨ Let be the angles between ⃗ and the unit axis vectors ̂ ̂ ̂ called the direction angles of ⃗
o o
‖ ⃗⃗‖ By cosine law ‖ ⃗⃗ ⃗‖ ‖ ⃗‖ ‖ ⃗⃗‖‖ ⃗‖ By property of the dot product, ‖ ⃗⃗ ⃗‖ ( ⃗⃗ ⃗) ( ⃗⃗ ⃗) ‖ ⃗⃗‖ ‖ ⃗‖ ⃗⃗ ⃗
45
The direction cosines of
‖ ⃗⃗‖
are defined as
*
+
In general, ⃗⃗|‖ ⃗⃗‖
⃗⃗
⃗
⃗⃗ ⃗⃗ ‖ ⃗⃗‖
⃗⃗
Note: 8.5. Cross Product Let o Define the determinant of the matrix 0 1 as | |
Orthogonal Components and Projections Let ⃗ ⃗⃗⃗ ⃗⃗⃗ be nonzero vectors, ⃗⃗⃗ ⃗⃗⃗ | ⃗⃗⃗ ⃗⃗ ‖ ⃗⃗ ‖ * + * + assuming that ⃗ ⃗⃗⃗ ⃗⃗⃗
+ 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
46
⃗ ⃗⃗
⃗⃗ ⃗⃗⃗
⟩ ⃗
o o
⃗⃗ ( ⃗⃗ ⃗) ⃗ ( ⃗⃗ ⃗) ⃗⃗ ⃗ ⃗⃗ ⃗ ‖ ⃗⃗ ⃗‖ ‖ ⃗⃗‖‖ ⃗‖ If ⃗⃗ ⃗ are the adjacent sides of a parallelogram, then ‖ ⃗⃗ ⃗‖ is its area If ⃗⃗ ⃗ are nonzero, then ‖ ⃗⃗ ⃗‖ ⃗⃗ * | +
Let
be the base, therefore, ‖ ⃗ ⃗⃗⃗‖
⃗⃗ ⃗⃗⃗ ⃗⃗ , ⃗⃗ ( ⃗⃗ ⃗⃗⃗)-( ⃗⃗ ⃗⃗⃗) ‖ ⃗⃗ ⃗⃗⃗‖
Since a scalar length as |
⃗⃗ ⃗⃗⃗ ⃗⃗|
is of same ⃗⃗ ⃗⃗⃗ ⃗⃗,
| ⃗⃗ ( ⃗⃗ ⃗⃗⃗)| ‖ ⃗⃗ ⃗⃗⃗‖
o
Note: Consider the vectors ̂ ̂ ̂ o ̂ ̂ ̂ o ̂ ̂ ̂ ̂ o ̂ ̂ Vectors ungrouped in a string are not associative The cross product is only defined in Remarks:
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 ⃗⃗ ( ⃗⃗ ⃗) ⃗ ( ⃗⃗ ⃗)
Theorem: Scalar Triple Product
⟩ ⃗ ⟨ ⟩ Let ⃗⃗ ⟨ ⟨ ⟩ ⃗⃗⃗ be vectors in o The number ⃗⃗ ( ⃗ ⃗⃗⃗) is called the scalar triple product, defined as | o |
8.6. Parametric and Vector Equations of Lines ⟩ be a vector parralel to a line Let ⃗ ⟨ ( ) and The parametric equations of are { In ⟨ ⟨ Extending ( o In ⟨ ) o vector ⟩ to { vector ⟩ form, ⟩ ⟨ ⟨ ⟩ ⟩ ⃗ ⟨ ⟩ form, ⟩ ⟨ ⟨ ⟩ ⟩
If the vectors are nonzero and the adjacent edges of a parallelepiped, then the volume of the | ⃗⃗ ( ⃗ ⃗⃗⃗)| parallelepiped is
47
o o o
Assume that a vector ⃗⃗
⟨
⟩
⟩ Form ⃗⃗⃗⃗⃗⃗ ⟨ ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗⃗⃗⃗ ( ) ( ) ⃗⃗ ⃗⃗⃗⃗⃗⃗ ( ) – the equation of the plane
Theorem: ⟩ and In any space, by letting ⃗ ⟨ ⃗ ⟨ ⟩, ⃗ ⃗ ⃗ o As is being exhausted, the terminal points of ⃗ ⃗ generate the line The parametric equations can be interpreted as { 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
48
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 o o
In the case in the figure above, ⃗( ) ⃗( ) is pointing opposite to
⃗( ) ⃗( )
By dividing
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 ( )⟩
⃗( ) ⟨ ( ) ( ) ( )⟩, ( )
o o
In this case in the next figure, ⃗( ) ⃗( ) is pointing to
⃗( ) ⃗( )
Continuity Let o be the curve defined by ⃗( ), and ⃗( ) is said to be continuous at and only if ⃗( ) ⃗( ) ⃗( ) ⃗( ) if
o
By dividing
is
the parralel vector that points to
⃗( ) ⃗( )
always points to the direction of
⃗ ( ) is called the tangent vector to the curve at | ⃗( )
Theorem: Let ⃗( ) , ⃗⃗( ) ⃗( )49
Derivatives The derivative of ⃗( ) at ⃗( ) Let
⃗( ) ⃗( )
⃗( ) ⟨ ( ) ( ) ( )⟩, ⟨ ( ) ( ) ( )⟩ ⃗⃗( ) ⃗ ( ) , ⃗⃗( ) ⃗( )⃗ ( )-, ⃗⃗ ( )
then
is defined as
, ⃗⃗( ) ⃗( )-
⃗⃗ ( ) ⃗( )
be the curve defined by ⃗
