Mathematics: Analysis and Approaches HL

Product Code: 9781925489590

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This book has been written for the IB Diploma Programme course Mathematics: Analysis and Approaches HL, for first assessment in May 2021.

This book is designed to complete the course in conjunction with the Mathematics: Core Topics HL textbook. It is expected that students will start using this book approximately 6-7 months into the two-year course, upon the completion of the Mathematics: Core Topics HL textbook.

This product has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate, Baccalaureát International, Bachillerato Internacional and IB are registered trademarks owned by the International Baccalaureate Organization.

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Product Code: 9781925489590

Brand: Haese Mathematics

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1	FURTHER TRIGONOMETRY		17
	A	Reciprocal trigonometric functions	18
	B	Inverse trigonometric functions	20
	C	Algebra with trigonometric functions	23
	D	Double angle identities	27
	E	Compound angle identities	31
		Review set 1A	38
		Review set 1B	39

2	EXPONENTIAL FUNCTIONS		41
	A	Rational exponents	42
	B	Algebraic expansion and factorisation	44
	C	Exponential equations	47
	D	Exponential functions	49
	E	Growth and decay	54
	F	The natural exponential	60
		Review set 2A	64
		Review set 2B	65

3	LOGARITHMS		67
	A	Logarithms in base $10$	68
	B	Logarithms in base $𝑎$	71
	C	Laws of logarithms	73
	D	Natural logarithms	76
	E	Logarithmic equations	80
	F	The change of base rule	81
	G	Solving exponential equations using logarithms	82
	H	Logarithmic functions	87
		Review set 3A	92
		Review set 3B	94

4	INTRODUCTION TO COMPLEX NUMBERS		97
	A	Complex numbers	99
	B	The sum of two squares factorisation	101
	C	Operations with complex numbers	102
	D	Equality of complex numbers	104
	E	Properties of complex conjugates	106
		Review set 4A	107
		Review set 4B	108

5	REAL POLYNOMIALS		109
	A	Polynomials	110
	B	Operations with polynomials	111
	C	Zeros, roots, and factors	114
	D	Polynomial equality	117
	E	Polynomial division	120
	F	The Remainder theorem	124
	G	The Factor theorem	127
	H	The Fundamental Theorem of Algebra	128
	I	Sum and product of roots theorem	131
	J	Graphing cubic functions	133
	K	Graphing quartic functions	139
	L	Polynomial equations	143
	M	Cubic inequalities	145
		Review set 5A	146
		Review set 5B	148

6	FURTHER FUNCTIONS		151
	A	Even and odd functions	152
	B	The graph of $𝑦 = [𝑓 (𝑥)]^{2}$	154
	C	Absolute value functions	156
	D	Rational functions	164
	E	Partial fractions	169
		Review set 6A	171
		Review set 6B	172

7	COUNTING		175
	A	The product principle	176
	B	The sum principle	178
	C	Factorial notation	179
	D	Permutations	181
	E	Combinations	186
		Review set 7A	190
		Review set 7B	191

8	THE BINOMIAL THEOREM		193
	A	Binomial expansions	194
	B	The binomial theorem for $𝑛 \in 𝑍^{+}$	198
	C	The binomial theorem for $𝑛 \in 𝑄$	202
		Review set 8A	206
		Review set 8B	207

9	REASONING AND PROOF		209
	A	Logical connectives	212
	B	Proof by deduction	213
	C	Proof by equivalence	217
	D	Definitions	219
	E	Proof by exhaustion	222
	F	Disproof by counter example	223
	G	Proof by contrapositive	225
	H	Proof by contradiction: reductio ad absurdum	227
		Review set 9A	230
		Review set 9B	231

10	PROOF BY MATHEMATICAL INDUCTION		233
	A	The process of induction	234
	B	The principle of mathematical induction	237
		Review set 10A	251
		Review set 10B	252

11	LINEAR ALGEBRA		253
	A	Systems of linear equations	255
	B	Row operations	257
	C	Solving $2 \times 2$ systems of linear equations	259
	D	Solving $3 \times 3$ systems of linear equations	261
		Review set 11A	266
		Review set 11B	267

12	VECTORS		269
	A	Vectors and scalars	270
	B	Geometric operations with vectors	273
	C	Vectors in the plane	279
	D	The magnitude of a vector	281
	E	Operations with plane vectors	282
	F	Vectors in space	285
	G	Operations with vectors in space	287
	H	Vector algebra	289
	I	The vector between two points	290
	J	Parallelism	296
	K	The scalar product of two vectors	299
	L	The angle between two vectors	301
	M	Proof using vector geometry	307
	N	The vector product of two vectors	309
		Review set 12A	318
		Review set 12B	320

13	VECTOR APPLICATIONS		323
	A	Lines in $2$ and $3$ dimensions	324
	B	The angle between two lines	328
	C	Constant velocity problems	330
	D	The shortest distance from a point to a line	333
	E	Intersecting lines	336
	F	Relationships between lines	338
	G	Planes	345
	H	Angles in space	353
	I	Intersecting planes	355
		Review set 13A	360
		Review set 13B	363

