HSC Maths Ext 1 flashcards

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

One-to-one function

A function where f(a)=f(b) always implies a=b, so no two distinct inputs share an output. Only such functions have inverse functions on their range.

Inverse function graph

The graph of y=f⁻¹(x) is the reflection of y=f(x) across the line y=x. Its domain equals the range of f and its range equals the domain of f.

Odd and even functions

An even function satisfies f(−x)=f(x) and is symmetric about the y-axis. An odd function satisfies f(−x)=−f(x) and has rotational symmetry about the origin.

Absolute value definition

|x| equals x when x is non-negative and equals −x when x is negative. Geometrically |a−b| is the distance between a and b on the number line.

Composite function domain

For (f∘g)(x)=f(g(x)) to exist, x must lie in the domain of g and g(x) must lie in the domain of f. Both conditions are required.

Pythagorean identities

The three identities sin²θ+cos²θ=1, tan²θ+1=sec²θ and 1+cot²θ=cosec²θ follow from dividing the first by cos²θ or sin²θ.

Auxiliary angle form

Any expression a sin x+b cos x can be written as R sin(x+α), where R=√(a²+b²) and tan α=b/a. Useful for solving trigonometric equations and finding extrema.

Double angle formulas

sin 2A=2 sin A cos A, while cos 2A equals cos²A−sin²A, 1−2sin²A or 2cos²A−1. The three cosine forms suit different substitution contexts.

t-formula substitution

Setting t=tan(θ/2) gives sin θ=2t/(1+t²) and cos θ=(1−t²)/(1+t²). This converts trigonometric equations into algebraic equations in t.

General solution for sine

If sin θ=k with |k|≤1, then θ=nπ+(−1)ⁿ arcsin k for integer n. The (−1)ⁿ factor alternates the two symmetric solutions on the circle.

Amplitude and period

For y=a sin(bx+c) the amplitude is |a| and the period is 2π/|b|. For y=a tan(bx+c) the period is π/|b| instead.

Product and quotient rules

The product rule states (uv)′=u′v+uv′. The quotient rule states (u/v)′=(u′v−uv′)/v², valid where v is nonzero.

Chain rule

The derivative of f(g(x)) is f′(g(x))·g′(x). Differentiate the outer function at the inner value, then multiply by the inner derivative.

Standard trigonometric derivatives

The derivatives are d/dx sin x=cos x, d/dx cos x=−sin x and d/dx tan x=sec²x. The negative sign in the cosine derivative is a common error source.

Fundamental theorem of calculus

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

Integration by substitution

Substitution reverses the chain rule: replace the inner expression by u, convert dx to du, and for definite integrals also convert the limits before evaluating.

Solid of revolution volume

Rotating y=f(x) about the x-axis from a to b gives volume V=π∫ₐᵇ y² dx. The squared radius accounts for the circular cross-sections.

Stationary point classification

Stationary points satisfy f′(x)=0. Test with a sign change of f′ across the point, or with f″: positive gives a minimum, negative a maximum.

Permutations formula

The number of ordered arrangements of r objects chosen from n distinct objects is ⁿPᵣ=n!/(n−r)!. Order matters, so different orderings count separately.

Combinations formula

The number of unordered selections of r objects from n distinct objects is ⁿCᵣ=n!/(r!(n−r)!). Order does not matter, so each subset is counted once.

Binomial theorem expansion

(a+b)ⁿ=ΣⁿCᵣ aⁿ⁻ʳbʳ for r from 0 to n. The general term ⁿCᵣ aⁿ⁻ʳbʳ locates a specific power without full expansion.

Circular arrangements

The number of distinct arrangements of n objects around a circle is (n−1)!, since rotations are identical. If reflections also count as the same, halve the result.

Stars and bars counting

The number of non-negative integer solutions of x₁+…+xₖ=n is ⁿ⁺ᵏ⁻¹Cₖ₋₁. It counts distributions of n identical items into k distinct boxes.

Proof by contradiction

To prove a statement, assume its negation and derive a logical impossibility. The contradiction shows the negation is false, so the original statement holds.

Proof by contrapositive

Proving ¬Q⇒¬P establishes P⇒Q, since a statement and its contrapositive are logically equivalent. This differs from proof by contradiction in structure.

Mathematical induction steps

Show the statement holds for the base case, assume it holds for n=k, then use that assumption to prove it holds for n=k+1. This establishes it for all integers beyond the base.

Disproof by counterexample

A single example where a universal statement fails disproves it completely. Counterexamples do not prove statements, only refute them.

Dot product formula

For vectors a and b, a·b=|a||b|cos θ also equals the sum of corresponding component products. A zero dot product means the vectors are perpendicular.

Vector projection

The scalar projection of a onto b is (a·b)/|b|. The vector projection is ((a·b)/|b|²)b, which lies along the direction of b.

Unit vector

The unit vector in the direction of a is a/|a|, obtained by dividing each component by the magnitude. It always has magnitude exactly one.

Angle between vectors

The angle satisfies cos θ=(a·b)/(|a||b|), giving θ between 0 and π. This follows directly from rearranging the geometric dot product formula.

Variance of a random variable

Variance equals E(X²)−[E(X)]², and standard deviation is its positive square root. Variance measures spread around the mean in squared units.

Binomial distribution mean

For X~Bin(n,p), the mean is np and the variance is np(1−p). The probability P(X=k) equals ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ.