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Showing papers on "Inverse trigonometric functions published in 1994"


Book
01 Dec 1994
TL;DR: In this article, the authors define the concept of limit and define a formal definition of limit in the context of the calculus of geometry, which is used to describe the relationship between velocity, growth rate, and area.
Abstract: Preface To the Student To the Instructor Acknowledgments What Is Calculus? Preliminaries P1 Real Numbers and the Real Line P2 Cartesian Coordinates in the Plane P3 Graphs of Quadratic Equations P4 Functions and Their Graphs P5 Combining Functions to Make New Functions P6 Polynomials and Rational Functions P7 The Trigonometric Functions 1 Limits and Continuity 11 Examples of Velocity, Growth Rate, and Area 12 Limits of Functions 13 Limits at Infinity and Infinite Limits 14 Continuity 15 The Formal Definition of Limit Chapter Review 2 Differentiation 21 Tangent Lines and Their Slopes 22 The Derivative 23 Differentiation Rules 24 The Chain Rule 25 Derivatives of Trigonometric Functions 26 The Mean-Value Theorem 27 Using Derivatives 28 Higher-Order Derivatives 29 Implicit Differentiation 210 Antiderivatives and Initial-Value Problems 211 Velocity and Acceleration Chapter Review 3 Transcendental Functions 31 Inverse Functions 32 Exponential and Logarithmic Functions 33 The Natural Logarithm and Exponential 34 Growth and Decay 35 The Inverse Trigonometric Functions 36 Hyperbolic Functions 37 Second-Order Linear DEs with Constant Coefficients Chapter Review 4 Some Applications of Derivatives 41 Related Rates 42 Extreme Values 43 Concavity and Inflections 44 Sketching the Graph of a Function 45 Extreme-Value Problems 46 Finding Roots of Equations 47 Linear Approximations 48 Taylor Polynomials 49 Indeterminate Forms Chapter Review 5 Integration 51 Sums and Sigma Notation 52 Areas as Limits of Sums 53 The Definite Integral 54 Properties of the Definite Integral 55 The Fundamental Theorem of Calculus 56 The Method of Substitution 57 Areas of Plane Regions Chapter Review 6 Techniques of Integration 61 Integration by Parts 62 Inverse Substitutions 63 Integrals of Rational Functions 64 Integration Using Computer Algebra or Tables 65 Improper Integrals 66 The Trapezoid and Midpoint Rules 67 Simpson's Rule 68 Other Aspects of Approximate Integration Chapter Review 7 Applications of Integration 71 Volumes by Slicing Solids of Revolution 72 More Volumes by Slicing 73 Arc Length and Surface Area 74 Mass, Moments, and Centres of Mass 75 Centroids 76 Other Physical Applications 77 Applications in Business, Finance, and Ecology 78 Probability 79 First-Order Differential Equations Chapter Review 8 Conics, Parametric Curves, and Polar Curves 81 Conics 82 Parametric Curves 83 Smooth Parametric Curves and Their Slopes 84 Arc Lengths and Areas for Parametric Curves 85 Polar Coordinates and Polar Curves 86 Slopes, Areas, and Arc Lengths for Polar Curves Chapter Review 9 Sequences, Series, and Power Series 91 Sequences and Convergence 92 Infinite Series 93 Convergence Tests for Positive Series 94 Absolute and Conditional Convergence 95 Power Series 96 Taylor and Maclaurin Series 97 Applications of Taylor and Maclaurin Series 98 The Binomial Theorem and Binomial Series 99 Fourier Series Chapter Review 10 Vectors and Coordinate Geometry in 3-Space 101 Analytic Geometry in Three Dimensions 102 Vectors 103 The Cross Product in 3-Space 104 Planes and Lines 105 Quadric Surfaces 106 A Little Linear Algebra 107 Using Maple for Vector and Matrix Calculations Chapter Review 11 Vector Functions and Curves 111 Vector Functions of One Variable 112 Some Applications of Vector Differentiation 113 Curves and Parametrizations 114 Curvature, Torsion, and the Frenet Frame 115 Curvature and Torsion for General Parametrizations 116 Kepler's Laws of Planetary Motion Chapter Review 12 Partial Differentiation 121 Functions of Several Variables 122 Limits and Continuity 123 Partial Derivatives 124 Higher-Order Derivatives 125 The Chain Rule 126 Linear Approximations, Differentiability, and Differentials 127 Gradients and Directional Derivatives 128 Implicit Functions 129 Taylor Series and Approximations Chapter Review 13 Applications of Partial Derivatives 131 Extreme Values 132 Extreme Values of Functions Defined on Restricted Domains 133 Lagrange Multipliers 134 The Method of Least Squares 135 Parametric Problems 136 Newton's Method 137 Calculations with Maple Chapter Review 14 Multiple Integration 141 Double Integrals 142 Iteration of Double Integrals in Cartesian Coordinates 143 Improper Integrals and a Mean-Value Theorem 144 Double Integrals in Polar Coordinates 145 Triple Integrals 146 Change of Variables in Triple Integrals 147 Applications of Multiple Integrals Chapter Review 15 Vector Fields 151 Vector and Scalar Fields 152 Conservative Fields 153 Line Integrals 154 Line Integrals of Vector Fields 155 Surfaces and Surface Integrals 156 Oriented Surfaces and Flux Integrals Chapter Review 16 Vector Calculus 161 Gradient, Divergence, and Curl 162 Some Identities Involving Grad, Div, and Curl 163 Green's Theorem in the Plane 164 The Divergence Theorem in 3-Space 165 Stokes's Theorem 166 Some Physical Applications of Vector Calculus 167 Orthogonal Curvilinear Coordinates Chapter Review 17 Ordinary Differential Equations 171 Classifying Differential Equations 172 Solving First-Order Equations 173 Existence, Uniqueness, and Numerical Methods 174 Differential Equations of Second Order 175 Linear Differential Equations with Constant Coefficients 176 Nonhomogeneous Linear Equations 177 Series Solutions of Differential Equations Chapter Review Appendix I Complex Numbers Appendix II Complex Functions Appendix III Continuous Functions Appendix IV The Riemann Appendix V Doing Calculus with Maple Answers to Odd-Numbered Exercises Index

