DOCSLIB.ORG

The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12) 1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ..................................... 6 Differentiating trigonometric functions ......................... 8 Applications of the derivatives .............................. 12 Integrating trigonometric functions ........................... 13 Simple harmonic motion .................................. 15 Links forward .......................................... 19 Inverse trigonometric functions .............................. 19 The hyperbolic functions ................................. 21 History and applications ................................... 22 Answers to exercises ..................................... 25 The calculus of trigonometric functions Assumed knowledge The content of the modules: • Trigonometric functions and circular measure • Introduction to differential calculus • Integration. Motivation It is an interesting exercise to sit back and think about how we have developed the topic of trigonometry during the earlier secondary school years. We commenced by looking at ratios of sides in a right-angled triangle. This enabled us to find unknown sides and angles. We extended this to include non-right-angled triangles using the sine and cosine rules. In the module Trigonometric functions and circular measure, we redefined the sine and cosine functions in terms of the coordinates of points on the unit circle. This enabled us to define the sine and cosine of angles greater than 90± and to plot the graphs of the trigonometric functions and discover their periodic nature. The sine and cosine functions are used to model periodic phenomena in nature, such as waves, tides and signals. Indeed, these functions are used to model all sorts of oscillatory motion arising in a range of subjects, including economics and ecology. In this module, we continue this development by applying the ideas and techniques of calculus to the trigonometric functions. For example, if we wish to analyse the motion of a particle modelled by a trigonometric function, we can use calculus to find its velocity and acceleration. The simplicity of the results obtained by doing this is amazing and has wide-ranging impact in physics and electrical engineering, and indeed in any area in which periodic motion is being modelled. A guide for teachers – Years 11 and 12 • {5} Content Review of radian measure We saw in the module Trigonometric functions and circular measure that angles can be naturally defined using arc length. We define 1 radian (written as 1c ) to be the angle subtended in the unit circle by an arc length of one unit. y (0,1) 1 1 x (1,0) O (1,0) � (0,1) Since the circumference of the unit circle is 2¼ and the angle in one revolution is 360±, c we can relate the two units by 2¼ 360± or Æ ¼ 180±. Æ (As usual, we will drop the superscript c when it is clear that the angle under discussion is in radians.) Many commonly occurring angles can be expressed in radians as fractions ¼ 11¼ of ¼. For example, 60± and 330± . Æ 3 Æ 6 We have also seen that the arc length ` of a sector of a circle of radius r , containing an angle θ (in radians), is given by ` r θ, ℓ Æ θ r while the area A of the sector is given by O 1 A r 2θ. Æ 2 {6} • The calculus of trigonometric functions An important limit In order to apply calculus to the trigonometric functions, we will need to evaluate the fundamental limit sinx lim , x 0 x ! which arises when we apply the definition of the derivative of f (x) sinx. Æ It must be stressed that, from here on in this module, we are measuring x in radians. sinx You can see that attempting to substitute x 0 into is fruitless, since we obtain the Æ x indeterminate form ‘zero over