IB Maths: Analysis and Approaches formulas

Number and algebra

Arithmetic sequences and series

• nth term = first term + (n − 1) × common difference (u_n = u₁ + (n − 1)d)
• Common difference: d = u₂ − u₁ = u₃ − u₂, the same for every pair of neighbours
• Sum of the first n terms: S_n = (n/2) × (2u₁ + (n − 1)d)
• If you know the last term: S_n = (n/2) × (u₁ + u_n)
• Two terms given: u_m − u_k = (m − k)d, which gives d straight away
• A single term from sums: u_n = S_n − S_{n − 1}
• Σ from k = 1 to n has n terms; put k = 1 to get u₁ and k = n to get the last term

Geometric sequences, series and financial applications

• nth term = first term × ratio to the power (n − 1) (u_n = u₁ × r^{n − 1})
• Common ratio: r = u₂ ÷ u₁ = u₃ ÷ u₂
• Sum of the first n terms: S_n = u₁(rⁿ − 1)/(r − 1) = u₁(1 − rⁿ)/(1 − r), r ≠ 1
• Sum to infinity: S_∞ = u₁/(1 − r), only when |r| < 1
• Two terms given: u_m ÷ u_k = r^{m − k}
• Compound interest: FV = PV × (1 + r/(100k))^kn, with r% the nominal annual rate, k compounding periods per year, n years
• Depreciation at r% a year for n years: value = start value × (1 − r/100)ⁿ

Standard form, exponents and logarithms

• a^m × aⁿ = a^{m + n}; a^m ÷ aⁿ = a^{m − n}; (a^m)ⁿ = a^mn
• a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^m/n = (n-th root of a)^m
• a^x = b ⇔ x = log_a b (a > 0, a ≠ 1, b > 0)
• log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y
• log_a(x^m) = m log_a x
• log_a 1 = 0; log_a a = 1; ln(e^x) = x; e^{ln x} = x
• Change of base: log_a x = (log_b x)/(log_b a), for example (ln x)/(ln a)

Simple deductive proof

• = is true for some values (3x + 2 = 11 only when x = 3); ≡ is true for all values (3(x + 2) ≡ 3x + 6)
• Even number: 2n. Odd number: 2n + 1 or 2n − 1, where n is an integer
• Consecutive integers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2
• A multiple of k is written k × (an integer); factorise to show it
• One of any two consecutive integers is even, so n(n + 1) is always even
• Layout: LHS ≡ ... ≡ ... ≡ RHS, one working step per line, then a concluding sentence

The binomial theorem

• (a + b)ⁿ = aⁿ + ⁿC₁ a^{n − 1}b + ⁿC₂ a^{n − 2}b² + ... + bⁿ
• General term: ⁿC_r × a^{n − r} × b^r, for r = 0, 1, ..., n
• ⁿC_r = n!/(r! × (n − r)!); ⁿC₀ = ⁿC_n = 1; ⁿC_r = ⁿC_{n − r}
• Pascal's triangle: row 4 is 1, 4, 6, 4, 1; row 5 is 1, 5, 10, 10, 5, 1
• The expansion of (a + b)ⁿ has n + 1 terms
• If b is negative the signs alternate: (1 − x)³ = 1 − 3x + 3x² − x³
• Term independent of x: choose r so that the powers of x cancel to x⁰

Functions

Equations of straight lines

• gradient = change in y ÷ change in x (m = (y₂ − y₁)/(x₂ − x₁))
• y = mx + c: gradient m, y-intercept (0, c)
• Through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁)
• From ax + by + d = 0: gradient = −a/b; rearrange to y = ... to check
• x-intercept: put y = 0. y-intercept: put x = 0
• Parallel: m₁ = m₂. Perpendicular: m₁ × m₂ = −1, so m₂ = −1/m₁
• Midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Functions, graphs, composites and inverses

• (f ∘ g)(x) = f(g(x)): work from the inside out; in general f ∘ g ≠ g ∘ f
• To find f⁻¹: write y = f(x), swap x and y, then make y the subject
• (f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x, the identity function
• Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f
• √(expression) needs expression ≥ 0; 1/(expression) needs expression ≠ 0
• Zeros: solve f(x) = 0. y-intercept: find f(0)
• Even function, f(−x) = f(x): graph symmetric about the y-axis

Quadratic functions, equations and inequalities

• Axis of symmetry: x = −b/(2a); from factorised form, x = (p + q)/2
• Vertex form a(x − h)² + k: vertex (h, k). Note the sign: (x − 3)² gives h = 3
• Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c
• Quadratic formula: x = (−b ± √(b² − 4ac))/(2a)
• Discriminant Δ = b² − 4ac: Δ > 0 two distinct real roots; Δ = 0 two equal real roots; Δ < 0 no real roots
• For a > 0 with roots p < q: ax² + bx + c > 0 when x < p or x > q; < 0 when p < x < q
• A line and a curve: set them equal; Δ < 0 no meeting, Δ = 0 tangent, Δ > 0 two points

