International Baccalaureate Diploma (IB) Group 5: Mathematics (Analysis and Approaches) Syllabus

State the quadratic formula and what the discriminant tells you.

x = (-b +/- sqrt(b^2 - 4ac))/(2a). Discriminant D = b^2 - 4ac: D > 0 two real roots, D = 0 one repeated root, D < 0 no real roots.

What is the factor theorem and the remainder theorem?

Remainder theorem: the remainder when P(x) is divided by (x - a) is P(a). Factor theorem: (x - a) is a factor of P(x) if and only if P(a) = 0.

What characterises a vertical asymptote and a horizontal asymptote of a rational function?

Vertical asymptote: x-value where the denominator is zero (numerator nonzero). Horizontal asymptote: the y-value the function approaches as x tends to +/- infinity (compare degrees of numerator and denominator).

State the key properties of the exponential function f(x) = a^x (a > 1).

Domain all real x, range y > 0, passes through (0, 1), increasing, with horizontal asymptote y = 0.

What is the natural exponential and natural logarithm relationship?

e^x and ln x are inverse functions: e^(ln x) = x (x > 0) and ln(e^x) = x; e is approximately 2.718.

Convert between degrees and radians.

pi radians = 180 degrees. To convert degrees to radians multiply by pi/180; to convert radians to degrees multiply by 180/pi.

State the formulas for arc length and area of a sector with angle theta in radians.

Arc length l = r theta. Sector area A = (1/2) r^2 theta.

State the Pythagorean trigonometric identity and the double angle formulas for sine and cosine.

sin^2 x + cos^2 x = 1. sin 2x = 2 sin x cos x. cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x.

Define tan x in terms of sin and cos, and give its identity with secant.

tan x = sin x / cos x; and 1 + tan^2 x = sec^2 x.

State the sine rule and the cosine rule for a triangle.

Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: c^2 = a^2 + b^2 - 2ab cos C.

State the formula for the area of a triangle given two sides and the included angle.

Area = (1/2) ab sin C, where a and b are two sides and C is the angle between them.

(HL) What is the scalar (dot) product of two vectors, and what does it indicate about the angle?

a . b = |a||b| cos theta = a1 b1 + a2 b2 + a3 b3. If a . b = 0 (and vectors nonzero), the vectors are perpendicular.

(HL) What is the vector (cross) product of two vectors and what does its magnitude represent?

a x b is a vector perpendicular to both a and b with magnitude |a||b| sin theta, which equals the area of the parallelogram they span.

(HL) Give the vector equation of a line in 3D.

r = a + t*d, where a is the position vector of a point on the line, d is the direction vector, and t is a scalar parameter.

What is the difference between the median, mean, and mode?

Mean: sum of values divided by count (arithmetic average). Median: middle value when ordered. Mode: most frequently occurring value.

How are quartiles and the interquartile range (IQR) defined?

Q1 is the lower quartile (25th percentile), Q3 the upper quartile (75th percentile), and IQR = Q3 - Q1, a measure of spread of the middle 50%.

What does Pearson's correlation coefficient r indicate, including its range and meaning of sign?

r measures the strength and direction of a linear relationship; -1 <= r <= 1. r near +1 strong positive, near -1 strong negative, near 0 little/no linear correlation.

What is the equation of a regression line of y on x, and what does it pass through?

y = a x + b (least squares line minimising vertical distances); it always passes through the mean point (x-bar, y-bar).

State the addition rule and the formula for conditional probability.

P(A or B) = P(A) + P(B) - P(A and B). Conditional: P(A|B) = P(A and B)/P(B).

What does it mean for two events to be independent, and what is the resulting multiplication rule?

Events A and B are independent if one occurring does not affect the other: P(A and B) = P(A) * P(B), equivalently P(A|B) = P(A).

State the mean and variance of a binomial distribution X ~ B(n, p).

Mean E(X) = np; Variance Var(X) = np(1 - p). P(X = r) = C(n,r) p^r (1-p)^(n-r).

What are the defining properties of a normal distribution and the standardisation formula?

Symmetric, bell-shaped, defined by mean mu and standard deviation sigma; about 68/95/99.7% of data lie within 1/2/3 sigma. Standardise with z = (x - mu)/sigma.

For a discrete random variable, how do you find the expected value E(X)?

E(X) = sum of x * P(X = x) over all values x. The probabilities must sum to 1.

State the formal limit definition of the derivative.

f'(x) = lim as h -> 0 of [f(x + h) - f(x)] / h.

State the power rule, product rule, quotient rule, and chain rule for differentiation.

Power: d/dx(x^n) = n x^(n-1). Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv')/v^2. Chain: d/dx f(g(x)) = f'(g(x)) g'(x).

Give the derivatives of sin x, cos x, e^x, and ln x.

d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(e^x) = e^x; d/dx(ln x) = 1/x.

How do you classify a stationary point using the first and second derivatives?

Stationary points occur where f'(x) = 0. If f''(x) > 0 it is a local minimum; if f''(x) < 0 a local maximum; if f''(x) = 0 test further (could be a point of inflexion).

State the power rule for integration and the role of the constant of integration.

Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1; C is the arbitrary constant added for indefinite integrals.

How do you find the area between a curve y = f(x) and the x-axis, and the volume of revolution about the x-axis?

Area = integral from a to b of f(x) dx. Volume about x-axis = pi * integral from a to b of [f(x)]^2 dx.

(HL) What is the method of solving a separable differential equation dy/dx = f(x)g(y)?

Separate variables and integrate both sides: integral of (1/g(y)) dy = integral of f(x) dx, then solve for y including the constant.

(HL) State the general Maclaurin series and give the expansion of e^x.

Maclaurin: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... . For e^x: e^x = 1 + x + x^2/2! + x^3/3! + ...