Time-Frequency Analysis: Fourier Transforms and Wavelets
|
|
- Amy Payne
- 5 years ago
- Views:
Transcription
1 Chapter 4 Time-Frequenc Analsis: Fourier Transforms and Wavelets 4. Basics of Fourier Series 4.. Introduction Joseph Fourier (768-83) who gave his name to Fourier series, was not the first to use Fourier series neither did he answer all the questions about them. These series had alread been studied b Euler, d Alembert, Bernoulli and others before him. Fourier also thought wrongl that an function could be represented b Fourier series. However, these series bear his name because he studied them etensivel. The first concise stud of these series appeared in Fourier s publications in 87, 8 and 8 in his Théorie analtique de la chaleur. He applied the technique of Fourier series to solve the heat equation. He had the insight to see the power of this new method. His work set the path for techniques that continue to be developed even toda. Fourier Series, like Talor series, are special tpes of epansion of functions. With Talor series, we are interested in epanding a function in terms of the special set of functions,,, 3,... or more generall in terms of, ( a), ( a), ( a) 3,... You will remember from calculus that if a function f has a power series representation at a then f () = n= f (n) (a) n! ( a) n (4.) Remember from calculus that a series is an infinite sum. We never use the full series, we usuall truncate it. In other words, if we call S N () = ( a) n, N f (n) (a) n! 39 n=
2 4CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS then we approimate f () b S n (). S n () is called a partial sum. A reason for using Talor series is that their partial sums are polnomials and polnomials are the easiest functions to work with. With Fourier series, we are interested in epanding a function f in terms of the special set of functions, cos π π 3π π π, cos, cos,..., sin, sin, sin 3π,... Thus, a Fourier series epansion of a function is an epression of the form f () = A + ( A n cos nπ + B n sin nπ ) for some positive constant. Finding the Fourier series for a given function f () (if it eists) amounts to finding the coeffi cients A n for n =,,,... and B n for n =,, 3,... In this section, we will not focus on theoretical considerations such as convergence, differentiation and integration of Fourier series. We will onl define Fourier series and give a few eamples and applications. 4.. Euler s Formulas for the Coeffi cients Definition 4.. The Fourier series of a function f () on the interval [, ] where > is given b f () = A + ( A n cos nπ + B n sin nπ ) (4.) The coeffi cients which appear in the Fourier series were known to Euler before Fourier, hence the bear his name. The are given b the following formulas. To find the coeffi cients, the following formulas pla an important role:.. nπ cos d = for n =,,... nπ sin d = for n =,, nπ sin 4. nπ sin cos mπ d = for ever m, n { mπ = if m n sin d = = if m = n 5. { nπ mπ cos cos d = = if m n = if m = n We give an outline on how to find these coeffi cients. integrate this Fourier series term b term. We assume we can
3 4.. BASICS OF FOURIER SERIES 4 Computation of A From equation 4., if we integrate each side from to we get ( f () d = A d+ A n cos nπ ) d + B n sin nπ d Using the properties listed above, we are left with Hence, f () d = = A A = f () d A d Computation of A m for m =,,... From equation 4. we multipl each side b cos mπ d and integrate each side from to we get f () cos mπ ( d = A cos mπ d+ A n cos nπ cos mπ d + B n From the properties listed above, A cos mπ d =, nπ sin d = and nπ mπ cos cos d = for ever n m and for the value of n = m we have nπ mπ cos cos d = mπ cos cos mπ d =. Hence, we are left with Thus A m = f () cos mπ d = A m f () cos mπ d for m =,,... mπ cos Computation of B m for m =,,... We proceed in a similar manner, but we multipl each side of equation 4. b sin mπ d. Theorem 4.. The coeffi cients in equation 4. are given b A n = A = f () d (4.3) f () cos nπ d for n =,,... (4.4) sin nπ ) mπ cos d
