Untitled Quiz

Basic Mathematics and Statistics (BPC2111)

• This course covers basic mathematical and statistical concepts.
• Subtopics include basic calculations, functions, domain, codomain, and range, quadratic and reciprocal functions, and equations.
• Also includes graphs of quadratic and reciprocal functions, polynomial functions, division algorithms, zeros of polynomials, and the rational zero theorem.
• Other subjects in the course are quadratic equations (factorization, completing the square and quadratic formula), reciprocal equations and their graphs.

Functions

• A function takes an input (x) and produces an output (f(x)).
• Every input has one and only one output.
• Functions can be described using algebraic expressions, tables, or graphs.

Domain, Codomain, and Range

• Domain: The set of all possible input values (x).
• Codomain: The set of all possible output values (y) a function can produce.
• Range: The set of all actual output values (y) a function does produce.

Quadratic Equations

• A quadratic equation is an equation of the form ax² + bx + c = 0.
• Methods to solve quadratic equations include factorization, completing the square, and using the quadratic formula.

Quadratic and Reciprocal Functions

• Quadratic functions are functions with a degree of 2 (x²). Reciprocal functions involve 1/x.
• Graphing involves analyzing properties like direction of opening, intercepts, and stationary points (maximum/minimum).

Reciprocal Function

• A reciprocal function is a function where the output is 1 divided by the input (e.g., 1/x).

Polynomial Functions

• Polynomial functions are mathematical expressions that involve non-negative integer powers of a variable (x).
• These often include the concepts of division and finding zeros (roots) of a polynomial.

Polynomial Functions: Division Algorithm

• The division algorithm states that when a polynomial P(x) is divided by a polynomial D(x), there exists a quotient Q(x) and a remainder R(x) with a degree of R(x) < degree of D(x).

Polynomial Functions: Factor Theorem

• If a polynomial f(x) has a root a, then x - a is a factor of f(x).

Polynomial Functions: Rational Zero Theorem

• If a polynomial has integer coefficients and a rational zero p/q, where p and q are integers with no common factors other than 1, then p is a factor of the constant term and q is a factor of the leading coefficient.

Domain, Codomain, and Range

• Domain: The set of all possible input values.
• Codomain: The set of all possible output values a function can produce.
• Range: The set of all actual output values a function does produce.

Real Numbers

• Integers: ... -3, -2, -1, 0, 1, 2, 3 ...
• Rational numbers: Numbers that can be expressed as a fraction p/q, with p and q as integers and q not equal to 0.
• Irrational numbers: Numbers that cannot be expressed as a fraction.
• Real numbers: The set of all rational and irrational numbers.

Significant Figure

• Significant figures are the digits in a number that carry meaning contributing to its precision.
• Rules for determining significant figures depend on the type of number (integers, decimal numbers, etc).

Percent

• Percent (%) means "per hundred".
• To convert a fraction to a percentage, divide the numerator by the denominator, and multiply the result by 100.

Variation

• Types of variation: Direct, inverse, and joint.
• Direct variation: A direct relationship between two quantities (e.g., y = kx, where k is a constant).
• Inverse variation: An inverse relationship between two quantities (e.g., y = k⁄x, where k is a constant).
• Joint variation: A relationship where a variable is dependent on several other variables.

Arithmetic Progression (AP)

• An arithmetic progression is a sequence of numbers where each term is obtained by adding a fixed value (common difference) to the previous term.
• Formulas for the nth term (tₙ) and sum of n terms (Sₙ) of an AP are given.

Geometric Progression (GP)

• A geometric progression is a sequence of numbers where each term is obtained by multiplying a fixed value (common ratio) to the previous term.
• Formulas for the nth term and the sum of n terms of a GP are given.

Standard Form

• Standard form or scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.
• Useful for very large or very small numbers.

Laws of Logarithm

• Logarithms are inverse operations of exponentiation.
• Laws exist to simplify and manipulate expressions involving logarithms.

Derivatives and Differentiation

• Procedures of finding a derivative or gradient of a function (f(x)) are explained.
• Differentiating functions from the first principles are explained
• Rules for differentiation, including the product rule, quotient rule, and chain rule, are provided.

Higher Derivatives and Stationary Points

• Second, or higher, derivatives reveal the nature of stationary points.

Exponential Growth and Decay

• Exponential growth and decay are characterized by functions of the form N = N₀e⁻^λt, where N₀ is the initial amount.

Definite Integral

• Integration can be used to find the area under a curve.
• Definitions and formulas for finding definite integrals are given.

Area Under a Curve (AUC)

• The area under a curve can be determined by calculating the definite integral of the function.

Area Between Two Curves

• Finding the area between the curves of two different functions often involves calculating the definite integrals.

Asymptotes and Continuity

• Vertical asymptotes occur when a limit to infinity exists.
• Horizontal asymptotes occur at the limit at infinity.
• A function is described to be continuous if the limit at a point equals the value of the function at that point.

Limits

• Describing the behavior of a function as the input value approaches a certain point.