zero’. On the other hand, if we try substituting say x 0.1 Æ (in radians), then the calculator gives 0.998..., and if we try x 0.01, we get 0.99998.... Æ sinx We guess that, as x approaches 0, the value of approaches 1. We will now give a nice x geometric proof of this fact. Consider a sector OAB of the unit circle containing an acute angle x, as shown in the following diagram. Drop the perpendicular BD to OA and raise a perpendicular at A to meet OB produced at C. Since OA OB 1, we have BD sinx and AC tanx. Æ Æ Æ Æ B C 1 x O D A 1 The area of the sector OAB is x. The sector OAB clearly contains the triangle OAB and 2 is contained inside the triangle OAC. Hence, comparing their areas, we have 1 1 1 1 Area OAB x Area OAC OA BD x OA AC 4 · 2 · 4 Æ) 2 ¢ · 2 · 2 ¢ 1 1 1 sinx x tanx Æ) 2 · 2 · 2 sinx x tanx. Æ) · · sinx ¼ Now, since tanx and since sinx 0 for 0 x , we can divide both inequalities Æ cosx È Ç Ç 2 by sinx to obtain x 1 1 . · sinx · cosx A guide for teachers – Years 11 and 12 • {7} Since cosx approaches 1 as x approaches 0, we see that x 1 as x 0. sinx ! ! Taking the reciprocal, it follows that sinx 1 as x 0. x ! ! (Here we have used the pinching theorem and the algebra of limits, as discussed in the module Limits and continuity.) Thus we can write sinx lim 1. x 0 x Æ ! Note that this means that, if the angle x is small, then sinx x. This fact is often used by ¼ physicists when analysing such things as a simple pendulum with small angle. Exercise 1 Using the limit above, prove that tanx lim 1. x 0 x Æ ! Other related limits can be found by manipulating this basic limit. Example Find sin2x lim . x 0 3x ! Solution We can write sin2x sin2x 2 . 3x Æ 2x £ 3 If we let u 2x, then as x 0, we have u 0. Hence Æ ! ! sin2x 2 sinu 2 lim lim . x 0 3x Æ 3 u 0 u Æ 3 ! ! {8} • The calculus of trigonometric functions Exercise 2 Find sin3x sin7x lim Å . x 0 5x ! Exercise 3 1 cosx 1 a Show that lim ¡ . x 0 x2 Æ 2 ! b Deduce that cosx 1 1 x2, for small x. ¼ ¡ 2 Differentiating trigonometric functions The derivative of sine Since the graph of y sinx is a smooth curve, we would like to find the gradient of the Æ tangent to the curve at any point on it. Before doing this, we derive a useful trigonometric identity that will assist us. Using the compound-angle formulas, we have sin(A B) sin(A B) sin A cosB cos A sinB (sin A cosB cos A sinB) Å ¡ ¡ Æ Å ¡ ¡ 2 cos A sinB. Æ If we put C A B and D A B, we can add these equations to obtain A 1 (C D) Æ Å Æ ¡ Æ 2 Å and subtract them to obtain B 1 (C D). Substituting these back, we obtain the sine Æ 2 ¡ difference formula: ³C D ´ ³C D ´ sinC sinD 2 cos Å sin ¡ . ¡ Æ 2 2 To find the derivative of sinx, we return to the first principles definition of the derivative of y f (x): Æ d y f (x h) f (x) lim Å ¡ . dx Æ h 0 h ! Substituting y sinx, we have Æ d y sin(x h) sinx lim Å ¡ . dx Æ h 0 h ! A guide for teachers – Years 11 and 12 • {9} Applying the sine difference formula, we have ¡ h ¢ ¡ h ¢ sin(x h) sinx 2 cos x sin lim Å ¡ lim Å 2 2 . h 0 h Æ h 0 h ! ! We can put u h . Then u 0 as h 0. So the limit becomes Æ 2 ! ! d y 2 cos(x u) sinu lim Å dx Æ u 0 2u ! ³ ´ ³ sinu ´ lim cos(x u) lim Æ u 0 Å £ u 0 u ! ! cosx 1 Æ £ cosx. Æ We can thus conclude that d (sinx) cosx. dx Æ This result is both simple and surprising, and students need to commit it to memory. The derivation above involved a number of ingredients and is often difficult for students the first time through. Derivatives of other trigonometric functions Now that the derivative of sine is established, we can use the standard rules of calculus — the chain, product and quotient rules — to proceed. Since cosx sin¡x ¼ ¢, we can apply the chain rule to see that Æ Å 2 d d ³ ¼ ´ (cosx) sin¡x ¢ dx Æ dx Å 2 ¼ cos¡x ¢ Æ Å 2 sinx.