4 4CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS B n = f () sin nπ d for n =,,... (4.5) Definition 4..3 For a positive integer N, we denote the N th partial sum of the Fourier series of f b S N (). So, we have S N () = A + N ( A n cos nπ We now illustrate what we did with some eamples. + B n sin nπ ) 4..3 Eamples of Fourier Series Eample 4..4 Find the Fourier series of f () = sin on [ π, π]. Using the formulas above along with equation 4., we find that B n = π A n = π π A = sin d = π π π sin cos nd = for all n sin sin nd = ecept when n = When n =,we have A =. Thus, a Fourier series of sin is sin. Of course, this was to be epected. Eample 4..5 Find the Fourier series of f () = sin on [ π, π]. Clearl, this function is π-periodic. Its graph is shown in figure 4... Computation of A. Using the formulas above along with equation 4., we find that A = π = π = π π sin d sin sin d since is even and sin on [, π]. Computation of A n. Remembering that sin a cos bd = cos (a + b) (a + b) cos (a b), we (a b)
5 4.. BASICS OF FOURIER SERIES Figure 4.: Graph of sin have A n = π π sin cos nd = sin cos nd π = cos n + cos n π n + n = ( π n + + ) n = n + n + π (n + ) ( n) 4 = π (4n ) π 3. Computation of B n. B n = sin sin nd π π = since sin sin n is odd
6 44CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS 4. In conclusion sin = π + 4 π (4n cos n ) To see how this series compares to the function, we will plot some of the partial sums. et S N () = π + N 4 π (4n cos n ) Figure 4.: Graph of sin and S () Eample 4..6 We now look at a π-periodic function with discontinuities and derive its Fourier series using the formulas of this section (assuming it is legitimate). This function is called the sawtooth function. It is defined b g () = { (π ) if < π g ( + π) otherwise In other words, the function repeats itself over ever interval of length π. But, it will have discontinuities when = nπ for ever integer n. Find the Fourier series for this function. Plot this function as well as S (), S 7 (), S () where S N () is the N th partial sum of its Fourier series. Since this function is π-periodic, we would compute its Fourier series on [ π, π]
7 4.. BASICS OF FOURIER SERIES Figure 4.3: Graph of sin and S 4 () Figure 4.4: Graph of sin and S ()
8 46CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS and all the integrals involved in computing the coeffi cients would be from π to π. However, in our case, f is π-periodic but described between and π. Using the fact that if a function f is T -periodic then for an real number a, we have T f () d = a+t f () d, we can compute the Fourier coeffi cients a integrating between and π instead of between π and π.. Computation of A. A = π = 4π = g () d (π ) d. Computation of A n. A n = π = π = π g () cos nd for n =,,... [ π (π ) cos nd cos nd ] cos nd The first integral is. The second can be evaluated b parts. cos nd = n sin n π n = π n cos n = sin nd so A n = 3. Computation of B n. B n = π = π = π g () sin nd for n =,,... [ π (π ) sin nd sin nd ] sin nd
9 4.. BASICS OF FOURIER SERIES 47 The first integral is. The second can be done b parts. sin nd = n = π n = π n π cos n + n + sin n n π cos nd Therefore B n = [ π ] π n = n 4. Conclusion. The Fourier series of the sawtooth function is g () = sin n n Below, we show the graphs of S (), S 7 (), S () Graph of the sawtooth function (black) and S () (red)
10 48CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS Graph of the sawtooth function (black) and S 7 () (red) Graph of the sawtooth function (black) and S () (red) Remark 4..7 Several important facts are worth noticing here.. The Fourier series seems to agree with the function, ecept at the points of discontinuit.
