Experiments In Mathematics Using Maple

We believe that to understand and be comfortable with mathematical con­ cepts and methods, it is necessary to do mathematics and, traditionally, doing meant with a pencil and paper. Now, a modest home computer can provide a platform for a computer algebra package like Maple, which can perform all of the operations encountered in secondary school mathematics and beyond, and pro­ vide graphical representations of functions, including animations. The capability of rendering accurate graphics for mathematical functions greatly enhances the learning experience, and helps intuition work in new situations, before beginning to do the algebra and calculus needed to solve a problem. For example, if you can see that a function has a clear minimum from its graph, then you are more likely to be able to identify its location precisely by analysis. This book is designed to support the interactive Maple worksheets that we have developed for the two senior secondary school years and made available via the Internet at the anonymous ftp site ftp. ut irc. utoronto . ca (/pub/ednet/maths/maple). This takes you from basic algebra, functions and sequences, to calculus and its applications. Thus, the book is a hardcopy version of the worksheets, with additional explanatory text, cross-referencing, appen­ dices containing worked solutions to all exercises, and indexing.

I Pre-Calculus Mathematics.- 1 Introduction to Maple.- 1.1 Documentation.- 1.2 First Maple Session.- 1.2.1 Basic Operations.- 1.2.2 Solving Equations.- 1.3 More Commands.- 1.3.1 Calling Function Packages.- 1.3.2 Isolating Variables.- 1.3.3 Variable Names.- 1.3.4 Syntax.- 1.3.5 Assigning Variables.- 1.3.6 Ditto.- 1.3.7 Notation of Mathematical Expressions.- 1.3.8 Commonly Used Commands.- 1.4 Plots.- 1.4.1 Options.- 1.4.2 Text.- 1.4.3 Implicit Plots.- 1.4.4 3D Plots.- 1.4.5 Animations.- 1.5 Preparing Worksheets for Printing.- 2 Functions.- 2.1 Relations and Functions.- 2.2 Multiplication by a Real Number.- 2.3 Addition and Subtraction of Functions.- 2.4 Multiplication and Division of Functions.- 2.5 Composition of Functions.- 2.6 Inverse Functions.- 3 Quadratic Functions.- 3.1 Quadratic Functions.- 3.2 Parabola.- 3.3 Vertex.- 3.4 Maximum and Minimum Values.- 3.5 Applications of Quadratic Functions.- 4 Solving Quadratic Equations.- 4.1 Roots of Quadratic Equations.- 4.2 Nature of Roots of Quadratic Equations.- 4.3 Quadratic Equations in Other Variables.- 4.4 Linear-Quadratic Systems.- 4.5 Slope of a Tangent to a Parabola.- 5 Polynomial Functions.- 5.1 Polynomial Functions.- 5.2 Division of Polynomials.- 5.3 Product and Sum of Roots.- 5.4 Related Roots.- 5.5 Roots of Higher Order Polynomials.- 6 Exponential Functions.- 6.1 Properties of Exponentials.- 6.2 Scientific Notation.- 6.3 Table of Values.- 6.4 Exponential Growth and Decay.- 7 Logarithmic Functions.- 7.1 Properties of Logarithms.- 7.3 pH Scale.- 7.4 Simple Interest.- 7.5 Compound Interest.- 7.6 Equivalent Rates.- 8 Circular Functions.- 8.1 Primary Trigonometric Functions.- 8.2 Reciprocal Trigonometric Functions.- 8.3 Inverse Circular Functions.- 8.4 Transformations.- 8.5 Addition of Circular Functions.- 8.6 Simple Harmonic Motion.- 9 Trigonometry.- 9.1 Basic Trigonometry.- 9.2 Trigonometric Identities.- 9.3 Trigonometric Equations.- 9.4 Power Series Expansions.- 9.5 Right-Angled Triangles.- 9.6 Law of Sines.- 9.7 Law of Cosines.- 9.8 Vectors.- 9.8.1 Dot Product.- 9.9 Bisector of a Triangle.- 10 Similar Figures.- 10.1 Similar Figures.- 10.1.1 Similar Triangles.- 10.2 Length of a Perpendicular.- 10.3 Areas of Similar Triangles.- 10.4 Dilatations and Similar Figures.- 11 Circles and Spheres.- 11.1 Circle.- 11.1.1 Intersection of a Line and a Circle.- 11.1.2 Tangent to a Circle.- 11.1.3 Arc Length.- 11.1.4 Area Bounded by a Circle.- 11.2 Sphere.- 11.2.1 Intersection of a Plane and a Sphere.- 11.2.2 Volume and Surface Area.- 12 Loci.- 12.1 Locus.- 12.2 Equations and Inequations of a Locus.- 12.3 Circles Associated with a Triangle.- 12.3.1 Circumcircle and Circumcentre.- 12.3.2 Inscribed Circle and Incentre.- 12.3.3 Bisectors of Interior Angles.- 12.3.4 Exscribed Circles and E-Centres.- 12.3.5 Centroid of a Triangle.- 12.3.6 Orthocentre of a Triangle.- 12.4 Equations of Loci After a Transformation.- 12.4.1 Translation.- 12.4.2 Stretch.- 12.4.3 Reflection.- 12.4.4 Dilatation.- 12.5 Simulations.- 13 Sequences and Series.- 13.1 Sequences.- 13.1.1 Arithmetic Sequences.- 13.1.2 Geometric Sequences.- 13.2 Series.- 13.2.1 Arithmetic Series.- 13.2.2 Geometric Series.- 14 Statistics and Probability.- 14.1 Organizing and Presenting Data.- 14.1.1 Plots.- 14.1.2 Mean, Median, Mode, and Standard Deviation.- 14.2 Probability of Events.- 14.2.1 Binomial Theorem.- II Beginning Calculus.- 15 Secants and Tangents.- 15.1 Slope of a Line.- 15.2 Slope of a Secant.- 15.3 Slope of a Tangent.- 15.4 Equation of the Tangent to a Curve.- 16 Sequences and Limits.- 16.1 Sequences.- 16.2 Limit of an Infinite Sequence.- 16.3 Sum of an Infinite Geometric Series.- 16.4 Limit of a Function.- 16.5 Rules for Limits.- 16.5.1 Constants Rule.- 16.5.2 Sum Rule.- 16.5.3 Product Rule.- 16.5.4 Quotient Rule.- 16.5.5 Exponential Rule.- 16.5.6 Inequality Rule.- 16.5.7 Sandwich Rule.- 16.5.8 Squeeze Rule.- 16.6 Continuous Functions.- 16.7 Limits Involving $$\frac{0}{0}$$.- 17 Derivatives of Functions.- 17.1 Derivative.- 17.2 Differentiating from First Principles.- 17.3 Rules of Differentiation.- 17.3.1 Derivative of a Constant.- 17.3.2 Power Rule.- 17.3.3 Derivative of a Sum of Functions.- 17.3.4 Chain Rule.- 17.3.5 Product Rule.- 17.3.6 Quotient Rule.- 17.4 Higher Order Derivatives.- 17.5 Notation.- 17.6 Implicit Differentiation.- 18 Functions and Graphs.- 18.1 Plotting Functions.- 18.2 X- and Y-Intercepts.- 18.3 Asymptotes.- 18.3.1 Vertical Asymptotes.- 18.3.2 Horizontal Asymptotes.- 18.3.3 Oblique Asymptotes.- 18.4 Symmetry.- 18.5 Increasing and Decreasing Functions.- 18.6 Concavity.- 18.7 Relative Maxima and Minima.- 18.8 Inflection Point.- 18.9 Plot of f(x).- 19 Rates.- 19.1 Position, Velocity, and Acceleration.- 19.2 Rate of Change.- 19.3 Related Rates of Change.- 20 Integration.- 20.1 Approximation of the Area Under a Curve.- 20.2 Definite Integral.- 20.3 Indefinite Integral.- 20.4 Fundamental Theorem of Calculus.- 20.5 Rules of Integration.- 20.5.1 Sum of Functions.- 20.5.2 Function Times a Constant.- 20.5.3 Power Rule.- 20.6 Area Between Curves.- 20.7 Differential Equations.- 20.8 Integration by Parts.- 20.9 Partial Fractions.- 20.10 Numerical Integration.- 20.10.1 Trapezoidal Rule.- 20.10.2 Simpson’s Rule.- 21 Trigonometry.- 21.1 Compound Angles.- 21.2 Graphs of Trigonometric Functions.- 21.3 Derivative of Sine Function.- 21.4 Derivatives of Trigonometric Functions.- 21.5 Maxima and Minima.- 21.6 Integrals of Trigonometric Functions.- 21.7 Areas Defined by Trigonometric Functions.- 22 Exponents and Logarithms.- 22.1 Base for Natural Logarithm.- 22.2 Exponential Growth and Decay.- 22.3 Derivatives.- 22.4 Integrals.- 23 Polar Coordinates.- 23.1 Converting Coordinates.- 23.2 Circles in Polar Coordinates.- 23.3 Area in Polar Coordinates.- 23.3.1 Sector of a Circle.- 23.3.2 Region Enclosed by a Curve.- Appendices.- A Solutions to Part I Exercises.- A.1 Functions.- A.2 Quadratics.- A.3 Solving Quadratics.- A.4 Polynomials.- A.5 Exponential Functions.- A.6 Logarithmic Functions.- A.8 Trigonometry.- A.9 Similar Figures.- A.10 Circles and Spheres.- A.11 Loci.- A.12 Sequences and Series.- A.13 Statistics and Probability.- B Solutions to Part II Exercises.- B.1 Secants and Tangents.- B.2 Sequences and Limits.- B.3 Derivatives of Functions.- B.4 Functions and Graphs.- B.5 Rates.- B.6 Integration.- B.7 Trigonometry.- B.8 Exponents and Logarithms.- B.9 Polar Coordinates.- C.1 angles.- C.2 showAntiderivative.