Load more

Recommended publications
  • Mathematics 412 Partial Differential Equations
    Mathematics 412 Partial Differential Equations c S. A. Fulling Fall, 2005 The Wave Equation This introductory example will have three parts.* 1. I will show how a particular, simple partial differential equation (PDE) arises in a physical problem. 2. We’ll look at its solutions, which happen to be unusually easy to find in this case. 3. We’ll solve the equation again by separation of variables, the central theme of this course, and see how Fourier series arise. The wave equation in two variables (one space, one time) is ∂2u ∂2u = c2 , ∂t2 ∂x2 where c is a constant, which turns out to be the speed of the waves described by the equation. Most textbooks derive the wave equation for a vibrating string (e.g., Haber- man, Chap. 4). It arises in many other contexts — for example, light waves (the electromagnetic field). For variety, I shall look at the case of sound waves (motion in a gas). Sound waves Reference: Feynman Lectures in Physics, Vol. 1, Chap. 47. We assume that the gas moves back and forth in one dimension only (the x direction). If there is no sound, then each bit of gas is at rest at some place (x,y,z). There is a uniform equilibrium density ρ0 (mass per unit volume) and pressure P0 (force per unit area). Now suppose the gas moves; all gas in the layer at x moves the same distance, X(x), but gas in other layers move by different distances. More precisely, at each time t the layer originally at x is displaced to x + X(x,t).
    [Show full text]
  • Refresher Course Content
    Calculus I Refresher Course Content 4. Exponential and Logarithmic Functions Section 4.1: Exponential Functions Section 4.2: Logarithmic Functions Section 4.3: Properties of Logarithms Section 4.4: Exponential and Logarithmic Equations Section 4.5: Exponential Growth and Decay; Modeling Data 5. Trigonometric Functions Section 5.1: Angles and Radian Measure Section 5.2: Right Triangle Trigonometry Section 5.3: Trigonometric Functions of Any Angle Section 5.4: Trigonometric Functions of Real Numbers; Periodic Functions Section 5.5: Graphs of Sine and Cosine Functions Section 5.6: Graphs of Other Trigonometric Functions Section 5.7: Inverse Trigonometric Functions Section 5.8: Applications of Trigonometric Functions 6. Analytic Trigonometry Section 6.1: Verifying Trigonometric Identities Section 6.2: Sum and Difference Formulas Section 6.3: Double-Angle, Power-Reducing, and Half-Angle Formulas Section 6.4: Product-to-Sum and Sum-to-Product Formulas Section 6.5: Trigonometric Equations 7. Additional Topics in Trigonometry Section 7.1: The Law of Sines Section 7.2: The Law of Cosines Section 7.3: Polar Coordinates Section 7.4: Graphs of Polar Equations Section 7.5: Complex Numbers in Polar Form; DeMoivre’s Theorem Section 7.6: Vectors Section 7.7: The Dot Product 8. Systems of Equations and Inequalities (partially included) Section 8.1: Systems of Linear Equations in Two Variables Section 8.2: Systems of Linear Equations in Three Variables Section 8.3: Partial Fractions Section 8.4: Systems of Nonlinear Equations in Two Variables Section 8.5: Systems of Inequalities Section 8.6: Linear Programming 10. Conic Sections and Analytic Geometry (partially included) Section 10.1: The Ellipse Section 10.2: The Hyperbola Section 10.3: The Parabola Section 10.4: Rotation of Axes Section 10.5: Parametric Equations Section 10.6: Conic Sections in Polar Coordinates 11.
    [Show full text]
  • Algebra/Geometry/Trigonometry App Samples
    Algebra/Geometry/Trigonometry App Samples Holt McDougal Algebra 1 HMH Fuse: Algebra 1- HMH Fuse is the first core K-12 education solution developed exclusively for the iPad. The portability of a complete classroom course on an iPad enables students to learn in the classroom, on the bus, or at home—anytime, anywhere—with engaging content that provides an individually-tailored learning experience. Students and educators using HMH Fuse: will benefit from: •Instructional videos that teach or re-teach all key concepts •Math Motion is a step-by-step interactive demonstration that displays the process to solve complex equations •Homework Help provides at-home support for intricate problems by providing hints for each step in the solution •Vocabulary support throughout with links to a complete glossary that includes audio definitions •Tips, hints, and links that enable students to acquire the help they need to understand the lessons every step of the way •Quizzes that assess student’s skills before they begin a concept and at strategic points throughout the chapters. Instant, automatic grading of quizzes lets students know exactly how they have performed •Immediate assessment results sent to teachers so they can better differentiate instruction. Sample- Cost is Free, Complete App price- $59.99 Holt McDougal HMH Fuse: Geometry- Following our popular HMH Fuse: Algebra 1 app, HMH Fuse: Geometry is the newest offering in the HMH Fuse series. HMH Fuse: Geometry will allow you a sneak peek at the future of mobile geometry curriculum and includes a FREE sample chapter. HMH Fuse is the first core K-12 education solution developed exclusively for the iPad.
    [Show full text]
  • Trigonometric Functions
    Trigonometric Functions This worksheet covers the basic characteristics of the sine, cosine, tangent, cotangent, secant, and cosecant trigonometric functions. Sine Function: f(x) = sin (x) • Graph • Domain: all real numbers • Range: [-1 , 1] • Period = 2π • x intercepts: x = kπ , where k is an integer. • y intercepts: y = 0 • Maximum points: (π/2 + 2kπ, 1), where k is an integer. • Minimum points: (3π/2 + 2kπ, -1), where k is an integer. • Symmetry: since sin (–x) = –sin (x) then sin(x) is an odd function and its graph is symmetric with respect to the origin (0, 0). • Intervals of increase/decrease: over one period and from 0 to 2π, sin (x) is increasing on the intervals (0, π/2) and (3π/2 , 2π), and decreasing on the interval (π/2 , 3π/2). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Cosine Function: f(x) = cos (x) • Graph • Domain: all real numbers • Range: [–1 , 1] • Period = 2π • x intercepts: x = π/2 + k π , where k is an integer. • y intercepts: y = 1 • Maximum points: (2 k π , 1) , where k is an integer. • Minimum points: (π + 2 k π , –1) , where k is an integer. • Symmetry: since cos(–x) = cos(x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. • Intervals of increase/decrease: over one period and from 0 to 2π, cos (x) is decreasing on (0 , π) increasing on (π , 2π). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Tangent Function : f(x) = tan (x) • Graph • Domain: all real numbers except π/2 + k π, k is an integer.
    [Show full text]
  • Lecture 5: Complex Logarithm and Trigonometric Functions
    LECTURE 5: COMPLEX LOGARITHM AND TRIGONOMETRIC FUNCTIONS Let C∗ = C \{0}. Recall that exp : C → C∗ is surjective (onto), that is, given w ∈ C∗ with w = ρ(cos φ + i sin φ), ρ = |w|, φ = Arg w we have ez = w where z = ln ρ + iφ (ln stands for the real log) Since exponential is not injective (one one) it does not make sense to talk about the inverse of this function. However, we also know that exp : H → C∗ is bijective. So, what is the inverse of this function? Well, that is the logarithm. We start with a general definition Definition 1. For z ∈ C∗ we define log z = ln |z| + i argz. Here ln |z| stands for the real logarithm of |z|. Since argz = Argz + 2kπ, k ∈ Z it follows that log z is not well defined as a function (it is multivalued), which is something we find difficult to handle. It is time for another definition. Definition 2. For z ∈ C∗ the principal value of the logarithm is defined as Log z = ln |z| + i Argz. Thus the connection between the two definitions is Log z + 2kπ = log z for some k ∈ Z. Also note that Log : C∗ → H is well defined (now it is single valued). Remark: We have the following observations to make, (1) If z 6= 0 then eLog z = eln |z|+i Argz = z (What about Log (ez)?). (2) Suppose x is a positive real number then Log x = ln x + i Argx = ln x (for positive real numbers we do not get anything new).
    [Show full text]
  • An Appreciation of Euler's Formula
    Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Larson, Caleb (2017) "An Appreciation of Euler's Formula," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 17. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 Rose- Hulman Undergraduate Mathematics Journal an appreciation of euler's formula Caleb Larson a Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj North Dakota State University Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 an appreciation of euler's formula Caleb Larson Abstract. For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, eix = cos x + i sin x, and as a result, Euler's Identity, eiπ + 1 = 0. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it.