11 4.. BASICS OF FOURIER SERIES 49. At the points of discontinuit, the series converges to, which is the average value of the function from the left and from the right. 3. Near the points of discontinuit, the Fourier series overshoots its limiting values. This is a well known phenomenon, known as Gibbs phenomenon Piecewise Continuous and Piecewise Smooth Functions After defining some useful concepts, we give a suffi cient condition for a function to have a Fourier series representation. Notation 4..8 We will denote f (c ) = lim f () and f (c+) = lim f () c c + Remembering that a function f is continuous at c if and onl if lim c f () = f (c), we see that a function f is continuous at c if and onl if f (c ) = f (c+) = f (c) Definition 4..9 (Piecewise Continuous) A function f is said to be piecewise continuous on the interval [a, b] if the following are satisfied:. f (a+) and f (b ) eist.. f is defined and continuous on (a, b) ecept possibl at a finite number of points in (a, b) where the left and right limit at these points eist. Such points are called jump discontinuities. Definition 4.. (Piecewise Smooth) A function f, defined on [a, b] is said to be piecewise smooth on [a, b] if both f and f are piecewise continuous on [a, b]. Eample 4.. The sawtooth function is piecewise smooth. Eample 4.. A simple eample of a function which is not piecewise smooth is 3 for. Its derivative does not eist at, neither do the one sided limits of its derivative at. Eample 4..3 The function f () = is not piecewise continuous on [, ] since it is not continuous at and lim f () does not eist. Definition 4..4 The average of f at c is defined to be f (c ) + f (c+) Clearl, if f is continuous at c, then its average at c is f (c). We are now read to state a fundamental result in the theor of Fourier series.
12 5CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS Theorem 4..5 Suppose that f is a piecewise smooth function on [, ]. Then, for all in [, ], we have f ( ) + f (+) = A + ( A n cos nπ + B n sin nπ ) (4.6) f ( ) + f (+) that is the Fourier series converges to, where the coeffi cients are given b equations 4.3, 4.4, and 4.5. In particular, if f is piecewise smooth and continuous at, then f () = A + ( A n cos nπ that is the Fourier series converges to f (). + B n sin nπ ) (4.7) Thus, at points where f is continuous, the Fourier series converges to the function. At points of discontinuit, the series converges to the average of the function at these points. This was the case in the eample with the sawtooth function. Remark 4..6 In the case f is -periodic, we have an even stronger result. Convergence of the Fourier series is for ever, not just for ever in [, ]. We do one more eample. Eample 4..7 (Triangular Wave) The π-periodic triangular wave is given on the interval [ π, π] b { π + if π h () = π if π. Find its Fourier series.. Plot h () as well as some partial sums of its Fourier series. 3. Show how this series could be used to approimate π ( actuall π ). Solution We begin b plotting h ()We see the function is piecewise smooth and continuous for all. Computation of A. A = π = π π = π π h () d
13 4.. BASICS OF FOURIER SERIES Figure 4.5: Plot of the triangular wave Computation of A n. A n = h () cos nd π π = [ (π + ) cos nd + π π ] (π ) cos nd = (π ) cos nd replacing b in the first integral π = [ π π sin n π n + ] sin nd n = [ π] cos n π n = [ ] cos nπ + π n n = [ ] π n ( )n n { if n even = 4 πn if n odd
14 5CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS Computation of B n. B n = π π h () sin nd = since the integrand is odd (wh?) Conclusion. h () = π + 4 π n= cos (n + ) (n + ). et S N () = π + 4 π N n= cos (n + ) (n + ). We plot S (), S 5 () Figure 4.6: Plot of the triangular wave and S () 3. From h () = π + 4 π n= cos (n + ) (n + ), if we let = then h () = h () =
15 4.. BASICS OF FOURIER SERIES Figure 4.7: Plot of the triangular wave and S 5 () π, hence we get F (, ) = π + π = π + 4 π π π 8 = 4 π = n= This allows us to approimate π. n= 4 π(4n ) cos n n= (n + ) (n + ) (n + ) = Fourier Series of Even and Odd Functions We finish this section b noticing that in the special cases that f is either even or odd, the series simplifies greatl. If f is even, then nπ f () sin is odd so that B n = and the series is simpl a cosine series. Similarl, if f is odd, then nπ f () cos is odd and A n = and the series is simpl a sine series. We summarize this in a theorem.
