GRAPHS of ELEMENTARY and SPECIAL FUNCTIONS

Transcription

1 GRAPHS of ELEMENTARY and SPECIAL FUNCTIONS HANDBOOK N.O. Virchenko I.I. Lyashko BEGELL HOUSE, INC. PUBLISHERS New York

2 Part I. Plots of graphs with elementary methods Chapter 1. Basic notions of numbers, variables and functions 1.1. Numbers. Variables. Functions Real numbers Constants and variables Concept of a function Methods of defining a function Classification of functions Composition of functions Inverse functions Elementary functions Single-valued and multivalued functions Bounded and unbounded functions Monotone functions Even and odd functions Periodic functions Limit of a function. Continuity of a function Limit of a number sequence Limit of a function Continuity of a function 28 Chapter 2. Analysis of functions for constructing graphs 2.1. Coordinate systems Cartesian coordinate system Polar coordinate system 32" Transformations of the Cartesian coordinate system , Analysis of functions in the Cartesian coordinate system Domain of a function Range of a function. Graph of a bounded function Even and odd functions. Properties of graphs Types of symmetry "? -,._ Graph of the inverse function 41 2:2.6. Periodicity of a function Zeroes and signs of functions Monotonicity of functions Convexity of functions Characteristic points of graph 50 III

3 Chapter 3. Graphs of principal elementary functions 3.1. Power function Exponential function Logarithmic function Trigonometric functions Inverse trigonometric functions 80 Chapter 4. Methods for plotting graphs by using arithmetic transformations. Transformations of graphs in the Cartesian coordinate system 4.1. Arithmetic transformations of graphs Addition and subtraction of graphs Multiplication and division of graphs Plotting graphs by using simplest transformations Transformations that do not change the scale Transformations that change the scale Plotting graphs of functions the analytic expression of which contains the absolute value sign 117 Chapter 5. Plotting of graphs of elementary functions 5.1. Plots of compositions of functions Plotting the graphs of functions y = (/[x)) r, where r e Q Plotting the graph of the function >> = I/fix) from the graph of y =/x) Plottingtliegraphsofthefunction - v = a^x),wherea a-l > Plottingthe graphs of the function^ = \og a j{x), where a\a- l > Plotting the graphs of the functions y = sin/(x),y = cosffx) Plotting the graphs of functions > = tan - /(x),y = cot^x) Plotting the graphs of the functions y = sacsiaj[x),mccosj{x) Plottingthe graphs of the functions >> = arctanxx),>> = arccot./(jf) Graphs of algebraic functions Graphs of integer functions Graphs of fractional rational functions Graphs of irrational functions Graphs of transcendental functions Hyperbolic functions > Inverse hyperbolic functions 235 Chapter 6. Plots of parametrically defined curves 6.1. A study of parametrically defined functions The method of eliminating the parameter/ Methods for finding points of intersection of the curve x = <p(t), y = \\i(t) and the straight line y = kx 244 IV

4 Chapter 7. Plots of graphs in polar coordinates 7.1. Functions in polar coordinates Plots of graphs in polar coordinates Plots of the curves p = a, p = asin cp, p = acos 9, p = acos (cp cp 0 ) Plots of the graphs of curves p = asin mp, p = acos mp (n e N, n > 1) Plots of the graphs of curves p = a(l ± sin cp), p = a(l ± cos 9) Plots of graphs of curves p = a(l ± sin mp), p = a(l ± cos mp) (n e N, n > 1) Plots of the graphs of curves p = atancp,p = acotcp Plots of the graphs of curves p = otan mp, p = acot mp (n e N, n > 1) Plots of the graphs of curves p = a(l ± tan 9), p = a(l ± cot 9) Plots of the graphs of curves p = a(l ± tan mp), p = a(l ± cot mp) (n e N, n > 1) Plotsofthegraphsofcurvesp = 6±asin9,p = b±acosq>(b*a) Plots of the graphs of curves p = b ± asin nq>, p = b± acos mp (/7eN,6*a,w>l) Plots of the graphs of curves p = b ±atanmp, p = b ±acotmp (6*a,neN,wkl) Plots of the graphs of curves p = a sin /19, p = acos mp(/j e N) Plots of the graphs of curves p = a sin 3 «9, p = a cos 3 n9 (/? e N) Plots of the graphs of curves p = a tan 2 «9, p = a cot 2 «9 («e N, n > 1) Plots of curves p = otan 3 mp, p = aco^mp (n e N, n> 1) 306 Plots of curves p = a sin^ 9, p = a cos^ 9, p = atan^ 9, p = a cot^ 9, p ( ^ 9 ), p = a(l ±cos^9), p = a( a p = a(l ± cot ^ 9), p = a sin" ^ 9, p = a cos" 9, p = a tan" ^ 9, p = OCot" 9(p,a,«eN) Plots of curves p = a/sin* mp, p = a/cos fc 779 («,& e N) Transformations of graphs in polar coordinates 338 Chapter 8. Plots of implicitly defined functions 8.1. Implicitly defined functions Plotting the graphs of implicitly defined functions Plotting the graphs by transforming an implicitly defined function into an explicit function Plotting the graphs by using the method of "linear relation" Plotting of graphs by using the method of "cells" Curves defined by second-degree algebraic equation 363

