Pancyprian Core Math flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

GCD–LCM product identity

For any two positive integers a and b, the greatest common divisor times the least common multiple equals the product a·b. Useful when one of the two quantities is known.

Change of base formula

For positive bases not equal to 1, log base b of x equals ln x divided by ln b. It lets you evaluate any logarithm using a calculator's natural log key.

Compound interest formula

A = P(1 + r/n)^(nt), where P is the principal, r the annual rate as a decimal, n the number of compounding periods per year, and t the number of years.

Absolute value inequality rule

For k greater than zero, the inequality |x| < k is equivalent to −k < x < k, while |x| > k splits into x < −k or x > k.

Quadratic discriminant

The expression b² − 4ac from ax² + bx + c. Positive gives two distinct real roots, zero gives one repeated real root, negative gives no real roots.

Factor theorem

A polynomial P(x) has factor (x − r) exactly when P(r) equals zero. Equivalently, the remainder on dividing P(x) by (x − r) is P(r).

Sum and product of roots

For ax² + bx + c = 0 with a not zero, the two roots sum to −b/a and multiply to c/a, without needing to solve the equation.

Geometric series sum

For a geometric sequence with first term a and ratio r not equal to 1, the sum of the first n terms is a(1 − r^n)/(1 − r).

Function transformation shifts

Replacing f(x) by f(x) + k shifts the graph up by k, while f(x + k) shifts it left by k. Horizontal changes act opposite to their sign.

Sector area in radians

When the central angle θ is measured in radians, a sector of radius r has area ½r²θ and arc length rθ. Degrees must be converted first.

Similar figure scaling ratios

If two similar figures have corresponding lengths in ratio k, their areas are in ratio k² and their volumes in ratio k³.

Perpendicular gradient condition

Two non-vertical lines are perpendicular exactly when the product of their gradients equals −1, so one gradient is the negative reciprocal of the other.

Sphere volume and surface area

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². The surface area equals the derivative of the volume with respect to r.

Regular polygon interior angle

A regular polygon with n sides has each interior angle equal to (n − 2)·180° divided by n, since the interior angles sum to (n − 2)·180°.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

Cosine rule

In any triangle, a² = b² + c² − 2bc·cos A, where A is the angle opposite side a. It handles side-angle-side and three-sides cases.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite the angle of the same letter. It also equals twice the circumradius.

Tangent function period

The graph of y = tan x repeats every 180° (π radians) rather than 360°, and has vertical asymptotes where cos x equals zero, at 90° plus multiples of 180°.

Quadrant signs for trigonometry

Sine is positive in quadrants I and II, cosine in I and IV, tangent in I and III. A reference angle plus the correct sign gives all solutions.

Standard deviation effect of shift

Adding the same constant to every data value shifts the mean and median by that constant but leaves the standard deviation, variance and range unchanged.

Conditional probability formula

The probability of A given B equals the probability of both A and B divided by the probability of B, provided B has nonzero probability.

Independent events test

Two events A and B are independent exactly when the probability of both occurring equals the product of their individual probabilities.

Binomial probability formula

For n independent trials each with success probability p, the probability of exactly k successes is nCk times p^k times (1 − p)^(n−k).

Sample variance divisor

The sample variance divides the sum of squared deviations from the mean by n − 1, while the population variance divides by n.

Chain rule

For a composite function, dy/dx equals dy/du multiplied by du/dx. Differentiate the outer function, then multiply by the derivative of the inner function.

Second derivative test

At a stationary point where the first derivative is zero, a positive second derivative indicates a local minimum and a negative one a local maximum.

Definite integral as signed area

The definite integral from a to b of f(x) dx gives the area above the x-axis minus the area below it, so regions under the axis contribute negatively.

Fundamental theorem of calculus

If F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a), linking areas to antiderivatives.

Quotient rule

The derivative of u/v is (u′v − uv′)/v², valid where v is nonzero. The numerator order matters because subtraction is not commutative.

Contrapositive equivalence

The statement not-q implies not-p is logically equivalent to p implies q. The converse and the inverse are not equivalent to the original.

Pigeonhole principle

If more than n objects are placed into n containers, at least one container holds two or more objects. Used to prove existence without construction.

Proof by mathematical induction

Establish the statement for a base case, then show that assuming it for n forces it for n + 1. Both steps are required for the conclusion.