Note: The cross product rule, unlike the real-valued Integrals product rule, is not commutative nor Let ⃗( ) ⟨ ( ) ( ) ( )⟩, then simultaneously associative ⃗( ) ⟨ ( ) ( ) ( ) ⟩ ⃗, where ⟩ is the constant vector of ⃗ ⟨ Tangent Lines integration 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 o If ⃗ ( ) ⃗⃗, then ⃗ ( ) is parallel Note: to the tangent line to the curve at Integration of vector-valued functions is also defined in The tangent line ⃗ ( ) is defined as { ( ) ( ) ( ) ( ) ( ) ( )
Recall: Dot Product Theorems The derivative of the dot product between two vector-valued functions is
⃗ ⃗ ,⃗ ⃗ -
9.3. Arc Length A vector-valued function ⃗( ) is said to be smooth if ⃗( ) ⃗( ) ⃗ Recall: Arc Length of Parametric Equations Let ⃗( ) o o
⃗ ⃗
⃗ If ⃗ ⃗ ⃗,
, ⃗ ⃗-
o
⃗
⃗
⃗( ) ⃗( ) ⟨
⟨ ( ) ( ) ( )⟩| ⟩
The dot product of the same vector is ⃗ ⃗ ‖ ⃗‖ o If ‖ ⃗‖ ⃗
⃗
,
, ⃗ ⃗-
‖ ‖
√. / ‖ ‖
⃗
. /
. /
‖ ⃗‖
The arc length of a vector-valued function form to is
Theorem: If ⃗( ) is a vector-valued function and if ‖ ⃗( )‖ is a constant, then ⃗( ) ⃗ ( )
50
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 ⃗( )
9.5. Unit Tangent, Normal, and Binormal Vectors Let ⃗( ) be a smooth function o Define ⃗⃗( ) tangent vector o Define ⃗⃗( )
⃗⃗ ( ) ‖ ⃗⃗ ( )‖ ⃗ ( ) , ‖ ⃗ ( )‖ ⃗ ( ) , ‖ ⃗ ( )‖
called the unit
o
o o Define
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 ⃗⃗( ) ⃗ ( ) ‖ ‖
⃗
called the unit normal vector Define ⃗⃗( ) ⃗⃗( ) ⃗⃗( ), called the unit binormal vector
as the arc length
o o
⃗⃗( ) ⃗⃗ ⃗⃗
parametrization of ⃗ with reference point
⃗⃗ ( ) ‖ ⃗⃗ ( )‖
⃗ ( ) ‖ ⃗ ( )‖
⃗⃗
⃗⃗
⃗⃗
⃗⃗
Properties: a. ‖ ‖ b. ‖ ‖
⃗ ⃗
‖ ⃗⃗‖ The unit normal vector points to the concavity of ⃗
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 ( )
‖ ⃗⃗ ( )‖ ‖ ⃗ ( )‖ { ) ( )‖ ⃗ ‖ ⃗( ‖ ⃗ ( )‖
51
Radius of Curvature Note: ̅ ( ) ( ) Let be a circle with radius , then ( )
o o
⃗( ) Since ‖
⃗⃗( )
⃗( ) ⃗⃗( ) ‖, then
.
⃗⃗( )
/ and
⃗⃗( )
⃗⃗⃗( )
is defined to be the radius of curvature
‖
⃗⃗⃗( )
‖
⃗⃗
⃗⃗( ), which ⃗⃗( ) ⃗( )
implies ⃗⃗ ( ) o With . / the ⃗⃗( )
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 , 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
o
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
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
52
Tangential
Componenents
of
o o o
⃗( ) ⃗ ⟨ ⟩ ⃗ ⃗( ) ⃗ ⃗ ) ⃗( ⃗ ⟨
⃗ , ⟨ ⟩, ⟩ ⃗
then
⃗( ) ⟩
⃗( ) ⟨
then
⃗( )
Parametric Equations of Projectile Motion
Let ‖⃗ ‖ ⃗( ) ‖⃗ ‖ { ⟨‖ ⃗ ‖
⃗
⟨
⟩; ‖⃗ ‖
then
⟩ ‖⃗ ‖ ‖⃗ ‖
53
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 * ( )| ( ) +
In
(
, if ( ),
) (
is defined by ( and if ( ) )
( ) and ), then
Level Curves and Surfaces ( ), then the projection of the Let trace of on the plane , onto the -plane is called the level curve of at Let ( at ( ) ), then the graph of , is called a surface of
( ( ) ( )) 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 ( ) ( ) (
10.2.
