Pre-U Mathematics CAIE

This subject is broken down into 39 topics in 4 modules:

  1. Algebra and Geometry 10 topics
  2. Trigonometry and Functions 10 topics
  3. Calculus 9 topics
  4. Probability and Statistics 10 topics
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This page was last modified on 28 September 2024.

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Mathematics

Algebra and Geometry

Algebraic expressions and equations

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Algebraic expressions and equations

Algebraic Expressions

  • An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide).
  • Variables are symbols used to represent unknown numbers or values. They are typically represented by letters, such as x, y, or z.
  • The terms of an algebraic expression are the parts of the expression that are added or subtracted. Each term can be a number, a variable, or a product or quotient of numbers and variables.
  • Algebraic expressions can be simplified by combining like terms, which are terms with exactly the same variables and powers.
  • The coefficient is the numerical factor in any term. For example, in 5x, the coefficient is 5.
  • To evaluate an algebraic expression, you replace the variable with a given number and then perform the operations in the expression.

Algebraic Equations

  • An algebraic equation is a statement of equality between two algebraic expressions.
  • The solution of an algebraic equation is the numerical value of the variable that makes the equation true.
  • An algebraic equation can be solved by isolating the variable, which often involves addition, subtraction, multiplication, or division.
  • Sometimes, you'll need to use the distributive property to remove parentheses before isolating the variable.
  • Quadratic equations are algebraic equations that have a degree of 2. Their solutions can be found using factoring, completing the square, or using the quadratic formula.
  • System of equations is a set of two or more equations that all contain the same set of variables. Systems can be solved by graphing, substitution, or elimination.
  • A linear equation forms a straight line when graphed. It has no exponents higher than 1, and its graph is a straight line. You can solve a linear equation by isolating the variable.

Algebraic Formulae

  • A formula is a type of equation that shows the relationship between different variables.
  • An algebraic formula can be used to calculate measurements, like the area of a circle or rectangle, or the volume of a cylinder or sphere.
  • The formula for a linear sequence is an + b, where a is the common difference and b is the first term.
  • The formula for a quadratic sequence is an^2+ bn + c, where a, b and c are constants.
  • The formula for the sum of an arithmetic series is Sn = n/2 * (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Course material for Mathematics, module Algebra and Geometry, topic Algebraic expressions and equations

Mathematics

Calculus

Limits and continuity of functions

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Limits and continuity of functions

Understanding Limits

  • A limit is a value that a function or sequence "approaches" as the input or index approaches some value.
  • Limits are essential in calculus and are used in defining continuity, derivatives, and integrals.
  • The notation 'lim{x→a} f(x) = L' is read as "the limit of f(x) as x approaches a is L".
  • When you see the symbol '∞' (which means infinity), it represents an exponential growth which never meets any particular line on the graph.

Evaluating Limits

  • To evaluate the limit of a function as x approaches a particular value, you can often simply plug in that value.
  • However, there may be some cases when this is indeterminate (0/0, ∞/∞) or undefined. In that case, use algebraic simplification, l'Hôpital’s Rule, or evaluate the limit from the left (

    lim{x→a-}

    ) and the right (

    lim{x→a+}

    ) separately.
  • If the left-hand limit equals the right-hand limit, the general limit exists. If they aren't equal, the limit does not exist.

Continuity of Functions

  • A function is said to be continuous at certain point if the limit of the function as x approaches that point is equal to the value of the function at that point.
  • Formally, we say that a function f is continuous at a point x=a if

    lim{x→a} f(x) = f(a)

    .
  • If a function is continuous at every point in an interval, then we simply say it is continuous on that interval.

Types of Discontinuity

  • In case the limit does not exist or does not equal the function value, the function is said to be discontinuous at that point.
  • Discontinuities are of three types: jump (where the function 'jumps' from one value to another), removable (a single point where the function is not defined), and infinite (where the function approaches infinity).

Intermediate Value Theorem

  • The Intermediate Value Theorem states that if a function is continuous on an interval [a, b] and k is any value between f(a) and f(b), then there's at least one number c in the interval [a, b] such that f(c) = k.
  • This theorem is basically saying that a continuous function takes all values between its minimum and maximum in any interval.

Extreme Value Theorem

  • The Extreme Value Theorem establishes that if a function is continuous on a closed interval, then it reaches both a minimum and maximum value on that interval.
  • This is an important theorem to understand, as it can be helpful when faced with optimisation problems involving calculus.

Course material for Mathematics, module Calculus, topic Limits and continuity of functions

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