Mathematics Methods Level 4 flashcards

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

Composite function domain

For f(g(x)), x must lie in the domain of g and g(x) must lie in the domain of f. Solve both conditions and intersect the resulting sets.

Inverse function condition

A function has an inverse only if it is one-to-one. Restricting the domain can make it one-to-one, and the inverse graph is the reflection in y = x.

Transformation order

For y = a·f(b(x − h)) + k, apply reflections and scalings first, then translations. Horizontal changes act inside the brackets and behave inversely.

Even and odd functions

Even means f(−x) = f(x), giving symmetry about the y-axis. Odd means f(−x) = −f(x), giving rotational symmetry of order two about the origin.

Remainder and factor theorem

Dividing polynomial P(x) by (x − a) leaves remainder P(a). The linear factor (x − a) divides P exactly when P(a) equals zero.

Multiplicity of a root

A root of even multiplicity makes the graph touch the x-axis and turn back. A root of odd multiplicity makes the graph cross the axis.

Rational function asymptotes

Vertical asymptotes occur where the denominator is zero but the numerator is not. Horizontal asymptotes depend on comparing the degrees of numerator and denominator.

Extraneous roots

Squaring both sides of an equation can create solutions that do not satisfy the original. Always substitute each candidate root back to check.

Logarithm product and power laws

log(xy) = log x + log y and log(xⁿ) = n log x, valid for positive x and y. These laws convert products into sums and powers into multiples.

Change of base formula

log_b x = ln x / ln b for positive bases not equal to one. This allows any logarithm to be evaluated with natural logarithms.

Exponential growth model

y = A·e^(kt) with k > 0 gives growth and k < 0 gives decay. A is the initial value and k is the continuous rate constant.

Half-life formula

For decay y = A·e^(kt) with k < 0, the half-life is t = ln(0.5)/k, a positive time because the numerator and k are both negative.

Solving logarithmic equations

Combine logarithms into one term, rewrite in exponential form, solve, then reject any solution that makes an original argument non-positive.

Pythagorean identity

sin²θ + cos²θ = 1 for all angles. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

Double angle formulas

sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ, also written as 1 − 2sin²θ or 2cos²θ − 1.

Amplitude and period

For y = a sin(bx + c) + d, amplitude is |a| and period is 2π/|b|. The phase shift is −c/b and the vertical shift is d.

Cosine rule

a² = b² + c² − 2bc cos A relates three sides to the angle opposite side a. Use it with three sides or two sides and the included angle.

General solution for sine

If sin θ = k, then θ = arcsin k + 2nπ or θ = π − arcsin k + 2nπ for integer n, covering both symmetric positions on the unit circle.

Derivative from first principles

f′(x) = lim_{h→0} [f(x + h) − f(x)]/h. It gives the gradient of the tangent and the instantaneous rate of change of the function.

Chain rule

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

Quotient rule

For y = u/v, the derivative is (u′v − uv′)/v². The numerator order matters, and the denominator is squared.

Second derivative test

At a stationary point where f′(x) = 0, f″(x) > 0 indicates a local minimum and f″(x) < 0 indicates a local maximum.

Derivatives of sine and cosine

With angles in radians, d/dx sin x = cos x and d/dx cos x = −sin x. Radians are essential for these formulas to hold.

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.

Area between two curves

The area between curves from a to b is the integral of upper minus lower. Find a and b by solving where the curves intersect.

Volume of revolution

Rotating y = f(x) about the x-axis from a to b gives volume V = π times the integral of [f(x)]² from a to b, for non-negative f.

Integration by substitution

Set u = g(x) so du = g′(x) dx, rewrite the integral entirely in u, integrate, then substitute back. For definite integrals also change the limits.

Conditional probability

P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. It rescales the sample space to event B and underlies independence testing.

Binomial distribution

P(X = k) = C(n, k)p^k(1 − p)^(n−k) for n independent trials with constant success probability p. Mean is np and variance np(1 − p).

Standardising a normal variable

Convert to z = (x − μ)/σ, which has mean zero and standard deviation one. Then use tables or technology to find required areas.

Linear regression slope

For least-squares line ŷ = a + bx, the slope is b = r·(s_y/s_x), and the line always passes through the mean point (x̄, ȳ).

Choosing a model family

Constant first differences suggest a linear model, constant ratios suggest an exponential model, and constant second differences suggest a quadratic model.