Limits and Continuity Let be a smooth parametric ( ) ( ) by suppose that at ( ) ( ) ( o Let
( ) ( )
Theorem: curve defined ( ), and ( ) ), ( then ) If ( ) is continuous at , and if ( ) is ) ( ) ( ) continuous at , then ( ) is continuous at ( ) is continuous at ( ) and if If ( ( ) is continuous at ( ), then )) is continuous at ( ) ( (
( ( ) ( ) ( ))
54
Remarks:
( (
( ( ) ( ) ( ) )
If |
( |
)
,
then
(
)
) ( ( ) ), then the limit
A partial derivative can be interpreted as the slope of the tangent line at the cross section ) of the surface at ( Notations include derivatives, and derivatives for for -partial -partial
) ( ) ( )
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 Similarly, the partial derivative of ) is with respect to at ( defined as
( ) ( )
are both ), then
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
55
10.5.
Local Linear Approximation 10.8. ( )( ) Let o Define the local linear approximation ) as of at point ( ( ) ( ) ( ) For those points ( ) that are very close to ( ), then ( ) In general, if ( ), then ∑
Multivariate Chain Rule ( ) ( ) ( ) Let o If the end-function of the general function is univariate, then define ( ∑ ) ( ),
In general, if then define
, provided that
the end-function is univariate
Remark: To aid in MCR, a tree diagram may be used o The use of the tree diagram exhausts all possible paths from the most general function to the most specific function with the variable of interest
10.6.
Differentiability A function differentiable
( ) ( ) √(
( at
( ) ) (
) is said to be ( ) if
)
In (
general, if
( ) a function ) is said to be differentiable at
( ) √∑ ( )
Note: ( ) 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
point
Note: ( ) is the error in the approximation if the local linear approximation is used
10.7.
Differentials ( ) Let o Define the total differential of ( ) as ( ) ( ) Define o o If ( ) ( and ) )
at
(
(
, then )
56
Elementary Analysis III
Note:
11. Behaviour and Analysis of Multivariate The directional derivative of is interpreted Functions as the rate of change of to the direction of 11.1. Directional Derivatives ⃗⃗ Suppose a surface, , and a plane of arbitrary Remark: direction, , has an intersection curve with slope ⃗⃗ The directional derivatives of in the directions of ̂ and ̂ are and , respectively
11.2.
Gradient The gradient vector, or simply the gradient, ), at any point is given by of ( ( ) ⟨ ( ) ( )⟩, given that ( ) The gradient operator can be extended to a finite number of variables, where it is defined as ⟨ ( )⟩
Note: o ) ( ) ⃗⃗ ⃗⃗ ( If the tangent line to the curve is ‖ ( )‖ projected onto the -plane, its ) may have several At a certain point, ( direction is the same of that of a unit directional derivatives which depend on ⃗⃗, tangent vector ⃗⃗ but the gradient is unique The slope of the curve at a point is called the directional derivative of Remarks: in the direction of ⃗⃗ ( ) increases most rapidly at ( ) in the ) is direction of the gradient; that is, ⃗⃗ ( maximum
⃗⃗
o
Theorem:
⟩ be a unit vector, and be a Let ⃗⃗ ⟨ ) differentiable function of and at ( o The directional derivative of at ( ) exists and is given by ) ( ) ( ) ⃗⃗ ( ⟩, o Similarly, since ⃗⃗ ⟨ ) ( ) then ⃗⃗ ( ( )
57
when
⃗⃗
‖
( (
) )‖
or
) ‖ ( )‖ ( ) decreases most rapidly at ( ) in the opposite direction of the gradient; that )is minimum when ⃗⃗ is, ⃗⃗ (
‖ ( ( ) )‖
(
or
⃗⃗
(
)
‖
(
)‖
(
) is constant when ⃗⃗
11.3.
Tangent Planes to Surfaces
11.4.
Extreme Values Consider the graph:
Suppose and are the slopes of tangent lines and , respectively, to the surface ) at ( o The plane containing and is the Theorem: ) tangent plane to at ( ( ) The points |[ ( ) Consider a surface with equation ( ) ( ) ( ) ] [ ( ) ] are o It can be shown that called the critical points of ; therefore, the gradient ) and if If has a local extremum at ( of can be taken to be the normal ( ) ( ) |( ) vector of the tangent plane to at any point Note: o The normal line to at any point is A critical point need not contain a local the line through the same point with extremum value the gradient at the point as the A critical point that is neither a local direction vector maximum nor a local minimum is called a saddle point
At the region that encloses the peak, the value of the function at the same location is the largest value, or the local maximum ( ) Let ( ) be defined on a region ( ) is a local maximum value of o if ( ) in an open disk centered ) ( ) ( ) at ( ( ) is a local minimum value of o if ( ) in an open disk centered at ( ) ( ) ( )
o
58
Theorem: Second Partials Test Let ( ) have continuous 1st and 2nd partial derivatives throughout a disk ) centered at a critical point ( Define | | (
o
In the Method of Lagrange Multipliers, the given plane is called a constraint
) , Theorem: Method of Lagrange Multipliers Let ( o If ) and ( ) be differentiable is subjected to the constraint ( ) and if has an extreme value not in , it is attained ) satisfying the following: at ( ( ) ( ) { ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) {
also known as the Hessian matrix o If | , then has a local ) maximum at ( o If | , then has a local ) minimum at ( o If , then has a saddle point ) at ( o If , no conclusion
11.5. Absolute Extreme Values on a Closed and Bounded Region Note: A closed and bounded region on the is called the Lagrange Multiplier plane is a finite region whose boundary The Method of Lagrange Multipliers can be points are part of the region extended to a finite number of variables, provided that the partial derivatives exist Theorem: Extreme Value Theorem Let ( ) be continuous on a closed and bounded region , then attains a maximum and minimum value in, or along,
11.6.
