NZ Scholarship Calculus flashcards

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

Inverse function condition

A function has an inverse on its domain exactly when it is one-to-one there; equivalently, every horizontal line meets its graph at most once.

Composition domain rule

The domain of f composed with g consists of all x in the domain of g for which g(x) lies in the domain of f.

Remainder theorem

When a polynomial P(x) is divided by x-a, the remainder equals P(a); hence x-a is a factor exactly when P(a)=0.

Conjugate root theorem

For a polynomial with real coefficients, any non-real complex root a+bi occurs with its conjugate a-bi; repeated roots have matching multiplicities.

Rational asymptote rule

For a rational function, vertical asymptotes occur at uncancelled denominator zeros; horizontal or oblique behaviour is determined by comparing numerator and denominator degrees.

Absolute value inequality

For a>0, |x|<a is equivalent to -a<x<a, while |x|>a means x<-a or x>a; split at critical points for products and quotients.

Logarithm change equation

To solve a^x=b with a>0 and b>0, take logarithms: x=ln b/ln a; check that any transformation preserves the original domain.

Partial fractions setup

A proper rational function factors into partial fractions: distinct linear factors get constants, repeated linear factors get ascending powers, and irreducible quadratics get linear numerators.

Derivative limit definition

The derivative at x is the limit as h tends to zero of [f(x+h)-f(x)]/h, provided this limit exists.

Chain rule

For differentiable functions, d/dx f(g(x)) equals f'(g(x)) times g'(x); identify the outer and inner functions before differentiating.

Product rule

The derivative of uv is u'v+uv'; for a product of three factors, differentiate one factor at a time and add the three resulting terms.

Parametric second derivative

For x=x(t), y=y(t), dy/dx=(dy/dt)/(dx/dt); the second derivative is (d/dt(dy/dx))/(dx/dt), not d²y/dt² divided by d²x/dt², with dx/dt nonzero.

Implicit differentiation

Differentiate both sides with respect to x, treating y as y(x) and applying the chain rule; solve the resulting equation for dy/dx.

Closed interval extrema

On a closed interval, compare f values at interior critical points where f'=0 or is undefined with values at both endpoints; the largest and smallest are global extrema.

Concavity test

If f''>0 on an interval, the graph is concave up there; if f''<0 it is concave down; an inflection point requires a change of concavity.

Logarithmic differentiation

For products, quotients, or powers with variable exponents, take ln of both sides, differentiate implicitly, then multiply by the original expression.

Fundamental theorem part one

If f is continuous, then F(x)=∫_a^x f(t)dt is differentiable and F'(x)=f(x); this links accumulation and instantaneous rate.

Integration by substitution

Set u=g(x), so du=g'(x)dx; rewrite the integrand entirely in u, or for definite integrals change the limits to g(a) and g(b).

Integration by parts

Use ∫u dv=uv-∫v du; choose u to simplify when differentiated and dv to be integrable, often ln x or a polynomial as u.

Trig integral strategy

For ∫sin^m x cos^n x dx, reserve one factor for substitution when an exponent is odd; otherwise use double-angle identities to reduce powers.

Improper integral criterion

An improper integral is evaluated as a limit at its infinite or singular endpoint; it converges exactly when that limit is finite.

Area between curves

For curves y=f(x) above y=g(x) on a≤x≤b, area is ∫_a^b [f(x)-g(x)]dx; find intersections to determine limits and which curve is above.

Disk method volume

Rotating y=f(x) about the x-axis gives volume π∫_a^b [f(x)]²dx; rotating about a horizontal line y=k uses radius |f(x)-k|.

Partial fractions integration

Decompose a proper rational integrand into simpler fractions, then integrate each term as a logarithm or an arctangent after completing the square.

Euler formula

For real θ, e^{iθ}=cosθ+i sinθ; this gives polar form z=r e^{iθ}, linking complex multiplication to adding arguments.

De Moivre theorem

For integer n, [r(cosθ+i sinθ)]^n=r^n(cos nθ+i sin nθ); this computes powers and roots by scaling modulus and multiplying argument.

Complex conjugate properties

The conjugate of a+bi is a-bi; conjugates distribute through sums, products, and quotients, and z times its conjugate equals |z|².

Modulus argument division

For nonzero complex numbers, |z/w|=|z|/|w| and arg(z/w)=arg z-arg w; arguments are interpreted modulo 2π when dividing.

General sine solution

If sinθ=s, then θ=nπ+(-1)^n arcsin s for integer n, provided |s|≤1; cosine and tangent use different periodic forms.

Double angle identities

The identities sin2θ=2sinθcosθ and cos2θ=cos²θ-sin²θ=2cos²θ-1=1-2sin²θ convert products to sums and reduce powers in trigonometric integration.

Circle locus

The equation |z-a|=r describes a circle centre a and radius r in the complex plane; |z-a|=|z-b| describes the perpendicular bisector of segment ab.

Argument ray locus

The equation arg(z-a)=θ describes a ray starting at a, excluding the point a, in direction θ; arguments differing by 2π represent the same ray.