16 54CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS Theorem 4..9 Suppose that on [, ] f has the Fourier series representation [ f () = A + A n cos nπ + B n sin nπ ] Then:. If f is even then B n = for all n and in this case f () = A + A n cos nπ. If f is odd then A n = for all n and in this case f () = B n sin nπ 4..6 Fourier Series in Comple Form Recall Euler s identit e ±i = cos ± i sin (4.8) The Fourier series of a function f on [, ] can be written where f () = n= C n e inπ (4.9) C n = f () e inπ d (4.) The relationship between A n and B n on one hand and C n on the other is given b and C = A C n = A n ib n C n = A n + ib n A = C A n = C n + C n B n = i (C n C n )
17 4.. BASICS OF FOURIER SERIES Some Applications Fourier series are found in the following applications:. One of the main uses of Fourier series is in solving some of the differential equations from mathematical phsics such as the wave equation or the heat equation. Fourier developed his theor b working on the heat equation.. Fourier series also have applications in music snthesis and image processing (signal processing). When we represent a signal f (t) b its Fourier [ series f (t) = A + A n cos nπt + B n sin nπt ], we are finding the con- tribution of each frequenc nπ to the signal. The value of the corresponding coeffi cients give us that contribution. The n th term of the partial sum of the Fourier series, A n cos nπt + B n sin nπt, is called the nth harmonic of f. Its amplitude is given b A n + B n. 3. Conversel, we can create a signal b using the Fourier series A + [ A n cos nπt for a given value of and plaing with the value of the coeffi cients. Audio signals describe air pressure variations captured b our ears and perceived as sounds. We will focus here on periodic audio signals also known as tones. Such signals can be represented b Fourier series. A pure tone can be written as (t) = a cos (ωt + φ) where a > is the amplitude, ω > is the frequenc in radians/seconds and φ is the phase angle. An alternative wa to represent the frequenc is in Hertz. The frequenc f in Hertz is given b f = ω π. The pitch of a pure tone is logarithmicall related to the frequenc. An octave is a frequenc range between f and f for a given frequenc f in Hertz. Tones separated b an octave are perceived b our ears to be ver similar. In western music, an octave is divided into notes equall spaced on the logarithmic scale. The ordering of notes in the octave beginning at the frequenc Hz are shown below + B n sin nπt ] Note A A# B C C# D D# E F F# G G# A Frequenc (Hz) A more complicated tone can be represented b a Fourier series of the form (t) = a cos (ωt + φ ) + a cos (ωt + φ ) +...
18 56CHAPTER 4. TIME-FREQUENCY ANAYSIS: FOURIER TRANSFORMS AND WAVEETS 4..8 Eercises. In this section, we learned how to epress a function f () as a Fourier series, that is we learned how to compute the coeffi cients A, A n and B n ( for n =,, 3,... such that f () = A + A n cos nπ + B n sin nπ ). Suppose that f () represents some one-dimensional signal and we are given its Fourier series representation, that is we are given A, A n and B n. How would one recover the signal f () from these coeffi cients?. Using question, give an idea how one might remove noise from a nois signal given b a function f (). Give some justification to our idea. Give some shortcomings our method might have, if an. 3. Using question, give an idea how one might compress a signal given b a function f (). Give some justification to our idea. Give some shortcomings our method might have, if an.