Lagrange Multipliers
Recall: Distance between a Point and a Plane Suppose a point ( ) is being located, such that it should be near another point o This is the same as minimizing the distance between the plane and the point
59
12. Multivariate Integral Calculus 12.1. Parametric Surfaces and Vector-valued Functions Recall: Parametric and Space Curves A parametric surface
Multivariate o
on the tangent plane to ) ( ) point where ( The tangent plane to ⃗( has a normal vector ⃗ ⃗
at the ) then
can be represented 12.3. ( ) ( ), where by the equations { ( ) ( ) are the parameters Similarly, it can be represented by the ) vector-valued function ⃗( ⟨ ( ) ( ) ( )⟩
Double Integrals Rectangular Regions , o Define | , , and suppose ( ) is countinuous over
12.2. Calculus of Multivariate Vector-valued Functions ) Let be a surface defined by ⃗( ⟨ ( ) ( ) ( )⟩ Partial Derivatives o Define ⃗ ( and ⃗( ) )
⃗ ⃗
⟨ ⟨
⟩, ⟩,
provided that the partial derivatives exist Tangent Planes o If is fixed at , then ) ⃗( ), which is a function ⃗( of alone, and hence, a parametric curve, and ⃗ is interpreted as the tangent vector when Similarly, if is fixed at , then ) ⃗( ), which is a ⃗( parametric curve with tangent vector ⃗ ) ⃗( Suppose ⃗( )| ⃗ ⃗ , then the two tangent vectors lie
60
o
o
o
The motivation for double integration is the problem of determining the volume of an arbitrary solid Divide first into a mesh of rectangles Considering an arbitrary rectangle , the approximation for its volume is by rectangular prisms Let be the area of the rectangle, and the volume of the prism is ( ) | ( )
, ∑ As ( )
then Properties: ( ) , then a. ∬ ( b. ∬ ∑ ) ) , c. ∬ ( ) ∬ ( ) ∑ ∬ ( ∑ ∬ ( )
) ⋃
∑ ( and this limit is called the double integral of over , Theorem: ) denoted by ∬ (
Non-Rectangular Regions ( ) Consider an integral o Suppse, however, that is an arbitrary region on the -plane, o This expression is called a partial ( ), below bounded above by integral, which is evaluated in the ( ), by , and , same way as partial derivatives where ( ) The double integral is an o The double integral of over is example of a iterated integral, which is defined similarly, but much like an evaluated by performing the inner partial arbitrary Riemann sum integral first Because of the arbitrary shape of the region, divide into a mesh of arbitrary rectangles, with ‖ ‖, called Theorem: Fubini’s Theorem the norm, the largest Weak Form rectangle o Let ( ) be continuous on a Considering an arbitrary rectangular region , given by rectangle , The approximation of its volume ) ∬ ( is by rectangular prisms The approximate volume is ( ) similar with the approximate ( ) volume of rectangular Strong Form regions o Let be the region on the As ‖ ‖ ( ), plane, bounded below by , then ( ), above by , and , where ∑ ( ) ‖ ‖ ) is continuous on If ( ) ∬ ( ) ,then ∬ (
( ) ( )
(
)
o
Let be the region on the plane, bounded to the left by
61
( ), right by ( ), , and , where ) is continuous on If ( ) , then ∬ (
( ) ( )
(
)
Note: For non-rectangular regions, switching the order of integration entails an appropriate change in the limits of integration Sometimes, one order of integration is easier to compute than the other, particularly if the integrand has no closed- or elementary-form antiderivative
o o
Divide into polar rectangles Suppose is the area of the rectangle, then ) ( ( )( ) ), the mean (
o
Define area, and
, the differential
Remarks: If ( over If ∬ ( ) , then the double integral is interpreted as a volume ( ) ( ) ( ), then ( ) ( ) )( ( ) ), only if it is of the Note: ∬ o
radius, then Therefore, in polar coordinates, ) ∬ (
( ) ( )
(
)
weak form of Fubini’s theorem ) ) If ( , then ∬ ( is the area of the region
Double integrals over polar regions are not open for change of order of integration
12.4.
Double Integrals over Polar Regions In rectangular coordinates, the region of integration is divided into small rectangles so that Suppose, however, that can be expressed in polar form
12.5.
