IB Math AI SL flashcards

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

Compound interest formula

Future value FV = PV(1 + r/n)^(nt), where r is the annual rate as a decimal, n the number of compounds per year and t the number of years.

Sum of infinite geometric series

S∞ = u₁/(1 − r), valid only when the common ratio satisfies |r| < 1; otherwise the series diverges and no finite sum exists.

Arithmetic sequence nth term

uₙ = u₁ + (n − 1)d, where u₁ is the first term and d the constant common difference between consecutive terms.

Change of base for logarithms

log_a b = log b / log a, using any common base. This lets you solve exponential equations such as a^x = b by writing x = log b / log a.

Binomial coefficient

C(n,k) = n!/(k!(n−k)!) counts ways to choose k items from n without regard to order; it is the coefficient of the k-th term in (a + b)^n.

Percentage change formula

Percentage change = (new value − old value)/old value × 100%. A positive result is an increase, a negative result a decrease.

Quadratic discriminant

For ax² + bx + c = 0, Δ = b² − 4ac. Two distinct real roots if Δ > 0, one repeated root if Δ = 0, no real roots if Δ < 0.

Vertex of a parabola

For y = ax² + bx + c the axis of symmetry is x = −b/(2a); substituting this x gives the vertex, which is a maximum if a < 0 and a minimum if a > 0.

Inverse function rule

To find f⁻¹, swap x and y then solve for y. The graph of f⁻¹ is f reflected in y = x, and its domain equals the range of f.

Exponential model y = ka^x

k is the value when x = 0. If a > 1 the model shows growth; if 0 < a < 1 it shows decay. The x-axis is a horizontal asymptote.

Composite function f∘g

(f∘g)(x) = f(g(x)): apply g first, then f. It is defined only for x where g(x) lies in the domain of f.

Vertical asymptote of a rational function

A value x = c where the denominator equals zero but the factor does not cancel with the numerator; the function is undefined and grows without bound there.

Sine rule

a/sin A = b/sin B = c/sin C in any triangle. Use it when you know a side with its opposite angle plus one more side or angle.

Cosine rule

a² = b² + c² − 2bc cos A. Use it when given two sides and the included angle, or all three sides, to find a missing side or angle.

Triangle area with sine

Area = ½ab sin C, where a and b are two sides and C is the angle between them. The angle must be the included one.

Arc length and sector area

With angle θ in radians: arc length s = rθ and sector area A = ½r²θ. Convert degrees to radians by multiplying by π/180.

Angle between line and plane

The angle between a line and a plane is the angle between the line and its perpendicular projection onto that plane, found using a right triangle.

Amplitude and period

For y = a sin(bx) + d, amplitude is |a|, period is 360°/b (or 2π/b in radians), and d is the vertical shift of the central axis.

Volume of a sphere

V = 4/3 πr³ and surface area S = 4πr². For a hemisphere, halve the volume and add the flat circular face to the surface area.

Standard deviation behaviour

Adding a constant to every value leaves the standard deviation unchanged; multiplying every value by k multiplies the standard deviation by |k|.

Interquartile range and outliers

IQR = Q3 − Q1. A value is commonly flagged as an outlier if it lies below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.

Histogram frequency density

For unequal class widths, frequency density = frequency / class width. The area of each bar equals its class frequency, so total area equals n.

Correlation coefficient r

A number between −1 and 1 measuring the strength and direction of a linear relationship. Its square, r², is the proportion of variation explained.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B), for P(B) > 0. Rearranging gives the multiplication rule P(A ∩ B) = P(A|B)P(B).

Independent events

Events A and B are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A): knowing one occurred does not change the other's probability.

Binomial distribution mean and variance

For X ~ B(n, p): mean E(X) = np and variance Var(X) = np(1 − p). The standard deviation is the square root of the variance.

Normal standardisation

If X ~ N(μ, σ²), then z = (x − μ)/σ follows N(0, 1). Use it, or the GDC normal functions, to find probabilities and inverse values.

Derivative as gradient

f′(a) = lim(h→0)[f(a + h) − f(a)]/h is the gradient of the tangent to y = f(x) at x = a.

Second derivative test

At a stationary point where f′(x) = 0: f″(x) > 0 gives a local minimum, f″(x) < 0 a local maximum, and f″(x) = 0 is inconclusive.

Equation of a tangent line

At x = a the tangent is y − f(a) = f′(a)(x − a). The normal at the same point has gradient −1/f′(a).

Definite integral and area

∫ from a to b of f(x) dx gives the signed area between the curve and the x-axis. For total area, split the integral where f(x) crosses the axis.

Kinematics with calculus

Velocity is the derivative of displacement and acceleration the derivative of velocity. Displacement is ∫v dt; total distance travelled is ∫|v| dt.