19 Bibliograph [] M. C, S. M, Y. Z, V. C., Big data: related technologies, challenges and future prospects, Springer, 4. [] J. D, Big data, data mining, and machine learning: value creation for business leaders and practitioners, John Wile & Sons, 4. [3]. G, What s this all about?, Time, 86 (5), pp [4] H. A. K, Big Data: techniques and technologies in geoinformatics, CRC Press, 4. [5] J. N. K, Data-Driven Modeling & Scientific Computation: Methods for Comple Sstems and Big Data, Oford Universit Press, 3. [6] F. R. I, Google le nouvel einstein, Science & Vie, 38 (), pp [7] K.-C., H. J,. T. Y, A. C, Big Data: Algorithms, Analtics, and Applications, CRC Press, 5. [8] T. M P, Will our data drown us, IEEE Spectrum, (5), pp [9] T. P, Giving our bod a "check engine" light, IEEE Spectrum, (5), pp []. S, Should ou get paid for our data, IEEE Spectrum, (5), pp [] E. S, Their prescription: Big data, IEEE Spectrum, (5), pp [] S. Q. Y, Big data analsis for bioinformatics and biomedical discoveries, CRC Press, 6. 3
Time-Frequency Analysis: Fourier Transforms and Wavelets
Chapter 4 Time-Frequenc Analsis: Fourier Transforms and Wavelets 4. Basics of Fourier Series 4.. Introduction Joseph Fourier (768-83) who gave his name to Fourier series, was not the first to use Fourier
More informationTime-Frequency Analysis
Time-Frequency Analysis Basics of Fourier Series Philippe B. aval KSU Fall 015 Philippe B. aval (KSU) Fourier Series Fall 015 1 / 0 Introduction We first review how to derive the Fourier series of a function.
More information3.3 Eigenvalues and Eigenvectors
.. EIGENVALUES AND EIGENVECTORS 27. Eigenvalues and Eigenvectors In this section, we assume A is an n n matrix and x is an n vector... Definitions In general, the product Ax results is another n vector
More informationNumerical Linear Algebra
Chapter 3 Numerical Linear Algebra We review some techniques used to solve Ax = b where A is an n n matrix, and x and b are n 1 vectors (column vectors). We then review eigenvalues and eigenvectors and
More information3.2 Iterative Solution Methods for Solving Linear
22 CHAPTER 3. NUMERICAL LINEAR ALGEBRA 3.2 Iterative Solution Methods for Solving Linear Systems 3.2.1 Introduction We continue looking how to solve linear systems of the form Ax = b where A = (a ij is
More informationMath 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy
Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide
More informationSection 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
More information16.5. Maclaurin and Taylor Series. Introduction. Prerequisites. Learning Outcomes
Maclaurin and Talor Series 6.5 Introduction In this Section we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see)
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationRepresentation of Functions by Power Series. Geometric Power Series
60_0909.qd //0 :09 PM Page 669 SECTION 9.9 Representation of Functions b Power Series 669 The Granger Collection Section 9.9 JOSEPH FOURIER (768 80) Some of the earl work in representing functions b power
More informationDifferentiation and Integration of Fourier Series
Differentiation and Integration of Fourier Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 12 Introduction When doing manipulations with infinite sums, we must remember
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
More information10.5 Graphs of the Trigonometric Functions
790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular (trigonometric functions as functions of real numbers and pick up where
More information5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS
CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationIn everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
More informationChapter 8 More About the Trigonometric Functions
Relationships Among Trigonometric Functions Section 8. 8 Chapter 8 More About the Trigonometric Functions Section 8. Relationships Among Trigonometric Functions. The amplitude of the graph of cos is while
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More information67. (a) Use a computer algebra system to find the partial fraction CAS. 68. (a) Find the partial fraction decomposition of the function CAS
SECTION 7.5 STRATEGY FOR INTEGRATION 483 6. 2 sin 2 2 cos CAS 67. (a) Use a computer algebra sstem to find the partial fraction decomposition of the function 62 63 Find the area of the region under the
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More informationThe Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More information7-6. nth Roots. Vocabulary. Geometric Sequences in Music. Lesson. Mental Math
Lesson 7-6 nth Roots Vocabular cube root n th root BIG IDEA If is the nth power of, then is an nth root of. Real numbers ma have 0, 1, or 2 real nth roots. Geometric Sequences in Music A piano tuner adjusts
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationCopyright PreCalculusCoach.com
Continuit, End Behavior, and Limits Assignment Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit as
More informationAnalytic Trigonometry
CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It
More informationNotes 7 Analytic Continuation
ECE 6382 Fall 27 David R. Jackson Notes 7 Analtic Continuation Notes are from D. R. Wilton, Dept. of ECE Analtic Continuation of Functions We define analtic continuation as the process of continuing a
More informationFOURIER ANALYSIS. (a) Fourier Series
(a) Fourier Series FOURIER ANAYSIS (b) Fourier Transforms Useful books: 1. Advanced Mathematics for Engineers and Scientists, Schaum s Outline Series, M. R. Spiegel - The course text. We follow their notation
More informationIntroduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationUniform continuity of sinc x
Uniform continuit of sinc Peter Haggstrom www.gotohaggstrom.com mathsatbondibeach@gmail.com August 3, 03 Introduction The function sinc = sin as follows: is well known to those who stud Fourier theor.
