# f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y

Save this PDF as:

Size: px
Start display at page:

Download "f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y"

## Transcription

1 Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions: f a n a n cos n b n sin n a a cos a cos a 3 cos 3 b sin b sin b 3 sin 3 Earlier, Daniel Bernoulli and eonard Euler had used such series while investigating problems concerning vibrating strings and astronom. The series in Equation is called a trigonometric series or Fourier series and it turns out that epressing a function as a Fourier series is sometimes more advantageous than epanding it as a power series. In particular, astronomical phenomena are usuall periodic, as are heartbeats, tides, and vibrating strings, so it makes sense to epress them in terms of periodic functions. We start b assuming that the trigonometric series converges and has a continuous function f as its sum on the interval,, that is, f a n a n cos n b n sin n Our aim is to find formulas for the coefficients a n and b n in terms of f. Recall that for a power series f c n a n we found a formula for the coefficients in terms of derivatives: c n f n an!. Here we use integrals. If we integrate both sides of Equation and assume that it s permissible to integrate the series term-b-term, we get But f d a d a n a n n a n cos n b n sin n d cos n d cos n d n sin n n sin n sinn n b n sin n d because n is an integer. Similarl,. So sin n d f d a

2 FOURIER SERIES Notice that a is the average value of f over the interval,. and solving for 3 a gives a f d To determine a n for n we multipl both sides of Equation b cos m (where m is an integer and m ) and integrate term-b-term from to : f cos m d a n cos n b n sin n cos m d a n a cos m d n a n cos n cos m d n b n sin n cos m d We ve seen that the first integral is. With the help of Formulas 8, 8, and 6 in the Table of Integrals, it s not hard to show that sin n cos m d cos n cos m d for all n and m for n m for n m So the onl nonzero term in () is a m and we get Solving for, and then replacing m b n, we have a m f cos m d a m 5 a n f cos n d n,, 3,... Similarl, if we multipl both sides of Equation b sin m and integrate from to, we get 6 b n f sin n d n,, 3,... We have derived Formulas 3, 5, and 6 assuming f is a continuous function such that Equation holds and for which the term-b-term integration is legitimate. But we can still consider the Fourier series of a wider class of functions: A piecewise continuous function on a, b is continuous ecept perhaps for a finite number of removable or jump discontinuities. (In other words, the function has no infinite discontinuities. See Section.5 for a discussion of the different tpes of discontinuities.)

3 FOURIER SERIES 3 7 Definition et f be a piecewise continuous function on,. Then the Fourier series of f is the series a a n cos n b n sin n where the coefficients and in this series are defined b a n b n n a f d a n f cos n d b n f sin n d and are called the Fourier coefficients of f. Notice in Definition 7 that we are not saing f is equal to its Fourier series. ater we will discuss conditions under which that is actuall true. For now we are just saing that associated with an piecewise continuous function f on, is a certain series called a Fourier series. EXAMPE Find the Fourier coefficients and Fourier series of the square-wave function f defined b f if if and f f So f is periodic with period and its graph is shown in Figure. Engineers use the square-wave function in describing forces acting on a mechanical sstem and electromotive forces in an electric circuit (when a switch is turned on and off repeatedl). Strictl speaking, the graph of f is as shown in Figure (a), but it s often represented as in Figure (b), where ou can see wh it s called a square wave. _ (a) FIGURE Square-wave function _ (b) SOUTION Using the formulas for the Fourier coefficients in Definition 7, we have a f d d d

4 FOURIER SERIES and, for n, a n f cos n d sin n n n d sin n sin cos n d b n f sin n d cos n n n d cos n cos sin d Note that cos n equals if n is even and if n is odd. n if n is even if n is odd The Fourier series of f is therefore a a cos a cos a 3 cos 3 b sin b sin b 3 sin 3 sin sin sin 3 Since odd integers can be written as n k, where k is an integer, we can write the Fourier series in sigma notation as k sin sin 5 7 sink k sin 3 sin 5 sin 7 sin 5 In Eample we found the Fourier series of the square-wave function, but we don t know et whether this function is equal to its Fourier series. et s investigate this question graphicall. Figure shows the graphs of some of the partial sums S n sin when n is odd, together with the graph of the square-wave function. 3 sin 3 n sin n

