DOCSLIB.ORG

Chapter 3 the Derivative

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 3 the Derivative Chapter 3 The Derivative In developing the S-I-R model in chapter 1 we took the idea of the rate of change of a population as intuitively clear. The rate at which one quantity changes with respect to another is a central concept of calculus and leads to a broad range of insights. The chief purpose of this chapter is to develop a fuller understanding—both analytic and geometric—of the connection between a function and its rate of change. To do this we will introduce the concept of the derivative of a function. 3.1 Rates of Change The Changing Time of Sunrise ◦ The sun rises at different times, depending on the date and location. At 40 The time of sunrise N latitude (New York, Beijing, and Madrid are about at this latitude) in the is a function year 1990, for instance, the sun rose at 7:16 on January 23, 5:58 on March 24, 4:52 on July 25. Clearly the time of sunrise is a function of the date. If we represent the time of sunrise by T (in hours and minutes) and the date by d (the day of the year), we can express this functional relation in the form T = T (d). For example, from the table above we find T (23) = 7:16. It is not obvious from the table, but it is also true that the the rate at which the time of sunrise is changing is different at different times of the year—T ′ varies as d varies. 101 Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 102 CHAPTER 3. THE DERIVATIVE We can see how the rate varies by looking at some further data for sunrise at the same latitude, taken from The Nautical Almanac for the Year 1990 : date time date time date time January 20 7:18 March 21 6:02 July 22 4:49 23 7:16 24 5:58 25 4:52 26 7:14 27 5:53 28 4:54 Let’s use this table to estimate the rate at which the time of sunrise is Calculate the rate changing on January 23. We’ll use the times three days earlier and three using earlier and days later, and compare them. On January 26 the sun rose 4 minutes earlier later dates than on January 20. This is a change of 4 minutes in 6 days, so the rate of change is − minutes 4 .67 minutes per day. − 6 days ≈ − We say this is the rate at which sunrise is changing on January 23, and we write minutes T ′(23) .67 . ≈ − day The rate is negative because the time of sunrise is decreasing—the sun is rising earlier each day. Similarly, we find that around March 24 the time of sunrise is changing approximately 9/6 = 1.5 minutes per day, and around July 25 the rate − − is 5/6 +.8 minutes per day. The last value is positive, since the time of sunrise≈ is increasing—the sun is rising later each day in July. Since March 24 is the 83rd day of the year and July 25 is the 206th, using our notation for rate of change we can write minutes minutes T ′(83) 1.5 ; T ′(206) .8 . ≈ − day ≈ day Notice that, in each case, we have calculated the rate on a given day by using times shortly before and shortly after that day. We will continue this pattern wherever possible. In particular, you should follow it when you do the exercises about a falling object, at the end of the section. Once we have the rates, we can estimate the time of sunrise for dates not given in the table. For instance, January 28 is five days after January 23, so the total change in the time of sunrise from January 23 to January 28 should be approximately minutes ∆T .67 5 days = 3.35 minutes. ≈ − day × − Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 3.1. RATES OF CHANGE 103 In whole numbers, then, the sun rose 3 minutes earlier on January 28 than on January 23. Since sunrise was at 7:16 on the 23rd, it was at 7:13 on the 28th. By letting the change in the number of days be negative, we can use this same reasoning to tell us the time of sunrise on days shortly before the given dates. For example, March 18 is 6 days away from March 24, so the change − in the time of sunrise should be minutes ∆T 1.5 6 days = +9 minutes. ≈ − day × − Therefore, we can estimate that sunrise occurred at 5:58 + 0:09 = 6:07 on March 18. Changing Rates Suppose instead of using the tabulated values for March we tried to use our January data to predict the time of sunrise in March. Now March 24 is 60 days after January 23, so the change in the time of sunrise should be approximately minutes ∆T .67 60 days = 40.2 minutes, ≈ − day × − and we would conclude that sunrise on March 24 should be at about 7:16 0:40 = 6:37, which is more than half an hour later than the actual time! − This is a problem we met often in estimating future values in the S-I-R model. implicitly assume that the time of sunrise changes at the fixed rate Predictions over of .67 minutes per day over the entire 60-day time-span. But this turns out long time spans − not to be true: the rate actually varies, and the variation is too great for us are less reliable to get a useful estimate. Only with a much