14	COMPLEX NUMBERS		367
	A	The complex plane	368
	B	Modulus and argument	371
	C	Geometry in the complex plane	375
	D	Polar form	379
	E	Euler's form	386
	F	De Moivre's theorem	388
	G	Roots of complex numbers	392
		Review set 14A	395
		Review set 14B	396

15	LIMITS		399
	A	Limits	401
	B	The existence of limits	404
	C	Limits at infinity	406
	D	Trigonometric limits	409
	E	Continuity	410
		Review set 15A	413
		Review set 15B	413

16	INTRODUCTION TO DIFFERENTIAL CALCULUS		415
	A	Rates of change	417
	B	Instantaneous rates of change	420
	C	The gradient of a tangent	423
	D	The derivative function	424
	E	Differentiation from first principles	426
	F	Differentiability and continuity	430
		Review set 16A	432
		Review set 16B	433

17	RULES OF DIFFERENTIATION		435
	A	Simple rules of differentiation	436
	B	The chain rule	441
	C	The product rule	444
	D	The quotient rule	446
	E	Derivatives of exponential functions	449
	F	Derivatives of logarithmic functions	454
	G	Derivatives of trigonometric functions	457
	H	Derivatives of inverse trigonometric functions	461
	I	Second and higher derivatives	463
	J	Implicit differentiation	466
		Review set 17A	469
		Review set 17B	471

18	PROPERTIES OF CURVES		475
	A	Tangents	476
	B	Normals	483
	C	Increasing and decreasing	485
	D	Stationary points	489
	E	Shape	494
	F	Inflection points	497
	G	Understanding functions and their derivatives	502
	H	L'Hôpital's rule	504
		Review set 18A	508
		Review set 18B	512

19	APPLICATIONS OF DIFFERENTIATION		517
	A	Rates of change	518
	B	Optimisation	524
	C	Related rates	533
		Review set 19A	538
		Review set 19B	540

20	INTRODUCTION TO INTEGRATION		543
	A	Approximating the area under a curve	544
	B	The Riemann integral	547
	C	Antidifferentiation	551
	D	The Fundamental Theorem of Calculus	553
		Review set 20A	558
		Review set 20B	559

21	TECHNIQUES FOR INTEGRATION		561
	A	Discovering integrals	562
	B	Rules for integration	565
	C	Particular values	570
	D	Integrating $𝑓 (𝑎 𝑥 + 𝑏)$	571
	E	Partial fractions	576
	F	Integration by substitution	577
	G	Integration by parts	583
		Review set 21A	585
		Review set 21B	587

22	DEFINITE INTEGRALS		589
	A	Definite integrals	590
	B	Definite integrals involving substitution	594
	C	The area under a curve	596
	D	The area above a curve	601
	E	The area between two functions	603
	F	The area between a curve and the $𝑦$ -axis	608
	G	Solids of revolution	610
	H	Problem solving by integration	616
	I	Improper integrals	620
		Review set 22A	623
		Review set 22B	626

23	KINEMATICS		629
	A	Displacement	631
	B	Velocity	633
	C	Acceleration	640
	D	Speed	644
		Review set 23A	649
		Review set 23B	651

24	MACLAURIN SERIES		653
	A	Maclaurin series	656
	B	Convergence	659
	C	Composite functions	661
	D	Addition and subtraction	663
	E	Differentiation and integration	664
	F	Multiplication	668
	G	Division	669
		Review set 24A	670
		Review set 24B	671

25	DIFFERENTIAL EQUATIONS		673
	A	Differential equations	674
	B	Euler's method for numerical integration	677
	C	Differential equations of the form $\frac{𝑑 𝑥}{𝑑 𝑦} = 𝑓 (𝑥)$	680
	D	Separable differential equations	684
	E	Logistic growth	690
	F	Homogeneous differential equations $\frac{𝑑 𝑥}{𝑑 𝑦} = 𝑓 (\frac{𝑦}{𝑥})$	694
	G	The integrating factor method	696
	H	Maclaurin series developed from a differential equation	697
		Review set 25A	702
		Review set 25B	704

26	BIVARIATE STATISTICS		707
	A	Association between numerical variables	708
	B	Pearson's product-moment correlation coefficient	713
	C	Line of best fit by eye	718
	D	The least squares regression line	722
	E	The regression line of $𝑥$ against $𝑦$	729
		Review set 26A	732
		Review set 26B	734

27	DISCRETE RANDOM VARIABLES		737
	A	Random variables	738
	B	Discrete probability distributions	740
	C	Expectation	745
	D	Variance and standard deviation	750
	E	Properties of $𝑎 𝑋 + 𝑏$	753
	F	The binomial distribution	756
	G	Using technology to find binomial probabilities	760
	H	The mean and standard deviation of a binomial distribution	763
		Review set 27A	765
		Review set 27B	766

28	CONTINUOUS RANDOM VARIABLES		769
	A	Probability density functions	771
	B	Measures of centre and spread	774
	C	The normal distribution	778
	D	Calculating normal probabilities	782
	E	The standard normal distribution	789
	F	Normal quantiles	793
		Review set 28A	799
		Review set 28B	800

	ANSWERS		803

	INDEX		910

Mathematics: Analysis and Approaches HL

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