5 FOURIER SERIES 5 S S S _ S S S _ FIGURE Partial sums of the Fourier series for the square-wave function We see that, as n increases, S n becomes a better approimation to the square-wave function. It appears that the graph of S n is approaching the graph of f, ecept where or is an integer multiple of. In other words, it looks as if f is equal to the sum of its Fourier series ecept at the points where f is discontinuous. The following theorem, which we state without proof, sas that this is tpical of the Fourier series of piecewise continuous functions. Recall that a piecewise continuous function has onl a finite number of jump discontinuities on,. At a number a where f has a jump discontinuit, the one-sided limits eist and we use the notation f a lim l a f f a lim l a f 8 Fourier Convergence Theorem If f is a periodic function with period and f and f are piecewise continuous on,, then the Fourier series (7) is convergent. The sum of the Fourier series is equal to f at all numbers where f is continuous. At the numbers where f is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is f f If we appl the Fourier Convergence Theorem to the square-wave function Eample, we get what we guessed from the graphs. Observe that f in f lim l f and f lim l f and similarl for the other points at which f is discontinuous. The average of these left and right limits is, so for an integer n the Fourier Convergence Theorem sas that k k sink f if n n if n (Of course, this equation is obvious for n.)

6 6 FOURIER SERIES Functions with Period If a function f has period other than, we can find its Fourier series b making a change of variable. Suppose f has period, that is f f for all. If we let t and tt f f t then, as ou can verif, t has period and corresponds to t. The Fourier series of t is a a n cos nt b n sin nt n where a tt dt a n tt cos nt dt b n tt sin nt dt If we now use the Substitution Rule with t, then t, dt d, and we have the following Notice that when these formulas are the same as those in (7). 9 If f is a piecewise continuous function on,, its Fourier series is where a na n cos n b n sin n a f d and, for n, a n f cos n d b n f sin n d Of course, the Fourier Convergence Theorem (8) is also valid for functions with period. EXAMPE Find the Fourier series of the triangular wave function defined b for and f f for all. (The graph of f is shown in Figure 3.) For which values of is f equal to the sum of its Fourier series? f FIGURE 3 The triangular wave function _

7 FOURIER SERIES 7 SOUTION We find the Fourier coefficients b putting in (9): Notice that a is more easil calculated as an area. a d ] ] d d and for n, a n cosn d cosn d because cosn is an even function. Here we integrate b parts with u and dv cosn d. Thus, Since sinn sinn d sinn cosn n cos n a n n n is an odd function, we see that n n We could therefore write the series as b n sinn d n cos n n cosn But cos n if n is even and cos n if n is odd, so a n n cos n n if n is even if n is odd Therefore, the Fourier series is cos cos3 n 9 k cosk The triangular wave function is continuous everwhere and so, according to the Fourier Convergence Theorem, we have 5 cos5 f n k cosk for all

8 8 FOURIER SERIES In particular, k k cosk for Fourier Series and Music One of the main uses of Fourier series is in solving some of the differential equations that arise in mathematical phsics, such as the wave equation and the heat equation. (This is covered in more advanced courses.) Here we eplain briefl how Fourier series pla a role in the analsis and snthesis of musical sounds. We hear a sound when our eardrums vibrate because of variations in air pressure. If a guitar string is plucked, or a bow is drawn across a violin string, or a piano string is struck, the string starts to vibrate. These vibrations are amplified and transmitted to the air. The resulting air pressure fluctuations arrive at our eardrums and are converted into electrical impulses that are processed b the brain. How is it, then, that we can distinguish between a note of a given pitch produced b two different musical instruments? The graphs in Figure show these fluctuations (deviations from average air pressure) for a flute and a violin plaing the same sustained note D (9 vibrations per second) as functions of time. Such graphs are called waveforms and we see that the variations in air pressure are quite different from each other. In particular, the violin waveform is more comple than that of the flute. t t FIGURE Waveforms (a) Flute (b) Violin We gain insight into the differences between waveforms if we epress them as sums of Fourier series: Pt a a cos t b sin t a cos t b sin t In doing so, we are epressing the sound as a sum of simple pure sounds. The difference in sounds between two instruments can be attributed to the relative sizes of the Fourier coefficients of the respective waveforms. The nth term of the Fourier series, that is, a n cos nt b n nt is called the nth harmonic of P. The amplitude of the nth harmonic is A n sa n b n and its square, A n a n b n, is sometimes called energ of the nth harmonic. (Notice that