smaller time-span does the rate not vary too much. Here is the same lesson in another context. Suppose you are travelling in a car along a busy road at rush hour and notice that you are going 50 miles per hour. You would be fairly confident that in the next 30 seconds (1/120 of an hour) you will travel about miles 1 5 ∆ position 50 hour = mile = 2200 feet. ≈ hour × 120 12 The actual value ought to be within 50 feet of this, making the estimate accurate to within about 2% or 3%. On the other hand, if you wanted to Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 104 CHAPTER 3. THE DERIVATIVE estimate how far you would go in the next 30 minutes, your speed would probably fluctuate too much for the calculation miles 1 ∆ position 50 hours = 25 miles ≈ hour × 2 to have the same level of reliability. Other Rates, Other Units In the S-I-R model the rates we analyzed were population growth rates. They told us how the three populations changed over time, in units of persons per day. If we were studying the growth of a colony of mold, measuring its size by its weight (in grams), we could describe its population growth rate in units of grams per hour. In discussing the motion of an automobile, the rate we consider is the velocity (in miles per hour), which tells us how the distance from some starting point changes over time. We also pay attention to the rate at which velocity changes over time. This is called acceleration, and can be measured in miles per hour per hour. While many rates do involve changes with respect to time, other rates do not. Two examples are the survival rate for a disease (survivors per thousand infected persons) and the dose rate for a medicine (milligrams per Examples of rates pound of body weight). Other common rates are the annual birth rate and the annual death rate, which might have values like 19.3 live births per 1,000 population and 12.4 deaths per 1,000 population. Any quantity expressed as a percentage, such as an interest rate or an unemployment rate, is a rate of a similar sort. An unemployment rate of 5%, for instance, means 5 unemployed workers per 100 workers. There are many other examples of rates in the economic world that make use of a variety of units—exchange rates (e.g., francs per dollar), marginal return (e.g., dollars of profit per dollar of change in price). Sometimes we even want to know the rate of change of one rate with The rate of change respect to another rate. For example, automobile fuel economy (in miles per of a rate gallon—the first rate) changes with speed (in miles per hour—the second rate), and we can measure the rate of change of fuel economy with speed. Take a car that goes 22 miles per gallon of fuel at 50 miles per hour, but only 19 miles per gallon at 60 miles per hour. Then its fuel economy is changing Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 3.1. RATES OF CHANGE 105 approximately at the rate ∆ fuel economy 19 22 miles per gallon = − ∆ speed 60 50 miles per hour − = .3 miles per gallon per mile per hour. − Exercises A falling object. These questions deal with an object that is held motion- less 10,000 feet above the surface of the ocean and then dropped. Start a clock ticking the moment it is dropped, and let D be the number of feet it has fallen after the clock has run t seconds. The following table shows some of the values of t and D.
Load more

Recommended publications
  • The Linear Algebra Version of the Chain Rule 1
    Ralph M Kaufmann The Linear Algebra Version of the Chain Rule 1 Idea The differential of a differentiable function at a point gives a good linear approximation of the function – by definition. This means that locally one can just regard linear functions. The algebra of linear functions is best described in terms of linear algebra, i.e. vectors and matrices. Now, in terms of matrices the concatenation of linear functions is the matrix product. Putting these observations together gives the formulation of the chain rule as the Theorem that the linearization of the concatenations of two functions at a point is given by the concatenation of the respective linearizations. Or in other words that matrix describing the linearization of the concatenation is the product of the two matrices describing the linearizations of the two functions. 1. Linear Maps Let V n be the space of n–dimensional vectors. 1.1. Definition. A linear map F : V n → V m is a rule that associates to each n–dimensional vector ~x = hx1, . xni an m–dimensional vector F (~x) = ~y = hy1, . , yni = hf1(~x),..., (fm(~x))i in such a way that: 1) For c ∈ R : F (c~x) = cF (~x) 2) For any two n–dimensional vectors ~x and ~x0: F (~x + ~x0) = F (~x) + F (~x0) If m = 1 such a map is called a linear function. Note that the component functions f1, . , fm are all linear functions. 1.2. Examples. 1) m=1, n=3: all linear functions are of the form y = ax1 + bx2 + cx3 for some a, b, c ∈ R.
