Mathematics and Statistics

Contents

1         What is programming, and why study it? 11

1.1      Programming 11

1.2      Summation 14

1.3      Examples of summation 15

1.4      A triangle with unknown angles 23

1.5      Summary 27

1.6      Challenges 27

2         Theoretical issues 29

2.1      Continuous functions 29

2.2      Differentiation 32

2.3      Rolle’s Theorem 33

2.4      The Mean Value Theorem  36

2.5      L’Hospital’s Rule 40

2.6      Summary 45

2.7      Challenges 45

3         Numbers 47

3.1      Differences in precision 49

3.2      Basic arithmetic methods with Python 51

3.3      Basic arithmetic 53

3.4      Composite or prime numbers 54

3.5      Numbers and variables 56

3.6      Fraction numbers 57

3.7      Percentage 60

3.8      The commands div and mod 61

3.9      The square root 62

3.10     The number π (pi) 64

3.11     Numbers in other bases 65

3.12     Converting measurements  68

3.13     Complex numbers  70

3.14     Second degree equations 74

3.15     Completing the square 77

3.16     Vector arithmetic 80

3.17     Summary 83

3.18     Challenges 84

4         Visualizing data and functions 85

4.1      Points and data sets 86

4.2      Relation and change: Apples 89

4.3      Three payment models for the indoor swimming pool 92

4.4      The decrease in value of a new car 94

4.5      Throwing a ball 97

4.6      Bar charts 99

4.7      Functions 100

4.8      Linear functions 101

4.9      Randomized linear functions 103

4.10     Quadratic functions 105

4.11     Graphics of quadratic functions 106

4.12     Randomized second-degree functions 108

4.13     The axis of symmetry of a parabola 110

4.14     Logarithms  113

4.15     Exponential functions 116

4.16     The relation between the logarithm and the exponential 118

4.17     Trigonometric functions 120

4.18     Some sine and cosine characteristics 121

4.19     Degrees or radians 123

4.20     Some characteristics of the sine function 124

4.21     The cosine function 126

4.22     Selecting the accurate domain 128

4.23     The tangent function 130

4.24     Expansion of the binomial distribution 134

4.25     The logistic function 135

4.26     Introduction to regression and interpolation 137

4.27     Implicit functions 143

4.28     Summary  147

4.29     Challenges 147

5         Sets and probability 149

5.1      Sets 149

5.2      Cardinality 151

5.3      The empty set 152

5.4      The number of values in a set  153

5.5      Subsets  154

5.6      The power set  156

5.7      Subsets and numbers 158

5.8      The union  159

5.9      The intersection (or the cut)  160

5.10     Union and intersection  161

5.11     The Cartesian product 162

5.12     Probability and sets 164

5.13     A definition of probability 167

5.14     Random numbers 170

5.15     Simulate the roll of a dice 170

5.16     Heads or tails 172

5.17     Binary trees 173

5.18     The roll of two dice 176

5.19     Estimate the value of π 177

5.20     Combinations and permutations 178

5.21     Create a seven-digit number 180

5.22     Buffon’s needle problem 182

5.23     The Venn diagram 185

5.24     Statistical inference and normal distribution 189

5.25     Summary 193

5.26     Challenges 193

6         Statistics 195

6.1      The mean value 195

6.2      The median and the mode 197

6.3      Histograms 203

6.4      The pie chart 205

6.5      Variance and variance spread 210

6.6      z-values 215

6.7      Correlation 219

6.8      Correlation and causality 225

6.9      Anscombe’s quartet 231

6.10     Summary  234

6.11     Challenges 234

7         Mathematical modelling 235

7.1      Linear regression and long jump results 236

7.2      Non-linear regression 239

7.3      Linear regression analysis 242

7.4      Non-linear regression analysis 244

7.5      The mathematical modelling process 245

7.6      Number patterns and polynomials 253

7.7      Number patterns again  255

7.8      From geometry to algebra  257

7.9      Summary  259

7.10     Challenges 259

8         Geometry, trigonometry and fractals 261

8.1      Geometry in GeoGebra, Python and WolframAlpha 261

8.2      Area and circumference of a circle 264

8.3      Area and circumference for a rectangle 265

8.4      A triangle 267

8.5      The ellipse 271

8.6      The constant π  279

8.7      Pythagorean triple 282

8.8      A surface area 284

8.9      A cube 284

8.10     Problem solving and geometry 286

8.11     Basic trigonometry 287

8.12     Area theorems 293

8.13     Heron’s formula 294

8.14     The radian 294

8.15     Chaos theory 296

8.16     Fractals  299

8.17     The Mandelbrot set 304

8.18     The Julia set 306

8.19     Summary  307

8.20     Challenges 308

9         Limits and Optimization 309

9.1      Mathematical definitions of limits 309

9.2      Limits and functions 311

9.3      Derivatives 314

9.4      Derivatives of trigonometrical functions 317

9.5      The derivative of a trigonometric product 319

9.6      Optimization 320

9.7      Summary  330

9.8      Challenges 331

10      Integral calculus 333

10.1     Introduction  334

10.2     A definite integral 335

10.3     Definite trigonometric integrals 336

10.4     The density function 337

10.5     Area between two curves 339

10.6     Area of a special domain 344

10.7     Area between three conditions 347

10.8     Area between trigonometric curves 348

10.9     Parametric form for curves 349

10.10   The folium of Descartes 351

10.11   The torus 353

10.12   The double folium 354

10.13   Fermat’s spiral 355

10.14   The lemniscate of Bernoulli 357

10.15   The Lissajous figure 359

10.16   The solid of revolution 361

10.17   Complex numbers and differential equations  365

10.18   Taylor series 369

10.19   Ordinary differential equations (ODEs) of the first order 371

10.20   The SIR model 375

10.21   Summary  376

10.22   Challenges 377

Index 379