HSC Maths Ext 2 key terms

The statement formed by swapping the hypothesis and conclusion of an implication. Its truth is independent of the original statement, so proving a converse does not prove the original.

The statement formed by negating both hypothesis and conclusion of an implication and swapping them. It is logically equivalent to the original implication, so proving it proves the original.

A single case in which the hypothesis of a statement holds but the conclusion fails, which is enough to disprove a universal statement.

A method that assumes the negation of the desired conclusion and derives an impossible consequence, thereby establishing the original claim.

A real number that cannot be written as a ratio of two integers. Classic proofs that certain roots are irrational proceed by contradiction from an assumed rational form in lowest terms.

A real number expressible as a fraction of two integers with nonzero denominator. Writing a number in lowest terms is a key step in many contradiction proofs.

A condition that must hold and whose holding guarantees the claim, giving logical equivalence. It corresponds to an if-and-only-if statement requiring proof in both directions.

A quantity with both magnitude and direction, represented by a directed segment or an ordered list of components and added componentwise.

A scalar formed from two vectors by summing products of corresponding components. It equals the product of magnitudes times the cosine of the angle between them.

A vector perpendicular to two given vectors in three dimensions, with magnitude equal to the area of the parallelogram they span and direction fixed by a right-hand rule.

The dot product of one vector with the cross product of two others. Its absolute value gives the volume of the parallelepiped spanned by the three vectors.

The component of one vector along the direction of another, found by taking the dot product with a unit vector in that direction.

A vector of length one, obtained by dividing any nonzero vector by its magnitude. It records direction only.

Vectors whose dot product is zero, meaning they meet at a right angle. This condition is used to find perpendicular directions and normals.

A description of a line as a fixed point plus a scalar multiple of a direction vector, with the scalar allowed to vary over all reals.

A vector perpendicular to every direction lying in the plane. Combined with one point on the plane it determines the plane's equation.

A number of the form a plus bi, where a and b are real and i squared equals negative one. It can be pictured as a point in the plane.

The distance of a complex number from the origin in the plane, equal to the square root of a squared plus b squared. It generalises absolute value.

The angle a complex number makes with the positive real axis, measured anticlockwise and defined up to whole turns unless a principal range is specified.

The complex number obtained by reversing the sign of the imaginary part. Multiplying a number by its conjugate gives the square of its modulus.

Writing a complex number as its modulus times a cosine-plus-i-sine expression of its argument, which makes multiplication and powers easy to compute.

The result that raising a complex number in polar form to a power raises the modulus to that power and multiplies the argument by it.

The complex solutions of z to the power n equals one. They are equally spaced around the unit circle and form the vertices of a regular polygon.

The set of all points in the plane satisfying a given condition, such as a fixed distance from a point or a fixed angle subtended by a segment.

A technique for finding a derivative when a relationship between variables is not solved for one of them, by differentiating both sides and treating one variable as a function of the other.

The rule for differentiating a composite function: differentiate the outer function at the inner function and multiply by the derivative of the inner function.

The rule stating that the derivative of a product of two functions is the first times the derivative of the second plus the second times the derivative of the first.

A rule for differentiating a ratio of two functions using the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.

The derivative of the derivative of a function. Its sign indicates concavity and it is used to classify stationary points and describe acceleration.

A point where the first derivative of a function is zero. It may be a local maximum, a local minimum, or a point of inflection.

The way a curve bends, upward or downward, as determined by the sign of the second derivative on an interval.

A technique derived from the product rule that rewrites an integral of a product as a boundary term minus another integral, useful when one factor simplifies on differentiation.

A technique that reverses the chain rule by changing the variable of integration, transforming the integrand and the limits into a simpler form.

Rewriting a rational expression as a sum of simpler fractions with linear or quadratic denominators, enabling term-by-term integration.

Motion in which acceleration is proportional to displacement from a fixed point and directed towards it, producing sinusoidal oscillation about that point.

Motion of a body under gravity alone after launch, analysed by resolving velocity into horizontal and vertical components and treating each independently.

Motion in which a resistance force, often proportional to velocity or its square, opposes movement, requiring a differential equation to be solved for velocity or displacement.

The constant speed reached when the resisting force balances the driving force, so net acceleration becomes zero and velocity stops changing.

A sequence in which consecutive terms differ by a fixed constant, so the nth term is linear in n and the sum of the first n terms is quadratic in n.

A sequence in which each term is a fixed multiple of the previous one. Its partial sums have a closed form and converge when the ratio has magnitude below one.

The value a series approaches as the number of terms grows without bound, existing for a geometric series only when the common ratio has magnitude less than one.

A sum in which most terms cancel in pairs after the general term is split into a difference, leaving only a few surviving terms.

The number of ways to choose a subset of a given size from a larger set, appearing in the expansion of a binomial power and computable from factorials.

The formula expanding a binomial raised to a positive integer power as a sum of terms involving binomial coefficients and descending powers of each part.

An expression for a typical term of a binomial expansion in terms of its position, used to locate a particular power or coefficient without expanding everything.

A triangular array in which each entry is the sum of the two above it, whose rows list the binomial coefficients for successive powers.

A proof technique for statements about positive integers: verify a base case, assume the statement for some value, then prove it for the next value.

The initial instance of a statement that is checked directly at the start of an induction proof, anchoring the chain of implications.

The part of an induction proof that derives the statement for one value from its assumed truth at the previous value, completing the chain from the base case.

A variant in which the inductive step assumes the statement holds for all values up to a given point, rather than for that point alone.