    [Show full text]
  • Developing Creative Thinking in Mathematics: Trigonometry
    Secondary Mathematics Developing creative thinking in mathematics: trigonometry Teacher Education through School-based Support in India www.TESS-India.edu.in http://creativecommons.org/licenses/ TESS-India (Teacher Education through School-based Support) aims to improve the classroom practices of elementary and secondary teachers in India through the provision of Open Educational Resources (OERs) to support teachers in developing student-centred, participatory approaches. The TESS-India OERs provide teachers with a companion to the school textbook. They offer activities for teachers to try out in their classrooms with their students, together with case studies showing how other teachers have taught the topic and linked resources to support teachers in developing their lesson plans and subject knowledge. TESS-India OERs have been collaboratively written by Indian and international authors to address Indian curriculum and contexts and are available for online and print use (http://www.tess-india.edu.in/). The OERs are available in several versions, appropriate for each participating Indian state and users are invited to adapt and localise the OERs further to meet local needs and contexts. TESS-India is led by The Open University UK and funded by UK aid from the UK government. Video resources Some of the activities in this unit are accompanied by the following icon: . This indicates that you will find it helpful to view the TESS-India video resources for the specified pedagogic theme. The TESS-India video resources illustrate key pedagogic techniques in a range of classroom contexts in India. We hope they will inspire you to experiment with similar practices. They are intended to complement and enhance your experience of working through the text-based units, but are not integral to them should you be unable to access them.
    [Show full text]
  • Chapter 9 Some Special Functions
    Chapter 9 Some Special Functions Up to this point we have focused on the general properties that are associated with uniform convergence of sequences and series of functions. In this chapter, most of our attention will focus on series that are formed from sequences of functions that are polynomials having one and only one zero of increasing order. In a sense, these are series of functions that are “about as good as it gets.” It would be even better if we were doing this discussion in the “Complex World” however, we will restrict ourselves mostly to power series in the reals. 9.1 Power Series Over the Reals jIn this section,k we turn to series that are generated by sequences of functions :k * ck x k0. De¿nition 9.1.1 A power series in U about the point : + U isaseriesintheform ;* n c0 cn x : n1 where : and cn, for n + M C 0 , are real constants. Remark 9.1.2 When we discuss power series, we are still interested in the differ- ent types of convergence that were discussed in the last chapter namely, point- 3wise, uniform and absolute. In this context, for example, the power series c0 * :n t U n1 cn x is said to be3pointwise convergent on a set S 3if and only if, + * :n * :n for each x0 S, the series c0 n1 cn x0 converges. If c0 n1 cn x0 369 370 CHAPTER 9. SOME SPECIAL FUNCTIONS 3 * :n is divergent, then the power series c0 n1 cn x is said to diverge at the point x0.
    [Show full text]
  • Bessel's Differential Equation Mathematical Physics
    R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R.
    [Show full text]
  • Meaning and Understanding of School Mathematical Concepts by Secondary Students: the Study of Sine and Cosine
    EURASIA Journal of Mathematics, Science and Technology Education, 2019, 15(12), em1782 ISSN:1305-8223 (online) OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/110490 Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine Enrique Martín-Fernández 1*, Juan Francisco Ruiz-Hidalgo 1, Luis Rico 1 1 Department of Mathematics Education, Faculty of Education, University of Granada, Campus Universitario Cartuja s/n, 18011, Granada, SPAIN Received 18 November 2018 ▪ Revised 5 April 2019 ▪ Accepted 12 April 2019 ABSTRACT Meaning and understanding are didactic notions appropriate to work on concept comprehension, curricular design, and knowledge assessment. This document aims to delve into the meaning of school mathematical concepts through their semantic analysis. This analysis is used to identify and establish the basic meaning of a mathematical concept and to value its understanding. To illustrate the study, we have chosen the trigonometric notions of sine and cosine of an angle. The work exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age; it collects the variety of emergent notions and elements related to the trigonometric concepts involved when answering on the categories of meaning which have been asked for. We gather the study data through a semantic questionnaire and analyze the responses using an established framework. The subjects provide a diversity of meanings, interpreted and structured by semantic categories. These meanings underline different understandings of the sine and cosine, according to the inferred themes, such as length, ratio, angle and the calculation of a magnitude.
    [Show full text]
  • The Standard Form of a Differential Equation
    Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process.
    [Show full text]
  • Special Functions
    Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering.
    [Show full text]