HKDSE Maths M2 flashcards

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

Principle of mathematical induction

To prove a statement for all positive integers: verify the base case n = 1, assume it holds for n = k, then use that assumption to prove it holds for n = k+1.

Binomial general term

In the expansion of (a+b)^n, the term containing b^r is C(n,r) a^(n−r) b^r, where C(n,r) denotes the binomial coefficient n choose r.

Term independent of x

In a binomial expansion, form the general term, collect the power of x, set that exponent equal to zero, solve for r, then substitute r back into the general term.

Factor theorem

A polynomial f(x) has (x−a) as a factor exactly when f(a) = 0; this links factorisation of polynomials to evaluating them at a single point.

Remainder theorem

When a polynomial f(x) is divided by (x−a), the remainder equals f(a); dividing by (ax−b) gives remainder f(b/a).

Polynomial division algorithm

For polynomials f and d with d not zero, there exist unique quotient q and remainder r with f = dq + r and deg(r) < deg(d).

Discriminant of quadratic

For ax²+bx+c = 0, the value b²−4ac is positive for two distinct real roots, zero for a repeated root, and negative for non-real complex roots.

Conjugate root pairs

If a polynomial has real coefficients and a non-real number p+qi is a root, then p−qi is also a root, so non-real roots occur in conjugate pairs.

Sum of geometric series

For a geometric series with first term a and ratio r ≠ 1, the sum of n terms is a(1−r^n)/(1−r); it converges to a/(1−r) when |r| < 1.

Infinite geometric series convergence

The infinite geometric series a + ar + ar² + … has a finite sum a/(1−r) exactly when |r| < 1; otherwise the series diverges.

Pascal's identity

The binomial coefficients satisfy C(n+1,r) = C(n,r−1) + C(n,r), which underlies the construction of Pascal's triangle and many coefficient identities.

Symmetry of binomial coefficients

For 0 ≤ r ≤ n, C(n,r) = C(n,n−r); consequently the coefficient of x^r and of x^(n−r) in (1+x)^n are equal.

Ranks and linear systems

A linear system has a unique solution when coefficient and augmented matrices have equal rank equal to the number of unknowns, and infinitely many solutions when the common rank is smaller.

Euclidean algorithm for polynomials

Repeatedly divide the divisor by the remainder until the remainder is zero; the last non-zero remainder, up to a constant factor, is the greatest common divisor.

Absolute value inequality

For a > 0, the inequality |x| < a is equivalent to −a < x < a, while |x| > a is equivalent to x > a or x < −a.

Derivative from first principles

The derivative at x is the limit as h tends to zero of [f(x+h) − f(x)]/h, provided this limit exists; it measures the instantaneous rate of change.

Product rule

If y = uv, where u and v are differentiable functions of x, then dy/dx = u(dv/dx) + v(du/dx).

Quotient rule

If y = u/v with v not zero, then dy/dx = [v(du/dx) − u(dv/dx)]/v²; the numerator order matters and cannot be reversed.

Chain rule

If y is a function of u and u is a function of x, then dy/dx = (dy/du)(du/dx), which handles composite functions such as (3x+1)^5.

Derivative of e^x

The exponential function e^x is its own derivative, so d/dx(e^x) = e^x; combined with the chain rule, d/dx(e^(kx)) = k e^(kx).

Derivative of ln x

For x > 0, d/dx(ln x) = 1/x; with the chain rule, d/dx(ln f(x)) = f′(x)/f(x) whenever f(x) is positive.

Derivatives of sine and cosine

The derivative of sin x is cos x, and the derivative of cos x is −sin x; the derivative of tan x is sec²x wherever cos x is non-zero.

Implicit differentiation

When y is defined implicitly by an equation in x and y, differentiate every term with respect to x, applying the chain rule to y-terms, then solve algebraically for dy/dx.

Second derivative test

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

Fundamental theorem of calculus

If F is an antiderivative of f on an interval, then the definite integral of f from a to b equals F(b) − F(a).

Reversing integration limits

Swapping the upper and lower limits of a definite integral changes the sign of the value: integrating from b to a equals minus integrating from a to b.

Area between two curves

If the upper curve is y = f(x) and the lower is y = g(x) between x = a and x = b, the enclosed area is the integral of f(x) − g(x) from a to b.

Integration by substitution

Replace an inner expression by u, rewrite dx as du divided by du/dx, integrate in u, then substitute back; for definite integrals also convert the limits.

Volume about the x-axis

Rotating the region under y = f(x) from x = a to x = b about the x-axis gives volume equal to π times the integral of y² from a to b.

Volume about the y-axis

Rotating the region between x = g(y) and the y-axis from y = c to y = d gives volume equal to π times the integral of x² from c to d.

Stationary points

Stationary points occur where the first derivative equals zero or fails to exist; they are classified as maxima, minima, or points of inflection.

Point of inflection

A point where the concavity of the curve changes sign; it is located by testing where the second derivative changes sign, not merely where it equals zero.