Surface Area Let be the parametric surface with ) equation ⃗( o The concept of surface area is the approximation of the patch of ∑ the surface, , so that
62
o
Therefore, ∬ √ ( ( )) ( ( ))
12.6.
o Define ) ⃗⃗ ⃗ ⃗⃗ ( ⃗ ⃗⃗ ), ) is ⃗ ⃗⃗ ( and
Mass and Moments of Mass of a Lamina A lamina is an idealized flat object that is thin enough to be viewed as a two-dimensional plane region, whose thickness is negligible A lamina is homogenous if its composition is uniform throughout, and inhomogenous if not The density of a homogenous lamina of mass and area is given by For an inhomogenous lamina, the composition may vary from point to point, and hence an appropriate definition of density must reflect this Suppose the density at a point ( ) is ), called a specified by a function ( density function, and that its area is fixed o To determine the mass, divide the region into smaller rectangles o For each small rectangle, the mass is ) , where approximated by ( is the area of the small rectangle o Asusming the dimensions of the rectangle tend to zero, the mass is finally given as ∬ ( ) Consider the -axis to be a weightless beam o If a mass point is located on the axis at , the tendency for that mass to produce a rotation of the beam about a point is measured by the quantity called the moment of mass about , ( ) given by o A moment that is positive of is to the right of
o
⃗⃗ ( ⃗ ⃗⃗ ( ) The surface area of ⃗⃗ ‖ ‖ ⃗ ⃗⃗‖ ‖ ⃗⃗ ⃗⃗ ‖ ‖ ⃗⃗
o
Thus, the surface area of the whole ∑ ‖ ⃗⃗ ( ) patch is ⃗⃗ ( )‖ ⃗⃗ ‖ As , ∑ ‖ ⃗⃗ converges to the surface of , that ⃗⃗ ‖ , where is is, ∬ ‖ ⃗⃗
o
the region for the parametric values However, in the special case that the surface ( ), then has an equation ⃗⃗ ( ) ⟨ ( )⟩ o ∬ ‖ ⃗⃗ area, and ⃗⃗ ⃗⃗ ⃗⃗ and √ ( ( where ⟨ ⟨ ( its )) ⃗⃗ ‖ ⃗⃗ ( ) is the surface ⟨ ( norm )) ( )⟩, ) ⟩, is )⟩, so that
( (
63
however, that masses Note: are located at For any function that exists in , or even on a coordinate axis possibly higher, its multiple integral is called and a fulcrum is positioed at point , a hypervolume then the rotation of the beam will o Such function can no longer be depend on the value of the sum expressed geometrically ∑ ( ) Suppose that the -plane is a weightless sheet supporting a mass point located at ), then the tendency for the Theorem: Fubini’s Theorem the point ( mass to produce a rotation of the sheet Weak Form ), called the about the line is ( o Let be the rectangular box given moment of about , and the one by ), corresponding to the line is ( called the moment of about ) ∭ ( Consider an inhomogenous lamina, for which ( ) every point has possibly different mass, ) determined by the density function ( Change in order of o The moment of this lamina about integration is also allowed the - and -axes are given by the for other combinations following: o Let be a solid bounded above by ( ) ∬ ( ), below by ( ), with as the projection ( ) ∬ of on the -plane o The center of gravity ( ̅ ̅) is given ) is continuous, If ( by ( ̅ ̅) . / ) then ∭ ( ∬
( ( ) )
o
Suppose,
(
)
12.7.
Triple Integrals Remark: The extension of triple integrals is analogous to that of double integrals If ( , Define | ∭ , , over solid ( ) o Much like the motivation for integration, will be divided into boxes o Thus, the triple integral of over ) is given by ∭ ( ∑ ( ) provided that the limit exists
‖ ‖
)
, then ∭
(
)
is interpreted as the volume of the
,
64
12.8.