    [Show full text]
  • Characterization of Non-Differentiable Points in a Function by Fractional Derivative of Jumarrie Type
    Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type Uttam Ghosh (1), Srijan Sengupta(2), Susmita Sarkar (2), Shantanu Das (3) (1): Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India; Email: [email protected] (2):Department of Applied Mathematics, Calcutta University, Kolkata, India Email: [email protected] (3)Scientist H+, RCSDS, BARC Mumbai India Senior Research Professor, Dept. of Physics, Jadavpur University Kolkata Adjunct Professor. DIAT-Pune Ex-UGC Visiting Fellow Dept. of Applied Mathematics, Calcutta University, Kolkata India Email (3): [email protected] The Birth of fractional calculus from the question raised in the year 1695 by Marquis de L'Hopital to Gottfried Wilhelm Leibniz, which sought the meaning of Leibniz's notation for the derivative of order N when N = 1/2. Leibnitz responses it is an apparent paradox from which one day useful consequences will be drown. Abstract There are many functions which are continuous everywhere but not differentiable at some points, like in physical systems of ECG, EEG plots, and cracks pattern and for several other phenomena. Using classical calculus those functions cannot be characterized-especially at the non- differentiable points. To characterize those functions the concept of Fractional Derivative is used. From the analysis it is established that though those functions are unreachable at the non- differentiable points, in classical sense but can be characterized using Fractional derivative. In this paper we demonstrate use of modified Riemann-Liouvelli derivative by Jumarrie to calculate the fractional derivatives of the non-differentiable points of a function, which may be one step to characterize and distinguish and compare several non-differentiable points in a system or across the systems.
    [Show full text]
  • 1 Think About a Linear Function and Its First and Second Derivative The
    Think about a linear function and its first and second derivative The second derivative of a linear function always equals 0 Anytime you are trying to find the limit of a piecewise function, find the limit approaching from the left and right ­ if they are equal then the limit exists Remember you can't differentiate at a vertical tangent line Differentiabilty implies continuity ­ the graph isn't continuous at x = 0 1 To find the velocity, take the first derivative of the position function. To find where the velocity equals zero, set the first derivative equal to zero 2 From the graph, we know that f (1) = 0; Since the graph is increasing, we know that f '(1) is positive; since the graph is concave down, we know that f ''(1) is negative Possible points of inflection ­ need to set up a table to see where concavity changes (that verifies points of inflection) (­∞, ­1) (­1, 0) (0, 2) (2, ∞) + ­ + + concave up concave down concave up concave up Since the concavity changes when x = ­1 and when x = 0, they are points of inflection. x = 2 is not a point of inflection 3 We need to find the derivative, find the critical numbers and use them in the first derivative test to find where f (x) is increasing and decreasing (­∞, 0) (0, ∞) ­ + Decreasing Increasing 4 This is the graph of y = cos x, shifted right (A) and (C) are the only graphs of sin x, and (A) is the graph of ­sin x, which reflects across the x­axis 5 Take the derivative of the velocity function to find the acceleration function There are no critical numbers, because v '(t) doesn't ever equal zero (you can verify this by graphing v '(t) ­ there are no x­intercepts).