252 citations


Journal ArticleDOI
TL;DR: A new algorithm for computing the complex logarithm and exponential functions is proposed, based on shift-and-add elementary steps, and it generalizes some algorithms by Briggs and De Lugish (1970), as well as the CORDIC algorithm.
Abstract: A new algorithm for computing the complex logarithm and exponential functions is proposed. This algorithm is based on shift-and-add elementary steps, and it generalizes some algorithms by Briggs and De Lugish (1970), as well as the CORDIC algorithm. It can easily be used to compute the classical real elementary functions (sin, cos, arctan, ln, exp). This algorithm is more suitable for computations in a redundant number system than the CORDIC algorithm, since there is no scaling factor when computing trigonometric functions. >

37 citations


Book
01 Jan 1994
TL;DR: In this paper, the authors present an algebraic approach for solving problems with trigonometric functions, including linear and quadratic functions with modeling and power functions with modeling.
Abstract: Chapter P Prerequisites P.1 Real Numbers P.2 Cartesian Coordinate System P.3 Linear Equations and Inequalities P.4 Lines in the Plane P.5 Solving Equations Graphically, Numerically, and Algebraically P.6 Complex Numbers P. 7 Solving Inequalities Algebraically and Graphically Chapter 1 Functions and Graphs 1.1 Modeling and Equation Solving 1.2 Functions and Their Properties 1.3 Twelve Basic Functions 1.4 Building Functions from Functions 1.5 Parametric Relations and Inverses 1.6 Graphical Transformations 1.7 Modeling With Functions Chapter 2 Polynomial, Power, and Rational Functions 2.1 Linear and Quadratic Functions with Modeling 2.2 Power Functions with Modeling 2.3 Polynomial Functions of Higher Degree with Modeling 2.4 Real Zeros of Polynomial Functions 2.5 Complex Zeros and the Fundamental Theorem of Algebra 2.6 Graphs of Rational Functions 2.7 Solving Equations in One Variable 2.8 Solving Inequalities in One Variable Chapter 3 Exponential, Logistic, and Logarithmic Functions 3.1 Exponential and Logistic Functions 3.2 Exponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic Functions 3.5 Equation Solving and Modeling 3.6 Mathematics of Finance Chapter 4 Trigonometric Functions 4.1 Angles and Their Measures 4.2 Trigonometric Functions of Acute Angles 4.3 Trigonometry Extended: The Circular Functions 4.4 Graphs of Sine and Cosine: Sinusoids 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 4.6 Graphs of Composite Trigonometric Functions 4.7 Inverse Trigonometric Functions 4.8 Solving Problems with Trigonometry Chapter 5 Analytic Trigonometry 5.1 Fundamental Identities 5.2 Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines Chapter 6 Applications of Trigonometry 6.1 Vectors in the Plane 6.2 Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5 Graphs of Polar Equations 6.6 De Moivre's Theorem and nth Roots Chapter 7 Systems and Matrices 7.1 Solving Systems of Two Equations 7.2 Matrix Algebra 7.3 Multivariate Linear Systems and Row Operations 7.4 Partial Fractions 7.5 Systems of Inequalities in Two Variables Chapter 8 Analytic Geometry in Two and Three Dimensions 8.1 Conic Sections and Parabolas 8.2 Ellipses 8.3 Hyperbolas 8.4 Translation and Rotation of Axes 8.5 Polar Equations of Conics 8.6 Three-Dimensional Cartesian Coordinate System Chapter 9 Discrete Mathematics 9.1 Basic Combinatorics 9.2 The Binomial Theorem 9.3 Probability 9.4 Sequences 9.5 Series 9.6 Mathematical Induction 9.7 Statistics and Data (Graphical) 9.8 Statistics and Data (Algebraic) Chapter 10 An Introduction to Calculus: Limits, Derivatives, and Integrals 10.1 Limits and Motion: The Tangent Problem 10.2 Limits and Motion: The Area Problem 10.3 More on Limits 10.4 Numerical Derivatives and Integrals Appendix A Algebra Review Appendix B Key Formulas Appendix C Logic