More information1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs
0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationRe(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by
F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square
More informationx c x c This suggests the following definition.
110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken
More informationContinuity, End Behavior, and Limits. Unit 1 Lesson 3
Unit Lesson 3 Students will be able to: Interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke Vocabular:
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationAdditional Topics in Differential Equations
6 Additional Topics in Differential Equations 6. Eact First-Order Equations 6. Second-Order Homogeneous Linear Equations 6.3 Second-Order Nonhomogeneous Linear Equations 6.4 Series Solutions of Differential
More informationThe Laplace Transform
The Laplace Transform Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Definition of the Laplace Transform Today 1 / 16 Outline General idea behind the Laplace transform and other
More informationChapter One. Chapter One
Chapter One Chapter One CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which of the following functions has its domain identical with its range?
More informationProperties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results
More informationFourier Analysis Fourier Series C H A P T E R 1 1
C H A P T E R Fourier Analysis 474 This chapter on Fourier analysis covers three broad areas: Fourier series in Secs...4, more general orthonormal series called Sturm iouville epansions in Secs..5 and.6
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More informationMore on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
More informationDerivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
More informationDerivatives of Multivariable Functions
Chapter 10 Derivatives of Multivariable Functions 10.1 Limits Motivating Questions What do we mean b the limit of a function f of two variables at a point (a, b)? What techniques can we use to show that
More informationLimits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
More information4 The Harmonics of Vibrating Strings
4 The Harmonics of Vibrating Strings 4. Harmonics and Vibrations What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task
More informationFunctions and Graphs TERMINOLOGY
5 Functions and Graphs TERMINOLOGY Arc of a curve: Part or a section of a curve between two points Asmptote: A line towards which a curve approaches but never touches Cartesian coordinates: Named after
More information4.2 The Fourier Transform
4.2. THE FOURIER TRANSFORM 57 4.2 The Fourier Transform 4.2.1 Inroducion One way o look a Fourier series is ha i is a ransformaion from he ime domain o he frequency domain. Given a signal f (), finding
More informationChapter 8 Notes SN AA U2C8
Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of
More informationLESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More informationMath Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
More informationIntegration. 5.1 Antiderivatives and Indefinite Integration. Suppose that f(x) = 5x 4. Can we find a function F (x) whose derivative is f(x)?