Triple Integrals in Cylindrical Coordinates
o
If the surface is in rectangular coordinates and can be transformed into cylindrical coordinates, then ) ∭ ( ) ∭ ( , where is the same region in cyindrical coordinates
Note: The differential volume in cylindrical ( ) coordinates is
Remark: A point in cylindrical coordinates is For simplicity, and are restricted for ), where: represented by ( , , ) ) , and o is the distance of the projection of from the origin o is the angle made by the projection of and the positive - 12.9. Triple Integrals in Spherical Coordinates axis o is the distance of from the plane of Cylindrical and Rectangular
Conversions Coordinates
Cylindrical to Rectangular ( ) ( Rectangular to Cylindrical ( ) .√
) /
Theorem: Let be the solid bounded below by the ( ), above by surface ( ), and whose projection on the plane is the polar region ) is continuous on , o If ( ) then ∭ ( ∬
( ( ) )
(
)
65
A point in spherical coordinates is ), where: represented by ( o is the distance of from the origin o is the angle made by the projection of from the positive axis
o
is the angle made by posiive -axis of Spherical and
from the
o
̅
∭ ∭
( (
) )
Conversions Coordinates
Rectangular
Spherical to Rectangular - ( ( ) Rectangular to Spherical - ( .√
) ) /
Theorem: ) is continuous on a solid If ( in spherical coordinates, then ) ∭ ( ∭ ( ) If the surface is in rectangular coordinates and can be transformed into spherical ) coordinates, then ∭ ( ∭ ( ( ) , where ), and is the same region in spherical coordinates
12.10. Mass and Moments of Mass of a Solid The concept of mass and moments of mass of a solid is just an extension of the mass and moments of mass of a lamina For a homogenous solid, its mass is given by the product of its density and its volume For an inhomogenous solid, the mass of a ) is solid whose density is defined as ( ) given by ∭ ( The center of mass of a solid is given by ( ̅ ̅ ̅), where o o ̅ ̅
∭ ∭ ∭ ∭ ( ( ( ( ) ) ) )
66
13. Analysis of Vector Fields Note: 13.1. Vector Fields If a vector field can be written as ⃗ , A vector field is a function that associates a then ⃗ is called a conservative vector field, unique vector ⃗ ( ) with each point in and is its potential function or ⃗ ⃗⃗ If ⃗ is conservative, then If ⃗ ( ) is a vector field in , then ⃗( ) ⟨ ( ) ( )⟩ Remarks: Similarly, if ⃗ ( ) is a vector field in , then A vector field with zero divergence is also ⃗( ) ⟨ ( ) ( ) ( )⟩ called a divergence-free vector field The gradient field of a differentiable function A vector field with zero vector curl is also is the field of gradient vectors called an irrotational vector field ⃗( ) o If , then ( o If ( ⟨ ( ) ) ) ⟨ ( , ( ) then ) ( ( ⃗( )⟩ )⟩ ) 13.2. Line Integrals of Scalar Fields Consider a smooth curve on the -plane
Operations on Vector Fields 1. Curl The curl operator measures the infinitesimal rotation of a vector field at a certain point ⃗ ⃗ The curl of ⃗ is defined as ̂ ̂ ̂ o If ⃗ ⟨ , ⃗ | ⟩ Consider the sheet that is swepth out by the vertical lines that extends upward from a ) ) and point ( to a height ( moves along on each point o To get the area of the sheet ( ) ( ) Divide into rectangles with increment The area of the sheet is ∑ given as , |
⃗ ( o If ⃗ , )̂ 2. Divergence The divergence operator measures the source or sink of a vector field ⃗ The divergence of ⃗ is defined as ⃗ o o If ⃗ If ⃗ , , ⃗ ⃗
67
where is the area of the ( ) Let ( ) be a smooth curve , rectangle over ) ( ) Suppose a point ( o lies on the surface, where ( ( ) ( )) ( ) ( ) is a point ( ) o corresponding to some ( ( ) ( )) ( ) ( The two line integrals usually come in pairs, ( ) ) ( ) ( ) that is, If is the length of arc, ( ) then , ( ) ( ) , which is called and furthermore, the differential form line integral ∑ ( ) As in such a way that Note: the intervals’ arc lengths go The parametric form line integral to zero, then ( ) is orientation-independent, ∑ ( ) while the differential form line integral is Let be a smooth curve on the -plane, orientation-dependent then the line integral of along is defined o Let ̃ be a curve which is the same by ∮ ( ) , provided that its as but whose orientation is corresponding limit exists opposite to that of , then the differential form line integral along ̃ ‖⃗⃗ ( )‖ Recall: is simply the negative of the differential form line integral along is the arc length parametrization of ⃗( ) In general, most line integrals along different By FTOC-I, the differential arc length ‖ ⃗ ( )‖ paths produce different results parametrization is given as ( ) ( ) is a Therefore, if Remarks: , -, then smooth curve from ( ) Line integrals can be extended to a finite ( ( ) ( ))√( ( )) ( ( )) number of variables o For a parametric form line integral, ( ) Differential Form ) along The line integral of ( with respect to and are defined as follows: ( ) o o ( ∑ ) ∑ ( ( ) )
68
. ( )/ √∑ o
. ( )/
,
( ) where For a differential form line integral, ∑ ( ) ∑ ( . ( )/ ( ) ), , where
13.3.
