Preparation for Mathematics 9709 CAIE examinations.

This is a class description for O-level math, a course that covers the basic concepts and skills of mathematics. The course aims to prepare students for further studies in mathematics, science, engineering, and other fields that require mathematical reasoning. The course covers topics such as algebra, geometry, trigonometry, statistics, and calculus. The course also develops students' problem-solving, logical thinking, and communication skills. The course is assessed by a written examination at the end of the year.

Chapter 1: Relationships The central question of this introductory chapter – which contains no calculus – is “What is a function?” The objective is to help students separate this concept from other relationships between varying quantities, and especially to separate the idea of function from such ideas as formula and equation. The concept of function is the basic building block of mathematics. A deep understanding of function will facilitate your future study of mathematics and computer science. Throughout this course, we will be working with multiple representations of functions. The authors of our text present functions verbally, numerically, and visually as well as algebraically. Chapter 2: Models of Growth: Rates of Change In this chapter, we will investigate some basic reasons for studying calculus. In particular we will investigate problem situations which can be modeled using differential equations. Topics introduced in this chapter include difference quotients, derivatives, slope fields, initial value problems whose solutions are functions and families of functions. The primary example of this chapter is natural population growth, the simplest ODE (ordinary differential equation) to solve. This example provides an immediate reason for moving beyond polynomials to other families of functions (e.g., to exponential and logarithmic functions). We will conclude this chapter by using tools of calculus to analyze the spread of the AIDS virus. Chapter 3: Initial Value Problems This short chapter builds on Chapter 2, introducing Newton’s Law of Cooling (exponential decay) to solve a murder mystery, then studying falling objects without air resistance (polynomial solutions). Chapter 4: Differential Calculus and Its Uses This chapter is the heart of first-semester calculus, consolidating what has been learned about derivatives to take up problems involving optimization, concavity, Newton’s Method (as an exercise in local linearity), and the basic formulas for differentiation. The product rule is introduced to study the growth rate of energy consumption, the chain rule to study reflection and refraction, and implicit differentiation to calculate derivatives of logarithmic functions and general powers. The process of zooming in on a graph is related to differentials and Leibniz notation. The chapter concludes with an interesting application of calculus to a problem in air-traffic control. Chapter 5: Modeling with Differential Equations This chapter builds on the problems introduced in Chapter 3, introducing air resistance to problems of falling bodies (e.g., raindrops and skydivers). The authors introduce problems of periodic motion, which are modeled using trigonometric functions and their derivatives.

After successfully completing the course, you will have a good understanding of the following topics and their applications: Systems of linear equations Row reduction and echelon forms Matrix operations, including inverses Block matrices Linear dependence and independence Subspaces and bases and dimensions Orthogonal bases and orthogonal projections Gram-Schmidt process Linear models and least-squares problems Determinants and their properties Cramer’s Rule Eigenvalues and eigenvectors Diagonalization of a matrix Symmetric matrices Positive definite matrices Similar matrices Linear transformations Singular Value Decomposition

This is a class description for Cambridge IGCSE math, a course that covers the basic concepts and skills of mathematics. The course aims to prepare students for further studies in mathematics, science, engineering, and other fields that require mathematical reasoning. The course covers topics such as algebra, geometry, trigonometry, statistics, and calculus. The course also develops students' problem-solving, logical thinking, and communication skills. The course is assessed by a written examination at the end of the year.