5 Integration 5. Antiderivatives and Indefinite Integration Suppose that f() = 5 4. Can we find a function F () whose derivative is f()? Definition. A function F is an antiderivative of f on an interval
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Eam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justif our answer b either computing the sum or b b showing which convergence test ou are
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More information5.6. Differential equations
5.6. Differential equations The relationship between cause and effect in phsical phenomena can often be formulated using differential equations which describe how a phsical measure () and its derivative
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationWe have examined power functions like f (x) = x 2. Interchanging x
CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function
More information(MTH5109) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES. 1. Introduction to Curves and Surfaces
(MTH509) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES DR. ARICK SHAO. Introduction to Curves and Surfaces In this module, we are interested in studing the geometr of objects. According to our favourite
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationA.5. Complex Numbers AP-12. The Development of the Real Numbers
AP- A.5 Comple Numbers Comple numbers are epressions of the form a + ib, where a and b are real numbers and i is a smbol for -. Unfortunatel, the words real and imaginar have connotations that somehow
More informationN coupled oscillators
Waves 1 1 Waves 1 1. N coupled oscillators towards the continuous limit. Stretched string and the wave equation 3. The d Alembert solution 4. Sinusoidal waves, wave characteristics and notation T 1 T N
More informationLimits and Continuity of Functions of several Variables
Lesson: Limits and Continuit of Functions of several Variables Lesson Developer: Kapil Kumar Department/College: Assistant Professor, Department of Mathematics, A.R.S.D. College, Universit of Delhi Institute
More informationINTRODUCTION TO DIFFERENTIAL EQUATIONS
INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationChapter 3: Three faces of the derivative. Overview
Overview We alread saw an algebraic wa of thinking about a derivative. Geometric: zooming into a line Analtic: continuit and rational functions Computational: approimations with computers 3. The geometric
More information1.7 Inverse Functions
71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,
More informationInfinite Limits. Let f be the function given by. f x 3 x 2.
0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and
More informationLesson 15: Piecewise Functions
Classwork Opening Eercise For each real number aa, the absolute value of aa is the distance between 0 and aa on the number line and is denoted aa. 1. Solve each one variable equation. a. = 6 b. = 4 c.
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan;
More informationA BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
More informationy = 3 2 x 3. The slope of this line is 3 and its y-intercept is (0, 3). For every two units to the right, the line rises three units vertically.
Mat 2 - Calculus for Management and Social Science. Understanding te basics of lines in te -plane is crucial to te stud of calculus. Notes Recall tat te and -intercepts of a line are were te line meets
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More information4Cubic. polynomials UNCORRECTED PAGE PROOFS
4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review
More informationBIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler
BIG IDEAS MATH Ron Larson Laurie Boswell Sampler 3 Polnomial Functions 3.1 Graphing Polnomial Functions 3. Adding, Subtracting, and Multipling Polnomials 3.3 Dividing Polnomials 3. Factoring Polnomials
More informationFitting Integrands to Basic Rules
6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration
More informationIncreasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video
SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl
More informationQUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
More information6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.
5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,
More informationVocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient.
CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing
More information8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
More informationSection J Venn diagrams
Section J Venn diagrams A Venn diagram is a wa of using regions to represent sets. If the overlap it means that there are some items in both sets. Worked eample J. Draw a Venn diagram representing the
More informationFitting Integrands to Basic Rules. x x 2 9 dx. Solution a. Use the Arctangent Rule and let u x and a dx arctan x 3 C. 2 du u.
58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration rules Fitting Integrands
More informationFixed Point Theorem and Sequences in One or Two Dimensions
Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask
More informationAdditional Topics in Differential Equations
0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential
More informationx y x 2 2 x y x x y x U x y x y
Lecture 7 Appendi B: Some sample problems from Boas Here are some solutions to the sample problems assigned for hapter 4 4: 8 Solution: We want to learn about the analyticity properties of the function
More informationExact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationPure Further Mathematics 2. Revision Notes
Pure Further Mathematics Revision Notes October 016 FP OCT 016 SDB Further Pure 1 Inequalities... 3 Algebraic solutions... 3 Graphical solutions... 4 Series Method of Differences... 5 3 Comple Numbers...
More informationPhysics Gravitational force. 2. Strong or color force. 3. Electroweak force
Phsics 360 Notes on Griffths - pluses and minuses No tetbook is perfect, and Griffithsisnoeception. Themajorplusisthat it is prett readable. For minuses, see below. Much of what G sas about the del operator
More information