Vector Fields and Let be a partition of piecewise, smooth 13.4. Conservative Independence of Path curves, that is, ⋃ o For a parametric form line integral, A connected set is a set where any two ( ) ∑ ( ) points in it can be connected by a curve that lies entirely in the set o For a differential form line integral, A simply connected set is a connected set ( ) ( ) where any closed curve in the set does not ∑ ( ) ( ) enclose points outside the set Geometrically, it has no holes; otherwise, it is called a multiply closed set Line Integrals of Vector Fields If ⃗ is a continuous vector field and is a Theorem: smooth parametric curve in or with ⃗⃗ unit tangent vector , then the work performed by ⃗ on a particle that moves along in the direction of increasing ⃗ ⃗⃗ parameter is Let o be a simply connected plane region, ) ⟨ ( ) ( )⟩ ⃗ is conservative if and only if ⃗ ⃗⃗, which implies is a simply connected with with ⃗ (
Note:
If
is parametrized, then
⃗
⃗⃗
Similarly, suppose space ⃗( ) ⟨ ( o ⃗
has coordinates ( ), then it is associated with a vector ⃗⃗ ( ) ⟨ ( ) ( )⟩ as its position vector, and hence, a vector If a point field ⃗ ( ) is also written as ⃗ . ⃗⃗ ( )/ ⃗ ⃗⃗
) ( ) ( )⟩ ⃗ is conservative if and only if ⃗⃗, which implies , and ,
Given the above note, ⃗ . ⃗⃗ ( )/ ⃗⃗ ( )
Theorem: Fundamental Theorem of Line Integrals ) is a conservative vector Suppose that ⃗ ( field in some open plane region containing ( ) and ( ), with points ) potential function ( o If is any piecewise, smooth curve ⃗ ⃗⃗ from to , then ( Remarks: If ⃗ is conservative, then its work said to be independent of path, that is, the line integral ) ( )
Remark: If ⃗ ( and ) is represented as ⟨ ( ) ( )⟩, ⃗⃗ ( ) ⟨ ( ) ( )⟩ , -, then ⃗ ⃗⃗ ⟨ ( ) ( )⟩ ( ) ( ) ,
⟨ ( ) ( )⟩
which is a differential form line integral
69
taken along any piecewise, smooth path will yield only one unique result If is any closed curve and if ⃗ is conservative, then ⃗ ⃗⃗
o
Suppose, ⟨ ⟩ By ∬ ∮ of
however, Green’s ∬ .
⃗(
)
Theorem, /
, which is the area however, ⃗( )
13.5.
Green’s Theorem
o
Suppose, ⟨ ⟩ By ∬ ∮
Theorem: Green’s Theorem Suppose is a simply connected set on a plane region with a smooth, closed boundary which is the curve , oriented counteclockwise ) ( )⟩ whose o If ⃗ ( ) ⟨ ( components are differentiable, then ∮ ⃗ Note: Green’s theorem does not need the assumption that ⃗ should be conservative; in fact, if ⃗ is conservative, then the trivial ⃗⃗ result would be ∮ ⃗ ⃗⃗ ∬ . / 13.6.
Green’s ∬ .
Theorem, / , which is
the area of
Recall: Area of a Plane Region with Double Integrals The area of a plane region double integrals is ∬ in terms of
By Green’s Theorem, the area of the simply connected plane region with as a smooth, closed boundary requires a vector field which satisfies o Suppose ⃗ ( By ∬ ∮ ) ⟨ ⟩ Green’s ∬ . Theorem, /
Surface Integrals Surface Integrals are integrals over a surface Let be a surface in with finite area, and ( ) be continuous on o If is divided into patches such that the patch has as its ) as a surface area, and ( point in the projection of the patch in the -plane whose height is ( ) ( ) o The sum ∑ approaches a limit, as , called a surface integral over , denoted by ) ∬ ( Suppose is the parametric ⃗⃗ ( ) surface ⟨ ( ) ( ) ( )⟩, where along its parameter space, then ( ) ∬ ∬ ⃗⃗ ‖ . ⃗⃗ ( )/ ‖ ⃗⃗ . ⃗⃗ ( )/
, which is the area of
, where
70
⃗ over is the parametrization of is ∬ ⃗ ⃗⃗ , where ⃗ ( ) represents the velocity of the fluid at a point Suppose, however, that ( ) ( ), where along the -plane, Recall: Surface Area ) then ∬ ( If the surface is expressed parametrically, ( )) , ∬ ( ⃗⃗ ‖ , then so that ‖ ⃗⃗ where √ , ⃗⃗ ( ⃗⃗
⃗⃗ ‖ ⃗⃗ ⃗⃗ ⃗⃗ ‖
‖ ⃗⃗ and
⃗⃗ ‖ thus, ∬ ⃗
which is the surface area of ( )
⃗⃗ )
,
13.7.
Flux An orientable surface is a surface whose points that lie in its separate sides cannot be connected with any wholly continuous line ), then its o If is the surface ⃗⃗ ( orientation is determined by the vector ⃗⃗ o
⃗⃗ ‖ ⃗⃗ ⃗⃗ ⃗⃗ ‖
⃗⃗ ) ( ⃗⃗ The same proof goes for surfaces of the form ( ), which yields ∬ ⃗
( ), then If is the surface its orientation is determined by the ( ) ( ) ⟩, vector ⟨ which is the gradient of Suppose that an oriented surface is immersed in an incompressible, steady-state fluid flow, and assume further that the surface is permeable so that the fluid can flow through freely in any direction o The motivation for this problem is to find the net volume of the fluid that passes through the surface per unit time, where the net volume is the volume that passes through the surface in the positive direction minus that in the negative direction, called the flux If represents the exact net volume of fluid that crosses in the direction of its orientation ⃗⃗ per unit time, then the flux of
71
14. Analysis of Sequences and Series 14.1. Sequences An (infinite) sequence is an unending succession of numbers called terms, and is of the form If the term can be expressed by a ( ), it is written as * + formula, A sequence * + is said to converge to a limit if the terms become arbitrarily close to , that is, given , | | is called the limit of the sequence, that is, If * + is not convergent, it is said to be divergent
Consider the series ∑
, and
o is called the partial sum The series ∑ is convergent is the sequence of partial sums converges; otherwise, it is divergent ∑ ∑ o If and are convergent series, then ∑ ( ) is also convergent o If ∑ is convergent, and , then ∑ is also convergent o A convergent series is still convergent if a finite number of terms have been deleted Convergence Tests
Notes: ̅ A sequence is a function The limit of a sequence is unique
14.3.