5 8.4. Curves defined by a third-degree polynomial Hyperbolic hyperbola Defective hyperbola Parabolic hyperbola Hyperbolisms of conic sections Trident (Hyperbola parabolism) Prolate parabola Cubic parabola Graphs of implicitly defined functions containing the absolute value sign Examples of implicitly defined functions for which it is convenient to plot the graphs in polar coordinates 412 Chapter 9. Plots of more complicated functions 9.1. Functions defined by several analytical expressions Function^ = sign/joe),.*'=./lsign*) Plots of functions defined by a recurrence relation Plots of functions defined in terms of limits Plots of functions y=lf(x)] Plots of functions y=/(m) Plots of functions.y = {/(*)} Plots of functions y =/({*}) 442 Part II. Plots of graphs with the use of calculus. Graphs of special functions Chapter 1. Derivatives and differentials. Their applications to plotting of graphs VI 1.1. Derivativeof afunctionof one variable. Properties Differential of a function of one variable Fundamental theorems of calculus Study of functions by using derivatives Condition for constancy Condition for monotonicity Maxima and minima of a function Determining extremums of a function by using the second derivative Determining extremums of a function by using Taylor's formula The maximal and minimal value of a function on an interval Convexity and concavity of a curve. Inflection points Graph plotting with the use of derivatives Plotting the graph of/' (x) and/" (x) from the graph off(x) Graphical differentiation Graphical integration The rule of Bernoulli-L'Hospital Approximations for solutions of equations 468

6 The chord method The tangent method (Newton's method) 469 Chapter 2. Graphs of all types of functions 2.1. Graphs of functions y=f(x) in Cartesian coordinates Plots of parametrically defined functions A study of parametrically defined functions by using derivatives Graphs of parametrically defined functions Graphs of implicitly defined functions Graphs of functions in polar coordinates 527 Chapter 3. Graphs of certain remarkable curves 3.1. Algebraic curves Curves of degree two The circle The ellipse The hyperbola The parabola > Curves of degree three Folium of Descartes Cissoid of Diodes Strophoid Ophiurida ; Trisectrix of Maclaurin Cubic of Tschirnhausen Versieraof Agnesi Semicubic parabola (parabola of Neil) Curves of degree four Conchoid The limacon of Pascal Cycloidal curves Transcendental curves Spiral of Archimedes Algebraic spirals Catenary : Tractrix > QuadratrixofDinostratus Cochleoid Exponential curves Cycloid Curves of Sturm Involute of a circle Pursuit curve Curves of Ribaucour 629 VII

7 Cornu's spiral Radial curves 622 Chapter 4. Graphs of special functions 4.1. Factorial function n\. Bernoulli numbers and polynomials. Euler numbers and polynomials. Binomial coefficients. Pochhammer polynomials Gamma function and related functions Gamma function T(x) Beta function B(x,y) (Euler integral of the first kind) Psi function (digamma function v /(x)) The function G(x) (Bateman function) Incomplete gamma functions Integral exponential and related functions Integral sine and cosine functions Fresnel integrals Probability integral and related functions Bessel functions Hypergeometric functions Orthogonal polynomials General properties of orthogonal polynomials Jacobi polynomials/^a>p) (x) Jacobi functions of the second kind Gegenbauer polynomials (ultra-spherical polynomials) Legendre polynomials (spherical polynomials) P n (x) Laguerre polynomials L n (x) Hermite polynomials H n (x) ' Chebyshev polynomials Legendre functions Elliptic integrals and elliptic functions Mathieu functions Zeta numbers and related functions Hurwitz function Heaviside function. Dirac's delta function 756 Bibliography 760 Index 761 VIII