34 citations


Proceedings ArticleDOI
19 Apr 1994
TL;DR: A 24 bit VLSI cell, capable of computing the (1) square root, (2) reciprocal, (3) sine/cosine, and (4) arctangent functions, is presented for single precision floating-point applications.
Abstract: This paper discusses an efficient interpolation method for nonlinear function generation. Based on this, a 24 bit VLSI cell, capable of computing the (1) square root, (2) reciprocal, (3) sine/cosine, and (4) arctangent functions, is presented for single precision floating-point applications. A 53 bit version, suitable for double precision computations, is also presented. Finally, an extension of the method to the two dimensional nonlinear functions is briefly addressed. >

23 citations



Proceedings Article
05 Oct 1994
TL;DR: A discovery system for trigonometric functions (DST) which has abilities to acquire new knowledge in the form of theorems and formulas in a plane geometry domain, including the Pythagorean theorem is described.
Abstract: This paper describes a discovery system for trigonometric functions (DST), which has abilities to acquire new knowledge in the form of theorems and formulas in a plane geometry domain. The system is composed of two subsystems: a plane geometry analysis system and a mathematical formula transformation system. The former changes the length and angles of a figure and extracts geometric relations, and the latter transforms the relations to acquire useful formulas. With little basic knowledge such as the definition of the congruence of triangles and the definition of fundamental trigonometric functions, our system has rediscovered many trigonometric formulas and geometric theorems, including the Pythagorean theorem.

8 citations


01 Jan 1994
TL;DR: The techniques used to bound the range of the arcsine, arccosine, Aragon, arctangent, arCCotangent and hyperbolic sine functions in the portable FORTRAN‐77 library INTLIB are presented.
Abstract: We present the techniques we have used to bound the range of the arcsine, arccosine, arctangent, arccotangent, and hyperbolic sine functions in our portable FORTRAN‐77 library INTLIB. The design of this library is based on a balance of simplicity and eciency, subject to rigor and portability.

4 citations



Patent
11 May 1994
TL;DR: In this article, a rotatable long indicating needle is connected with the center of a circle on the back of a base board; the other end is folded towards the front face at the edge of the base board, and is connected in a through groove on a vertical rod by screws at a unit circle; the graduated vertical rod is fixedly connected with a trapezoidal block, bisected into two equal parts, taking the fixing point as the boundary.
Abstract: The utility model relates to mathematical teaching aid, which is an ideal teaching (learning) aid and can quickly and accurately find out the values of the sine, the cosine, the tangent and the cotangent of the direct trigonometric function and inverse trigonometric function the utility model is characterized in that a rotatable long indicating needle (2) is connected with the center of a circle on the back of a base board; the other end of the indicating needle is folded towards the front face at the edge of the base board, and is connected in a through groove (3) on a vertical rod by screws at a unit circle; the graduated vertical rod is fixedly connected with a trapezoidal block (4); the vertical rod is bisected into two equal parts, taking the fixing point as the boundary; the trapezoidal block is movably connected with a groove (5) positioned on the X coordinate axis on the base board.

2 citations


Book ChapterDOI
01 Jan 1994
TL;DR: In this article, the magnitude of an angle α is given by the length l of the arc, intercepted by the arms of the angle α on the unit circle with centre at the vertex of the angles.
Abstract: If theoretical problems are under consideration, angles are not measured in degrees, but in radians (circular measure): The magnitude of an angle α is given by the length l of the arc, intercepted by the arms of the angle α on the unit circle with centre at the vertex of the angle (Fig. 2.1). We shall denote the magnitude of the angle α in circular measure again by α; sometimes, instead of α, the notation arc α° is employed, α° denoting the magnitude of the angle α expressed in degrees (in the sexagesimal system).

1 citations


Journal ArticleDOI
TL;DR: This article will show how a simple adjustment can significantly reduce execution times of the APL functions when calculating samples of the maximum possible order for linear combinations and products of functions.
Abstract: In a recent article [2], it was shown that function samples for all functions composed of the elementary functions can be computed using two operators, By and Times, which deliver samples of the maximum possible order for linear combinations and products of functions. The article highlighted the simplicity of the exposition, but made no attempt to modify algorithms to improve the speed of calculations. In particular, it was noted that the APL functions took around three seconds to obtain the values of the first ten derivatives of the test function (x csc x)/ log arctan exp x while Mathematica 2.0 obtained the same results in about a second. The APL functions, however, were also able to obtain these derivatives at any number of points, and the calculation for 100 points took approximately 18 seconds. This article will show how a simple adjustment can significantly reduce these execution times.