NZ Scholarship Calculus key terms

The set of all input values for which a function is defined. For a formula, exclude values that cause division by zero, even roots of negative numbers, or logarithms of non-positive numbers.

The set of all output values a function actually produces as its input runs through the domain. Finding it usually needs the graph, or solving y = f(x) for x and checking which outputs occur.

A function built by feeding the output of one function into another, written f(g(x)). Its domain contains only those inputs for which g(x) lies inside the domain of f.

The function that undoes another, sending each output back to its input. It exists only when the original is one-to-one, and its graph is the reflection in the line y = x.

A straight line that a curve approaches arbitrarily closely as the variable grows without bound, or as the function value tends to infinity near a point where the function is undefined.

A polynomial p(x) has (x − a) as a factor exactly when p(a) = 0. It turns the search for factors into evaluating the polynomial, and underpins factorising cubics and higher degree expressions.

Rewriting a single rational expression as a sum of simpler fractions whose denominators are linear factors or irreducible quadratics, so each piece can be integrated or expanded separately.

For the quadratic ax² + bx + c, the quantity b² − 4ac. Its sign shows whether the equation has two distinct real roots, one repeated real root, or a conjugate pair of complex roots.

The value a function's output approaches as the input approaches a chosen point, whether or not the function is defined there. Limits underpin both the derivative and the definite integral.

The instantaneous rate of change of a function, defined as the limit of [f(x + h) − f(x)]/h as h tends to zero. Geometrically it is the gradient of the tangent at that point.

The rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x)) multiplied by g′(x). In words, differentiate the outer function and multiply by the derivative of the inner.

The rule that the derivative of a product uv is u′v + uv′. It is not the product of the separate derivatives, and it extends to products of three or more factors.

The rule that the derivative of u/v is (u′v − uv′)/v², valid wherever v is nonzero. The order of the subtraction matters, so writing the numerator out carefully avoids sign errors.

Differentiating both sides of an equation relating x and y with respect to x, treating y as an unknown function of x and applying the chain rule to every term containing y.

Finding the gradient dy/dx of a curve given by x = f(t) and y = g(t) by dividing dy/dt by dx/dt, provided dx/dt is not zero at that value of the parameter.

A point on a curve where the derivative is zero or fails to exist. Such points are candidates for local maxima, minima, or horizontal inflections, confirmed by a sign test on the derivative.

A point where a curve changes from concave up to concave down, or the reverse. The second derivative is zero or undefined there, but that condition alone does not guarantee an inflection.

Finding the greatest or least value a quantity can take by writing it as a function of one variable, differentiating, and checking stationary points along with endpoints and any domain restrictions.

A function whose derivative is the given function. Any two antiderivatives of the same function differ by a constant, which is why an indefinite integral carries a constant of integration.

The signed accumulated area between a curve and the x-axis from one limit to another. Regions below the axis count as negative, so the value can be much smaller than the total geometric area.

The theorem linking the two branches of calculus: for any antiderivative F of f, the integral of f from a to b equals F(b) − F(a). It turns area problems into antiderivative evaluations.

Reversing the chain rule by replacing part of the integrand with a new variable, converting dx into a multiple of du, so the transformed integral becomes simpler to evaluate.

The product rule run backwards: the integral of u dv equals uv minus the integral of v du. It suits products such as x·e^x, x·ln x, and integrands consisting of ln x alone.

An integral taken over an infinite interval, or whose integrand becomes unbounded somewhere within the interval. It is defined as a limit and may converge to a finite value or diverge.

The volume swept out when a plane region is rotated about an axis. It is computed by slicing into discs or washers with cross-sectional area πy², or by cylindrical shells.

The angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn is 2π radians, so converting between radians and degrees uses a factor of π/180.

The complete family of solutions to a trigonometric equation, obtained by adding whole multiples of the function's period to a base solution found within one cycle.

An identity giving a trigonometric function of a sum or difference, for example sin(A + B) = sin A cos B + cos A sin B. These expand expressions and reduce equations to a single angle.

Writing a cos θ + b sin θ as one wave R sin(θ + α), where R is the square root of a² + b². This reveals the amplitude, the phase shift, and the maximum and minimum values.

The distance from the origin to the point representing z = a + bi in the Argand diagram, written |z| and equal to the square root of a² + b².

The angle measured anticlockwise from the positive real axis to the line joining the origin to a complex number. It is conventionally chosen in the interval from −π to π.

Expressing a complex number as r(cos θ + i sin θ), where r is its modulus and θ its argument. This form makes multiplication, division, powers, and roots straightforward to compute.

A coordinate plane used to picture complex numbers, with a + bi plotted at the point (a, b). Addition then looks like vector addition, and modulus becomes ordinary distance.

For a complex number in polar form, raising cos θ + i sin θ to the power n gives cos nθ + i sin nθ. Running the result backwards produces the n distinct nth roots.

The number a − bi obtained from z = a + bi by flipping the sign of the imaginary part. A number multiplied by its conjugate equals the square of the modulus, a real value.

The n complex solutions of z^n = 1. They lie equally spaced around the unit circle at arguments 2kπ/n, and for n greater than one their sum is zero.

The identity e^(iθ) = cos θ + i sin θ, which links exponential and trigonometric functions and lets any complex number be written compactly as r·e^(iθ).

The inverse of sine, cosine, or tangent after restricting its domain so the inverse is a function, with principal ranges such as −π/2 to π/2 for arcsine.