9 FOURIER SERIES 9 A n b n for a Fourier series with onl sine terms, as in Eample, the amplitude is and the energ is A n b n.) The graph of the sequence A n is called the energ spectrum of P and shows at a glance the relative sizes of the harmonics. Figure 5 shows the energ spectra for the flute and violin waveforms in Figure. Notice that, for the flute, A n tends to diminish rapidl as n increases whereas, for the violin, the higher harmonics are fairl strong. This accounts for the relative simplicit of the flute waveform in Figure and the fact that the flute produces relativel pure sounds when compared with the more comple violin tones. n n FIGURE 5 Energ spectra n 6 8 (a) Flute 6 8 (b) Violin n In addition to analzing the sounds of conventional musical instruments, Fourier series enable us to snthesize sounds. The idea behind music snthesizers is that we can combine various pure tones (harmonics) to create a richer sound through emphasizing certain harmonics b assigning larger Fourier coefficients (and therefore higher corresponding energies). Eercises 6 A function f is given on the interval, and f is periodic with period. (a) Find the Fourier coefficients of f. (b) Find the Fourier series of f. For what values of is f equal to its Fourier series? ; (c) Graph f and the partial sums S, S, and S 6 of the Fourier series f f 3. f. f f cos f if if if if if if if if if 7 Find the Fourier series of the function f f f. f,. f t sin3t, t. A voltage E sin t, where t represents time, is passed through a so-called half-wave rectifier that clips the negative part of the wave. Find the Fourier series of the resulting periodic function f t E sin t if if if if if if if if if t t f f f f f 8 f f f f t f t

10 FOURIER SERIES 3 6 Sketch the graph of the sum of the Fourier series of without actuall calculating the Fourier series. 3.. f 3 f 5. f 3, 6. f e, if if if if 7. (a) Show that, if, then f 8. Use the result of Eample to show that Use the result of Eample to show that Use the given graph of f and Simpson s Rule with n 8 to estimate the Fourier coefficients a, a, a, b, and b. Then use them to graph the second partial sum of the Fourier series and compare with the graph of f. 8 3 n (b) B substituting a specific value of, show that n n n cosn n 6.5

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### Analyzing the Graph of a Function

SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

### A Summary of Curve Sketching. Analyzing the Graph of a Function

0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### Pulsed Fourier Transform NMR The rotating frame of reference. The NMR Experiment. The Rotating Frame of Reference.

Pulsed Fourier Transform NR The rotating frame of reference The NR Eperiment. The Rotating Frame of Reference. When we perform a NR eperiment we disturb the equilibrium state of the sstem and then monitor

### Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions

Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,

### 3 Unit Circle Trigonometry

0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### 4.1 Piecewise-Defined Functions

Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

### Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### 6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

### y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### f increasing (tangent lines have positive slope)

smi9885_ch3b.qd 5/7/ : PM Page 7 7 Chapter 3 Applications of Differentiation Debt loan, our indebtedness will decrease over time. If ou plotted our debt against time, the graph might look something like

### C3: Functions. Learning objectives

CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

### Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

### Section 6-4 Product Sum and Sum Product Identities

480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 6-4 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing

### Investigation of Chebyshev Polynomials

Investigation of Chebyshev Polynomials Leon Loo April 25, 2002 1 Introduction Prior to taking part in this Mathematics Research Project, I have been responding to the Problems of the Week in the Math Forum

### 5.2 Inverse Functions

78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

### LIMITS AND CONTINUITY

LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Modelling musical chords using sine waves

Modelling musical chords using sine waves Introduction From the stimulus word Harmon, I chose to look at the transmission of sound waves in music. As a keen musician mself, I was curious to understand

About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is

### THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it

### Q (x 1, y 1 ) m = y 1 y 0

. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

### SECTION 5-1 Exponential Functions

354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

### Sine and Cosine Series; Odd and Even Functions

Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing

### Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions

SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life

### Section 0.2 Set notation and solving inequalities

Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.

### y 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x

Strateg for Integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should appl. But it

### When I was 3.1 POLYNOMIAL FUNCTIONS

146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

### Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis

Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### dy dx and so we can rewrite the equation as If we now integrate both sides of this equation, we get xy x 2 C Integrating both sides, we would have

Linear Differential Equations A first-der linear differential equation is one that can be put into the fm 1 d P y Q where P and Q are continuous functions on a given interval. This type of equation occurs

### Exponential Functions: Differentiation and Integration. The Natural Exponential Function

46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### 2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4

2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle

### 1.2 GRAPHS OF EQUATIONS

000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

### Mathematics of Music

Mathematics of Music Student Author: Janelle K. Hammond Faculty Sponsor: Dr. Susan Kelly, UW-L Department of Mathematics Music is the pleasure the human soul experiences from counting without being aware

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### THE POWER RULES. Raising an Exponential Expression to a Power

8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

### y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C

Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

### Trigonometric Identities and Equations

LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now

### Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

### Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.

Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve

### SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

### Core Maths C3. Revision Notes

Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

### 10.2 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

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### GRAPHS OF RATIONAL FUNCTIONS

0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 6.3 Polar Coordinates

6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

### In this unit, students study arithmetic and geometric

Unit Planning the Unit In this unit, students stud arithmetic and geometric sequences and implicit and eplicit rules for defining them. Then the analze eponential and logarithmic patterns and graphs as

### Pythagoras, Trigonometry, and The Cell Phone

Presentation to Clark State Communit College 03 Ma 2006 Pthagoras, Trigonometr, and The Cell Phone B 2 C 2 A 2 C 2 =A 2 +B 2 Poem: Love Triangle Consider ol Pthagorus, A Greek of long ago, And all that

### Scientific Programming

1 The wave equation Scientific Programming Wave Equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond,... Suppose that the function h(x,t)

### Systems of Linear Equations: Solving by Substitution

8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

### Semester 2, Unit 4: Activity 21

Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Section 0.3 Power and exponential functions

Section 0.3 Power and eponential functions (5/6/07) Overview: As we will see in later chapters, man mathematical models use power functions = n and eponential functions =. The definitions and asic properties

### SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Stud Course MODULE 27 FURTHER APPLICATIONS TO ELECTRICAL CIRCUITS Module Topics 1. Inverse of a matri using elimination 2. Mesh analsis of

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### r(x) = p(x) q(x), 4. r(x) = 2x2 1

Chapter 4 Rational Functions 4. Introduction to Rational Functions If we add, subtract or multipl polnomial functions according to the function arithmetic rules defined in Section.5, we will produce another

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

### DIODE CIRCUITS LABORATORY. Fig. 8.1a Fig 8.1b

DIODE CIRCUITS LABORATORY A solid state diode consists of a junction of either dissimilar semiconductors (pn junction diode) or a metal and a semiconductor (Schottky barrier diode). Regardless of the type,

### Implicit Differentiation

Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

### Introduction to Complex Fourier Series

Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function

### LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

### Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

### SECTION A-3 Polynomials: Factoring

A-3 Polynomials: Factoring A-23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### 1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### TMA4213/4215 Matematikk 4M/N Vår 2013

Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

### Solving Systems of Equations

Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that

### Taylor Polynomials. for each dollar that you invest, giving you an 11% profit.

Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.

### Section 6-3 Double-Angle and Half-Angle Identities

6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

### Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

### I think that starting

. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries