IB Math AI HL flashcards

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

Geometric series sum to infinity

For a geometric series with common ratio r where |r|<1, the sum converges to u₁/(1−r). If |r|≥1 the series diverges and no finite sum exists.

Change of base formula

log_b x equals ln x divided by ln b, or log x divided by log b in any common base. It lets you evaluate logarithms on a calculator and solve equations with mixed bases.

Binomial coefficient

C(n,r)=n!/(r!(n−r)!) counts the number of ways to choose r objects from n without regard to order. It appears as the coefficient of a^(n−r)b^r in the expansion of (a+b)^n.

Modulus and argument of z

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

Compound interest model

An amount P invested at nominal annual rate r compounded k times per year grows to P(1+r/k)^(kt) after t years. The effective annual rate is (1+r/k)^k−1.

Arithmetic sequence nth term

The nth term is u₁+(n−1)d, where u₁ is the first term and d is the common difference. The sum of the first n terms is n/2 times the first plus last term.

Inverse function domain and range

The domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f. Graphically, f⁻¹ is the reflection of f in the line y=x.

Composite function domain

For (f∘g)(x)=f(g(x)), the domain consists of all x in the domain of g for which g(x) lies in the domain of f.

Quadratic discriminant

For ax²+bx+c=0, the discriminant b²−4ac determines the roots: positive means two distinct real roots, zero means one repeated root, negative means no real roots.

Vertex of a parabola

For y=ax²+bx+c, the vertex has x-coordinate −b/(2a); substituting gives the y-coordinate. The vertex is a maximum when a<0 and a minimum when a>0.

Horizontal asymptote rule

For a rational function, compare degrees: numerator lower gives y=0; degrees equal gives the ratio of leading coefficients; numerator higher gives no horizontal asymptote.

Exponential function properties

y=e^x has domain all real numbers, range y>0, and horizontal asymptote y=0. Its inverse y=ln x has domain x>0 and vertical asymptote x=0.

Graph transformation order

Apply reflections and stretches first, then translations. y=af(b(x−c))+d stretches vertically by a, horizontally by 1/b, then shifts right by c and up by d.

Sine and cosine rules

The sine rule states a/sin A=b/sin B=c/sin C. The cosine rule states a²=b²+c²−2bc cos A, used for two sides with included angle or three sides.

Arc length and sector area

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

Scalar product of vectors

a·b=|a||b|cos θ, where θ is the angle between the vectors. If a·b=0 the vectors are perpendicular, which is used to test orthogonality.

General solution for sine

If sin θ=k, then θ=arcsin k+2πn or θ=π−arcsin k+2πn for integer n. Both families are needed because sine is positive in two quadrants.

Double angle identities

sin 2θ=2 sin θ cos θ, and cos 2θ=cos²θ−sin²θ, which also equals 2cos²θ−1 or 1−2sin²θ. These are used to simplify and solve trigonometric equations.

Triangle area with sine

The area of a triangle equals ½ab sin C, where a and b are two sides and C is the included angle. For a circular segment, subtract this from the sector area.

Line equation in 3D

A line is written r=a+tb, where a is a position vector of a point on the line and b is a direction vector. The angle between lines uses their direction vectors.

Conditional probability formula

P(A|B)=P(A∩B)/P(B), valid when P(B)>0. Rearranging gives the multiplication rule P(A∩B)=P(B)P(A|B), useful for tree diagrams.

Binomial distribution mean and variance

For X~B(n,p), the mean is np and the variance is np(1−p), with standard deviation √(np(1−p)). The probability of exactly r successes is C(n,r)p^r(1−p)^(n−r).

Standardising a normal variable

Convert X~N(μ,σ²) to Z=(X−μ)/σ, which follows N(0,1). Then use standard normal tables or a calculator's normal cumulative function to find probabilities.

Effect of linear transformation

If y=ax+b is applied to every data value, the mean becomes a x̄+b and the standard deviation becomes |a|σ. Adding b shifts the data without changing spread.

Hypothesis test rejection rule

Reject the null hypothesis H₀ when the p-value is less than the significance level α. For a two-tailed test, compare the p-value with α directly and state the conclusion in context.

Coefficient of determination

The value r² gives the proportion of the variation in the dependent variable explained by the regression line. It ranges from 0 to 1, with values near 1 indicating a strong fit.

Chi-squared test statistic

The statistic is Σ(O−E)²/E, where E=(row total×column total)/grand total. Degrees of freedom are (r−1)(c−1) for an r by c table.

Derivative from first principles

The derivative is the limit as h approaches zero of [f(x+h)−f(x)]/h. It gives the gradient of the tangent and the instantaneous rate of change.

Chain rule

If y is a function of u and u is a function of x, then dy/dx=dy/du·du/dx. It is needed for composite functions such as sin(3x) or e^(x²).

Classifying stationary points

At a stationary point f′(x)=0. If f″(x)>0 it is a minimum; if f″(x)<0 it is a maximum; if f″(x)=0 the test is inconclusive and a sign change of f′ must be checked.

Definite integral as area

The definite integral from a to b of f(x) dx gives the signed area between the curve and the x-axis. Regions below the axis contribute negative values, so use absolute value for total area.

Kinematics with calculus

Velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity. Displacement is the integral of velocity; distance travelled integrates the speed |v|.