Theorem: A geometric series ∑ converges if and only if | | , and diverges if and only if | |
Remarks:
Evaluating the limit of a sequence is the Remark: same with evaluating ( ) If a geometric series converges, then it A recursive sequence is a sequence whose converges to a sum , where is the term depends on its previous terms An alternating sequence is a sequence first term and is the common ratio whose signs of terms alternate o The alternating sequence for which converges if Theorem: Divergence Test If * then +, * +, * + are sequences such that , , , , by squeeze theorem If the series ∑ as If as divergent converges, then , then ∑ is
14.2.
Series A series is the sum ∑ , where , - are called the terms of the series
Remark: If as converge or not , then ∑ may
72
Theorem: Integral Test
Let ∑ be a series of positive terms, and let ( ) be a function that results when is replaced by in the term Theorem: Ratio Test o If is decreasing and continuous on , ), then if: Let ∑ be a series of positive terms, ( ) converges, then and let ∑ ∑ also converges ( ) diverges, then also diverges o o o If If If , the series converges , the series diverges , no conclusion
If ∑
, then if ∑ diverges
diverges, then
Remark: The p-series ∑ and diverges for converges for Theorem: Root Test Let ∑ and let o If o If o If be a series of positive terms, √ , the series converges , the series diverges , no conclusion
Theorem: Comparison Test
Let ∑ and ∑ be an arbitrary series of positive terms o Suppose If ∑ diverges, then 14.4. Alternating Series and Conditional ∑ diverges Convergence If ∑ converges, then A series whose terms alternate between ∑ converges positive and negative terms is called an alternating series In general, an alternating series has one of the following forms: o ∑ ( ) o ∑ ( )
Theorem: Limit Comparison Test Let ∑ and ∑ positive terms o If and ∑ diverges Notes: If ∑ , then if ∑ converges converges, then be series of , then ∑
both converges or Theorem: Alternating Series Test An alternating series converges if the following hold: o * + is a decreasing sequence o
73
A series ∑ is said to be absolutely convergent if the corresponding series of absolute values ∑ | | converges If a series is convergent but not absolutely, it is said to be conditionally convergent
At , the series may be absolutely or conditionally convergent, or divergent
Theorem: If a series is absolutely convergent, any rearrangement of the terms of the series will produce the same sum
Power Series Representation Consider the geometric series ∑ o By the ratio test, the converges to the sum o Thus, the geometric series ∑ is said to be equal to ( ) A convergent power series, in general, has a power series representation that is a function of along its corresponding interval of convergence
series | |
Remark: If the series is absolutely convergent, then it is convergent
14.5.
Power Series A power series in is the infinite sum ∑ , while a power series in is Differentiation and Integration of Power Series ( ) the infinte sum ∑ Theorem: If ( ) For a power series in , exactly one of the , then following is true: o o The series is convergent only when The series is convergent The series is convergent when ( ) At , the series may be absolutely or confitionally 14.6. convergent, or divergent For a power series in , exactly one of the following is true: o The series is convergent only when o o The series is convergent The series is convergent when ( )
74
Theorem:
∑ ( ( ) ∑ ( ) )
( ∑ ( ∑
) ): ( ) ∑ (
|| ) ( )
|
o o
o
Taylor and Maclaurin Series If a function is infinitely differentiable at , then it has a power series in ( ) Let ( ) ∑ o The derivative at yields ( , and thus ( ) ∑ ) ∑
( )(
)
(
) ,
and
this is called the Taylor series of ( ) about
In ( )
the ∑
case
( )(
that is called
, Remark: the If , the Taylor polynomial is called the local linear approximation If , the Taylor polynomial is called the local quadratic approximation In general, the Taylor polynomial is the generalization of approximations about o The larger is, the more accurate the approximation is, as well
)
Maclaurin series of ( ) Some Taylor/Maclaurin Series Expansions 1. 2. ∑ ∑ ∑ 3. 4. 5. ( ∑ ∑ ) (
(
( ) (
) ( )
) ) ( (
) ) ( ) ( )
) )
( ( (
)
( ( ( )
)
∑
)
Binomial Theorem Define . /
( )
) Let ( ) ( o If it is expanded with a Maclaurin Series, it will then become ( ) o ∑ . / have zero ( ) Since all terms after derivatives, then ∑ . / Binomial Theorem
, which is called the
14.7.
Taylor Polynomials and Approximations Let ( ) have a power series representation o The Taylor polynomial of about is the partial sum of its Taylor Series, that is, ( ) ∑
( )(
)
(
)
75