NCEA Level 3 Calculus flashcards

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

Power rule for differentiation

For y = x^n, the derivative is n·x^(n−1). It applies to any real exponent n, including negative and fractional powers, provided x^n is defined there.

Chain rule

If y = f(g(x)), then dy/dx = f′(g(x))·g′(x). Differentiate the outer function, keep the inner unchanged, then multiply by the derivative of the inner function.

Product rule

For y = u(x)v(x), the derivative is u′v + uv′. Each factor is differentiated in turn while the other is held constant, and the two results are added.

Quotient rule

For y = u/v with v ≠ 0, dy/dx = (u′v − uv′)/v². The order in the numerator matters: the derivative of the top comes first.

Integral of 1/x

The integral of 1/x with respect to x is ln|x| + C, not x⁰/0. The absolute value is needed because 1/x exists for negative x but ln x does not.

Fundamental theorem of calculus

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

Area between two curves

Integrate (upper curve − lower curve) between the x-values where the curves intersect. Intersections are found by solving f(x) = g(x) before integrating.

Second derivative test

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

Equation of a circle

A circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r². Expanding and completing the square converts general form back to this.

Parabola focus and directrix

The parabola y² = 4ax has focus (a, 0) and directrix x = −a. Every point on it is equidistant from the focus and the directrix.

Ellipse foci relation

For x²/a² + y²/b² = 1 with a > b, the foci are at (±c, 0) where c² = a² − b². The sum of focal distances is constant at 2a.

Hyperbola asymptotes

For x²/a² − y²/b² = 1, the asymptotes are y = (b/a)x and y = −(b/a)x. The foci satisfy c² = a² + b², with the plus sign.

Classifying a general conic

For Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant B² − 4AC is zero for a parabola, negative for an ellipse and positive for a hyperbola.

Perpendicular gradients

Two non-vertical lines are perpendicular exactly when the product of their gradients is −1, so the perpendicular gradient is the negative reciprocal of the original.

Point-to-line distance

The perpendicular distance from (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| divided by √(a² + b²). Absolute value keeps the distance positive.

Tangent to a circle

At the point (x₁, y₁) on x² + y² = r², the tangent has equation x·x₁ + y·y₁ = r². Its gradient is the negative reciprocal of the radius gradient.

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ. Dividing through by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Compound angle formula for sine

sin(A ± B) = sin A cos B ± cos A sin B. The sign in the middle matches the sign in the bracket, unlike the cosine formula where the signs differ.

Double angle for cosine

cos 2θ = cos²θ − sin²θ, which also equals 2cos²θ − 1 and 1 − 2sin²θ. The last two forms are used to reduce powers in integration.

General solution for sine

The equation sin θ = k has solutions θ = nπ + (−1)^n·arcsin k for integer n. The (−1)^n factor alternates the reflection needed in each period.

General solution for cosine

The equation cos θ = k has solutions θ = 2nπ ± arccos k for integer n. The ± accounts for the symmetry of cosine about the horizontal axis of the unit circle.

Amplitude and period

For y = A sin(Bx + C) + D, the amplitude is |A| and the period is 2π/|B|. The graph is shifted vertically by D and horizontally by −C/B.

Harmonic form R sin(θ + α)

a cos θ + b sin θ can be written as R sin(θ + α), where R = √(a² + b²) and tan α = a/b. This reduces the equation to a single sine to solve.

Exact values at 30 and 60

sin 30° = cos 60° = 1/2; cos 30° = sin 60° = √3/2; tan 30° = 1/√3 and tan 60° = √3. These come from the half-equilateral triangle.

Reciprocal trigonometric functions

sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ = cos θ/sin θ. Each is undefined wherever its denominator equals zero.

Imaginary unit

The symbol i is defined by i² = −1. Powers cycle with period four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the pattern repeats.

Complex conjugate

The conjugate of z = a + bi is z̄ = a − bi. The product z·z̄ = a² + b² is real and equals the square of the modulus of z.

Modulus and argument

For z = a + bi, |z| = √(a² + b²) and arg z is the angle from the positive real axis, found with arctan(b/a) and corrected for the quadrant.

Polar form of a complex number

z = r(cos θ + i sin θ), written r cis θ or re^(iθ), where r = |z| and θ = arg z. Multiplying two such forms multiplies r and adds θ.

De Moivre's theorem

(r cis θ)^n = r^n cis(nθ) for integer n. It converts powers of complex numbers into a modulus power and an argument multiple.

Roots of a complex number

The nth roots of r cis θ are r^(1/n) cis((θ + 2kπ)/n) for k = 0 to n − 1. They form n equally spaced points on a circle of radius r^(1/n).

Conjugate root theorem

If a polynomial has real coefficients and a + bi is a root, then a − bi is also a root. Non-real roots therefore always occur in conjugate pairs.