NCEA Level 3 Calculus key terms

The instantaneous rate of change of a function, found as the limit of the average rate of change as the interval shrinks to zero; geometrically it is the gradient of the tangent to the curve.

Finding a derivative by evaluating the limit of (f(x+h) minus f(x)) divided by h as h approaches zero, using the formal definition rather than shortcut rules.

The rule that the derivative of x to the power n is n times x to the power n minus one, valid for any real constant n, applied term by term to polynomials.

A rule for differentiating a composite function: the derivative of f(g(x)) equals f prime of g(x) multiplied by g prime of x, working from the outside inwards.

A rule for differentiating a product of two functions: the derivative of uv equals u prime v plus u v prime, where u and v are both functions of the same variable.

A rule for differentiating a fraction: the derivative of u over v equals (u prime v minus u v prime) divided by v squared, where v is not zero.

The straight line touching a curve at a point with the same gradient as the curve there; its gradient equals the derivative evaluated at that point.

The line through a point on a curve that is perpendicular to the tangent there, so its gradient is the negative reciprocal of the derivative at that point.

A point on a curve where the derivative equals zero, giving a local maximum, local minimum, or point of inflection; located by solving the derivative equal to zero.

A method classifying a stationary point by the sign of the second derivative there: negative means local maximum, positive means local minimum, and zero is inconclusive.

The derivative of the derivative of a function, written f double prime of x; it measures how the gradient itself is changing and relates to concavity.

The way a curve bends: concave up where the second derivative is positive, concave down where it is negative, with inflection points where the sign changes.

A point where the concavity of a curve changes sign; it may or may not be stationary, and is found by testing where the second derivative changes sign.

The value a function or expression approaches as the input or variable approaches some target; the foundation of both differentiation and integration.

A function whose derivative is the given function; because constants differentiate to zero, antiderivatives differ by an arbitrary constant, written with plus C.

The family of all antiderivatives of a function, written with an integral sign and a constant of integration, representing every possible antiderivative.

The integral of a function between two limits, evaluated by subtracting the antiderivative at the lower limit from its value at the upper limit; it gives signed area.

The result linking the two branches of calculus: differentiating an integral of a function returns the function, and a definite integral is computed from an antiderivative's values at the limits.

The arbitrary constant added when finding an indefinite integral, reflecting that infinitely many antiderivatives differ only by a constant; fixed using an initial condition.

The region between a curve and an axis, found by integrating; portions below the axis count as negative, so separate integrals or absolute values are needed for total area.

A technique reversing the chain rule: replace part of the integrand with a new variable, adjust the differential, and change limits for definite integrals.

A known value of a function or its derivative at a particular point, used to determine the constant of integration or a particular solution to a differential equation.

An equation involving derivatives of an unknown function; solving it means finding functions satisfying the equation, often with a constant fixed by an initial condition.

How quickly one quantity changes relative to another; the derivative gives instantaneous rate of change, while average rate of change uses two points.

Using calculus to find maximum or minimum values of a quantity, typically by forming a function, differentiating, solving for stationary points, and justifying the optimum.

A function formed by applying one function to the output of another, written f(g(x)); differentiating it requires the chain rule.

An expression made of terms of the form a times x to a non-negative whole-number power, added together; differentiated term by term using the power rule.

A curve obtained by slicing a double cone with a plane: circle, ellipse, parabola, or hyperbola, each described by a standard quadratic equation in two variables.

The set of points a fixed distance, the radius, from a fixed centre; its equation is (x minus a) squared plus (y minus b) squared equals r squared.

The set of points whose distances from two fixed foci add to a constant; its standard equation has squared x and y terms with unequal positive denominators.

The set of points whose distances from two fixed foci differ by a constant; its standard equation has a minus sign between the squared terms, giving two branches.

The set of points equidistant from a fixed focus and a fixed line called the directrix; its equation has one squared variable and one linear variable.

A fixed point used to define a conic: a parabola is equidistant from focus and directrix, while ellipses and hyperbolas have two foci.

A fixed line used together with a focus to define a parabola: every point on the parabola is the same distance from the focus and the directrix.

A number describing how elongated a conic is: zero for a circle, between zero and one for an ellipse, exactly one for a parabola, greater than one for a hyperbola.

A line a curve approaches ever more closely without meeting; hyperbolas have two asymptotes through their centre, found from the standard equation.

A pair of equations expressing x and y separately in terms of a third variable called a parameter, useful for describing curves such as circles and ellipses.

An equation true for all values of the variables involved, such as the Pythagorean identity sine squared plus cosine squared equals one, used to simplify expressions.

Formulas expanding trigonometric functions of sums or differences, for example sine of (A plus B) equals sine A cosine B plus cosine A sine B.

Identities expressing trigonometric functions of twice an angle in terms of the single angle, such as sine 2A equals 2 sine A cosine A.

The natural angular measure where one radian subtends an arc equal to the radius; a full turn is two pi radians, and calculus formulas for trigonometric derivatives require radians.

Half the difference between the maximum and minimum values of a periodic function; for a sine or cosine curve it is the coefficient in front of the function.

The smallest positive interval after which a periodic function repeats; for sine and cosine of kx the period is two pi divided by k.

The complete set of solutions to a trigonometric equation, expressed using the periodic nature of the functions, for example adding multiples of pi or two pi.

A number of the form a plus bi, where a and b are real and i squared equals minus one; a is the real part and b the imaginary part.

The symbol i defined by i squared equals minus one, allowing square roots of negative numbers and generating the complex number system.

For a complex number a plus bi, the number a minus bi, with the imaginary part's sign reversed; multiplying a number by its conjugate gives a real value.

The distance of a complex number from the origin on the Argand diagram, written as the square root of a squared plus b squared for a plus bi.

The angle a complex number makes with the positive real axis on the Argand diagram, measured anticlockwise, usually given in radians between minus pi and pi.

A plane where a complex number a plus bi is plotted as the point (a, b), with the real part on the horizontal axis and imaginary part on the vertical axis.

Writing a complex number as r times (cosine theta plus i sine theta), where r is the modulus and theta the argument, convenient for multiplication and powers.

The result that (cosine theta plus i sine theta) to the power n equals cosine n theta plus i sine n theta, used to find powers and roots of complex numbers.

A quadratic equation whose discriminant is negative, giving a conjugate pair of non-real roots found using the quadratic formula with i.

The quantity b squared minus 4ac for a quadratic; its sign shows whether roots are real and distinct, repeated, or a complex conjugate pair.