    [Show full text]
  • Algebra Vocabulary List (Definitions for Middle School Teachers)
    Algebra Vocabulary List (Definitions for Middle School Teachers) A Absolute Value Function – The absolute value of a real number x, x is ⎧ xifx≥ 0 x = ⎨ ⎩−<xifx 0 http://www.math.tamu.edu/~stecher/171/F02/absoluteValueFunction.pdf Algebra Lab Gear – a set of manipulatives that are designed to represent polynomial expressions. The set includes representations for positive/negative 1, 5, 25, x, 5x, y, 5y, xy, x2, y2, x3, y3, x2y, xy2. The manipulatives can be used to model addition, subtraction, multiplication, division, and factoring of polynomials. They can also be used to model how to solve linear equations. o For more info: http://www.stlcc.cc.mo.us/mcdocs/dept/math/homl/manip.htm http://www.awl.ca/school/math/mr/alg/ss/series/algsrtxt.html http://www.picciotto.org/math-ed/manipulatives/lab-gear.html Algebra Tiles – a set of manipulatives that are designed for modeling algebraic expressions visually. Each tile is a geometric model of a term. The set includes representations for positive/negative 1, x, and x2. The manipulatives can be used to model addition, subtraction, multiplication, division, and factoring of polynomials. They can also be used to model how to solve linear equations. o For more info: http://math.buffalostate.edu/~it/Materials/MatLinks/tilelinks.html http://plato.acadiau.ca/courses/educ/reid/Virtual-manipulatives/tiles/tiles.html Algebraic Expression – a number, variable, or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents and/or roots. o For more info: http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11 _simp.htm http://www.math.com/school/subject2/lessons/S2U1L1DP.html Area Under the Curve – suppose the curve y=f(x) lies above the x axis for all x in [a, b].
    [Show full text]
  • Generic Continuous Functions and Other Strange Functions in Classical Real Analysis
    Georgia State University ScholarWorks @ Georgia State University Mathematics Theses Department of Mathematics and Statistics 4-17-2008 Generic Continuous Functions and other Strange Functions in Classical Real Analysis Douglas Albert Woolley Follow this and additional works at: https://scholarworks.gsu.edu/math_theses Part of the Mathematics Commons Recommended Citation Woolley, Douglas Albert, "Generic Continuous Functions and other Strange Functions in Classical Real Analysis." Thesis, Georgia State University, 2008. https://scholarworks.gsu.edu/math_theses/44 This Thesis is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Theses by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A. Woolley Under the direction of Mihaly Bakonyi ABSTRACT In this paper we examine continuous functions which on the surface seem to defy well-known mathematical principles. Before describing these functions, we introduce the Baire Cate- gory theorem and the Cantor set, which are critical in describing some of the functions and counterexamples. We then describe generic continuous functions, which are nowhere differ- entiable and monotone on no interval, and we include an example of such a function. We then construct a more conceptually challenging function, one which is everywhere differen- tiable but monotone on no interval. We also examine the Cantor function, a nonconstant continuous function with a zero derivative almost everywhere. The final section deals with products of derivatives. INDEX WORDS: Baire, Cantor, Generic Functions, Nowhere Differentiable GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A.
    [Show full text]
  • Chapter 3. Linearization and Gradient Equation Fx(X, Y) = Fxx(X, Y)
    Oliver Knill, Harvard Summer School, 2010 An equation for an unknown function f(x, y) which involves partial derivatives with respect to at least two variables is called a partial differential equation. If only the derivative with respect to one variable appears, it is called an ordinary differential equation. Examples of partial differential equations are the wave equation fxx(x, y) = fyy(x, y) and the heat Chapter 3. Linearization and Gradient equation fx(x, y) = fxx(x, y). An other example is the Laplace equation fxx + fyy = 0 or the advection equation ft = fx. Paul Dirac once said: ”A great deal of my work is just playing with equations and see- Section 3.1: Partial derivatives and partial differential equations ing what they give. I don’t suppose that applies so much to other physicists; I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful ∂ mathematical relations If f(x, y) is a function of two variables, then ∂x f(x, y) is defined as the derivative of the function which maybe don’t have any physical meaning at all. Sometimes g(x) = f(x, y), where y is considered a constant. It is called partial derivative of f with they do.” Dirac discovered a PDE describing the electron which is consistent both with quan- respect to x. The partial derivative with respect to y is defined similarly. tum theory and special relativity. This won him the Nobel Prize in 1933. Dirac’s equation could have two solutions, one for an electron with positive energy, and one for an electron with ∂ antiparticle One also uses the short hand notation fx(x, y)= ∂x f(x, y).
    [Show full text]
  • 1 Uniform Continuity
    Math 0450 Honors intro to analysis Spring, 2009 Notes 17 1 Uniform continuity Read rst: 5.4 Here are some examples of continuous functions, some of which we have done before. 1. A = (0; 1]; f : A R given by f (x) = 1 . ! x Proof. To prove that f is continuous at c (0; 1], suppose that " > 0, and 2 2 let = min c ; c " : If x c < , then rst of all, x > c and so 0 < 1 < 2 . 2 2 j j 2 x c Hence, n o 1 1 c x 1 1 2 1 c2" = = c x < = ": x c xc x c j j c c 2 2. f : R R given by f (x) = x2: ! Proof. Observe that x2 c2 = x c (x + c) : j j j j " If c = 0; let = p": If c = 0; let = min c ; 2 c If c = 0 and x c < , 6 j j j j j j then x < p" and x2 c2 = x2 < ", Ifnc = 0; theno j j j j j j 6 " x2 c2 = x + c x c < 2 c = ": j j j j j j 2 c j j 3. A = [1; ); f (x) = 1 . 1 x Proof. This makes use of item 1 above. Suppose that c A. From item 1, 2 2 we see that if " > 0, and x c < min c ; c " , then 1 1 < ". Hence, j j 2 2 x c 2 let = " : If x c < , and x and c aren in A, theno x c < min c ; c " ; so 2 j j j j 2 2 1 1 < ": n o x c 1 1 4.
    [Show full text]
  • Section 14.4 Tangent Planes and Linear Approximations
    Section 14.4 Tangent planes and linear approximations. Tangent planes. • Suppose a surface S has equation z = f(x, y), where f has continuous first partial derivatives, and let P (x0,y0,z0) be a point on S. • Let C1 and C2 be the curves obtained by intersecting the vertical planes y = y0 and x = x0 with the surface S. P lies on both C1 and C2. • Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P . • The tangent plane to the surface S at the point P is defined to be the plane that containts both of the tangent lines T1 and T2. An equation on the tangent plane to the surface z = f(x, y) at the point P (x0,y0,z0) is z − z0 = fx(x0,y0)(x − x0)+ fy(x0,y0)(y − y0) Example 1. Find the equation of the tangent plane to the surface z = ln(2x + y) at the point (−1, 3, 0). 1 Linear Approximations. An equation of a tangent plane to the graph of the function f of two variables at the point (a,b,f(a,b)) is z = f(a,b)+ fx(a,b)(x − a)+ fy(a,b)(y − b) The linear function L(x, y)= f(a,b)+ fx(a,b)(x − a)+ fy(a,b)(y − b) is called the linearization of f(a,b) and the approximation f(x, y) ≈ f(a,b)+ fx(a,b)(x − a)+ fy(a,b)(y − b) is called the linear approximation ot the tangent line approximation of f at (a,b).
    [Show full text]
  • Multivariate Approximation and Matrix Calculus
    Multivariate Approximation and Matrix Calculus Introduction Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. In single variable calculus, Taylor polynomial of n degrees is used to approximate an (n+1)-order differentiable function and the error of the approximation can be estimated by the (n+1)-th term of the Taylor series. By introducing vector and matrix calculus notations, we can express the same idea for multivariate functions and vector functions. Applications Because all numbers that can be represented by finite digits are rational numbers, the numerical computation of an irrational function at a particular point is almost always approximated. The first order and second order of Taylor polynomials are most frequently selected as the proper rational function to approximate irrational functions. This idea is called linear and quadratic approximation in calculus, respectively. In addition, the quadratic approximation is also used to in optimization because local maximum or minimum occurs at the critical points where the second term (first derivatives) of the Taylor polynomial is zero and the third term (second derivatives) are definitely positive or negative. In order to obtain the first or second order Taylor polynomial, we compute the coefficients of Taylor series by calculating the first and second derivatives of the original function. When we move towards the advanced mathematical applications (temperature in 4 dimensional temporal-spatial space and vector field of moving hurricane centers), we need to use multivariate (vector) functions, instead of single variable functions.
    [Show full text]
  • Convergent Factorial Series Solutions of Linear Difference Equations*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF DIFFERENTIAL EQUATIONS 29, 345-361 (1978) Convergent Factorial Series Solutions of Linear Difference Equations* W. J. FITZPATRICK+ AND L. J. GRIMM Depal-tment of Mathematics, University of Missowi-Rolla, Rolh, Missotwi 65801 Received July 27, 1977 1. IPU’TRODUCTION Although the analytic theory of linear difference equations dates back to an 1885 memoir of Poincare, continuing work essentially began around 1910 with publication of several papers of G. D. Birkhoff, R. D. Carmichael, J. Horn, and N. E. Norlund; see, for instance, [l, 2, 4, 51. Since that time, C. R. Adams, M7. J. Trjitzinsky, and others have carried on the development of the theory; numerous contributions have been made in recent years by TV. A. Harris, Jr., P. Sibuya, and H. L. Turrittin; see, for instance, [7-9, 16-171. The monographs of Norlund [14] and L. M. Milne-Thomson [13] give a comprehensive account of earlier work, and references to more recent literature are given in the surrey paper of Harris [S]. The similarity between difference equations and differential equations has been noted by many investigators; for instance, Birkhoff [3] pointed out that the analytic theory of linear difference equations provides a “methodological pattern” for the “essentially simpler but analogous” theory of linear differential equations. In this paper, we apply a projection method which has been useful in the treat- ment of analytic differential equations [6, 101 to obtain existence theorems for convergent factorial series solutions of analytic difference equations.
    [Show full text]
  • Gradients Math 131 Multivariate Calculus D Joyce, Spring 2014
    Gradients Math 131 Multivariate Calculus D Joyce, Spring 2014 @f Last time. Introduced partial derivatives like @x of scalar-valued functions Rn ! R, also called scalar fields on Rn. Total derivatives. We've seen what partial derivatives of scalar-valued functions f : Rn ! R are and what they mean geometrically. But if these are only partial derivatives, then what is the `total' derivative? The answer will be, more or less, that the partial derivatives, taken together, form the to- tal derivative. Figure 1: A Lyre of Ur First, we'll develop the concept of total derivative for a scalar-valued function f : R2 ! R, that is, a scalar field on the plane R2. We'll find that that them. In that sense, the pair of these two slopes total derivative is what we'll call the gradient of will do what we want the `slope' of the plane to be. f, denoted rf. Next, we'll slightly generalize that We'll call the vector whose coordinates are these to a scalar-valued function f : Rn ! R defined partial derivatives the gradient of f, denoted rf, on n-space Rn. Finally we'll generalize that to a or gradf. vector-valued function f : Rn ! Rm. The symbol r is a nabla, and is pronounced \del" even though it's an upside down delta. The The gradient of a function R2 ! R. Let f be word nabla is a variant of nevel or nebel, an an- a function R2 ! R. The graph of this function, cient harp or lyre. One is shown in figure 1.
    [Show full text]
  • Sections 3.2: Recurrence Relations
    Sections 3.2: Recurrence Relations February 8, 2021 Abstract Recurrence relations are defined recursively, and solutions can sometimes be given in “closed-form” (that is, without re- course to the recursive definition). We will solve one type of linear recurrence relation to give a general closed-form solution, the solution being verified by induction. We’ll be getting some practice with summation notation in this section. Have you seen it before? 1 Solving Recurrence Relations Vocabulary: • linear recurrence relation: S(n) depends linearly on previous S(r), r<n: S(n)= f1(n)S(n − 1) + ··· + fk(n)S(n − k)+ g(n) That means no powers on S(r), or any other functions operat- ing on S(r). The relation is called homogeneous if g(n) = 0. (Both Fibonacci and factorial are examples of homogeneous linear recurrence relations.) • first-order: S(n) depends only on S(n − 1), and not previous terms. (Factorial is first-order, while Fibonacci is second-order, depending on the two previous terms.) • constant coefficient: In the linear recurrence relation, when the coefficients of previous terms are constants. (Fibonacci is constant coefficient; factorial is not.) • closed-form solution: S(n) is given by a formula which is sim- ply a function of n, rather than a recursive definition of itself. (Both Fibonacci and factorial have closed-form solutions.) The author suggests an “expand, guess, verify” method for solving recurrence relations. Example: The story of T (a) Practice 1, p. 159 (from the previous section): T (1) = 1 T (n)= T (n − 1)+3, for n ≥ 2 (b) Practice 9, p.
    [Show full text]