Reciprocal and rational functions

• For y = (ax + b)/(cx + d): vertical asymptote where cx + d = 0, so x = −d/c
• Horizontal asymptote: y = a/c (the ratio of the coefficients of x)
• x-intercept: numerator = 0, so x = −b/a. y-intercept: put x = 0, so y = b/d
• Domain: x ≠ −d/c. Range: y ≠ a/c
• y = 1/x: asymptotes x = 0 and y = 0; domain x ≠ 0; range y ≠ 0
• Self-inverse: f(f(x)) = x, and the graph is symmetric in the line y = x

Exponential and logarithmic functions; solving equations

• y = a^x: passes through (0, 1); asymptote y = 0; domain all real x; range y > 0
• y = log_a x: passes through (1, 0) and (a, 1); asymptote x = 0; domain x > 0; range all real y
• Inverses: log_a(a^x) = x and a^{log_a x} = x; ln(e^x) = x and e^{ln x} = x
• y = e^x + c has horizontal asymptote y = c
• A = A₀e^kt: A₀ is the value at t = 0; k > 0 growth, k < 0 decay
• Doubling time = (ln 2)/k for growth e^kt; half-life = (ln 2)/k for decay e^−kt
• After solving a log equation, reject any answer that makes the inside of a log zero or negative

Transformations of graphs

• y = f(x) + b: translation b units up (down if b is negative); (x, y) → (x, y + b)
• y = f(x − a): translation a units to the right; y = f(x + a): a units to the left
• y = −f(x): reflection in the x-axis; (x, y) → (x, −y)
• y = f(−x): reflection in the y-axis; (x, y) → (−x, y)
• y = pf(x): vertical stretch with scale factor p; (x, y) → (x, py)
• y = f(qx): horizontal stretch with scale factor 1/q; (x, y) → (x ÷ q, y)
• For y = pf(x) + b, do the stretch or reflection first and the vertical translation last

Geometry and trigonometry

Three-dimensional geometry

• distance = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
• midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
• volume of a pyramid = ⅓ × base area × vertical height
• cone: volume = ⅓πr²h; curved surface area = πrl, where l = √(r² + h²)
• sphere: volume = (4/3)πr³; surface area = 4πr²
• hemisphere: volume = (2/3)πr³; curved area = 2πr²; total area with the flat face = 3πr²
• angle between a line and a plane = angle between the line and its projection on the plane

Trigonometry of triangles and its applications

• sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent
• sine rule: a/sin A = b/sin B = c/sin C
• cosine rule for a side: c² = a² + b² − 2ab cos C
• cosine rule for an angle: cos C = (a² + b² − c²)/(2ab)
• area of a triangle = ½ab sin C, where C is the angle between sides a and b
• The largest angle is opposite the longest side
• Keep the calculator in degree mode when the angles are in degrees

Radians, the unit circle and exact values

• π radians = 180°; degrees to radians: × π/180; radians to degrees: × 180/π
• arc length = radius × angle (l = rθ) and sector area = ½r²θ, with θ in radians
• On the unit circle the point is (cos θ, sin θ), and tan θ = sin θ ÷ cos θ
• sin of 0, π/6, π/4, π/3, π/2 = 0, ½, √2/2, √3/2, 1; cos takes the same values in reverse order
• tan of 0, π/6, π/4, π/3 = 0, 1/√3, 1, √3; tan(π/2) is undefined
• Positive ratios: all in quadrant 1, only sin in quadrant 2, only tan in quadrant 3, only cos in quadrant 4
• Ambiguous case: the second angle 180° − C is valid only if it and the given angle add to less than 180°

Trigonometric identities and equations

• sin²θ + cos²θ = 1 and tan θ = sin θ ÷ cos θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• sin x = k: if one solution is α, another is π − α
• cos x = k: if one solution is α, another is 2π − α
• tan x = k: if one solution is α, another is α + π
• For sin bx, cos bx or tan bx, find all values of bx in the interval multiplied by b, then divide each by b

Circular functions and their graphs

• y = sin x and y = cos x: amplitude 1, period 2π; y = tan x: period π, asymptotes at x = π/2, 3π/2, ...
• For y = a sin(b(x + c)) + d: amplitude = |a| and period = 2π ÷ b (for tan, period = π ÷ b)
• The middle line (principal axis) is y = d; +c is a translation of c units to the left
• maximum = d + |a| and minimum = d − |a|
• From a graph: a = (maximum − minimum) ÷ 2 and d = (maximum + minimum) ÷ 2
• y = sin(bx) is a horizontal stretch of y = sin x with scale factor 1/b
• sin x is greatest at x = π/2; cos x is greatest at x = 0 and x = 2π

Statistics and probability

Collecting data and sampling

• Simple random: every possible sample of that size has the same chance of being chosen
• Systematic: random starting point, then every kth member of a list
• Stratified: random samples from each subgroup, in proportion to the size of the subgroup
• Quota: a fixed number from each subgroup, chosen by the interviewer and not at random
• Convenience: the people who are easiest to reach
• number from a subgroup in a stratified sample = (subgroup size ÷ population size) × sample size
• Outlier: a value below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR

Presenting and summarising data

• mean = sum of values ÷ number of values (x̄ = Σfx ÷ n); for grouped data use the class midpoints as x
• Median: the middle value of the ordered data; Q₁ and Q₃ are the medians of the lower and upper halves
• range = largest − smallest; interquartile range = Q₃ − Q₁
• variance = Σf(x − x̄)² ÷ n; standard deviation σ = √variance
• Add k to every value: mean, median and mode increase by k; range, IQR, standard deviation and variance do not change
• Multiply every value by k: mean and standard deviation are multiplied by k (|k| for the spread); variance is multiplied by k²
• Histogram with equal class widths: the height of a bar is the frequency of the class

Correlation and regression

• −1 ≤ r ≤ 1; r = 1 is perfect positive, r = −1 is perfect negative, r = 0 is no linear correlation
• r close to 1 or −1 is strong; r close to 0 is weak
• Both regression lines, and a line of best fit drawn by eye, pass through the mean point (x̄, ȳ)
• To estimate y from x use the line of y on x; to estimate x from y use the line of x on y
• In y = ax + b, the gradient a is the change in y for each increase of 1 in x; b is the value of y when x = 0
• gradient of a line through two points = (y₂ − y₁)/(x₂ − x₁)
• Correlation does not show cause; a third variable may affect both

Probability and combined events

• P(A) = n(A)/n(U): outcomes in A ÷ total number of equally likely outcomes
• relative frequency = number of times the event happens ÷ number of trials
• complement: P(A′) = 1 − P(A)
• expected number of occurrences = number of trials × P(A)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)
• Tree diagram: multiply along the branches, add the results of different routes; without replacement the second probabilities change

Conditional probability and independence

• P(A | B) = P(A ∩ B) ÷ P(B)
• Rearranged: P(A ∩ B) = P(B) × P(A | B)
• Tree diagram: multiply along the branches, add between different routes
• Independent events: P(A ∩ B) = P(A) × P(B), and P(A | B) = P(A)
• To test for independence, work out P(A) × P(B) and compare it with P(A ∩ B)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive means P(A ∩ B) = 0, which is not the same as independent

Discrete random variables and the binomial distribution

• ΣP(X = x) = 1 for every probability distribution
• Expected value: E(X) = Σ x × P(X = x)
• Fair game: expected gain = 0
• Binomial: P(X = r) = ⁿC_r × p^r × (1 − p)^{n − r}
• Binomial mean: E(X) = np
• Binomial variance: Var(X) = np(1 − p); standard deviation = √(np(1 − p))
• P(X ≥ 1) = 1 − P(X = 0)

The normal distribution

• Total area under the curve = 1; P(X < μ) = P(X > μ) = 0.5
• Symmetry: P(X < μ − k) = P(X > μ + k)
• About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ
• Standardise: z = (x − μ)/σ, where Z ~ N(0, 1)
• Rearranged: x = μ + zσ
• A value below the mean has a negative z; a probability below 0.5 for P(X < x) means x is below the mean
• Unknown μ or σ: find z from the probability with inverse normal, then use z = (x − μ)/σ

Calculus

Introduction to differentiation, tangents and normals

• If y = axⁿ then dy/dx = anx^{n − 1}; differentiate a sum term by term
• The derivative of a constant is 0; the derivative of ax is a
• Rewrite first: 1/xⁿ = x⁻ⁿ
• Increasing where f′(x) > 0, decreasing where f′(x) < 0, stationary where f′(x) = 0
• Tangent at (x₁, y₁): y − y₁ = m(x − x₁), where m = f′(x₁)
• Normal: same point, gradient −1/m
• Parallel lines have equal gradients

Further differentiation

• d/dx(xⁿ) = nx^{n − 1} for any rational n, so d/dx(√x) = 1/(2√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
• Chain rule: dy/dx = (dy/du) × (du/dx)
• So d/dx(e^u) = u′e^u, d/dx(ln u) = u′/u, d/dx(sin u) = u′cos u, d/dx(cos u) = −u′sin u
• Product rule: if y = uv then dy/dx = u(dv/dx) + v(du/dx)
• Quotient rule: if y = u/v then dy/dx = [v(du/dx) − u(dv/dx)]/v²

Second derivative, stationary points and optimisation

• Stationary point: solve f′(x) = 0, then substitute into f(x) to find y
• f′(a) = 0 and f″(a) < 0: local maximum at x = a
• f′(a) = 0 and f″(a) > 0: local minimum at x = a
• If f″(a) = 0 the test fails: check the sign of f′(x) on each side instead
• Point of inflexion: f″(x) = 0 and f″(x) changes sign
• Graph of f′ crossing the x-axis from + to − means a maximum of f; from − to + means a minimum
• Optimisation: use the constraint to remove one variable before differentiating

Anti-differentiation and indefinite integrals

• ∫axⁿ dx = ax^{n + 1}/(n + 1) + C, for n ≠ −1
• ∫(1/x) dx = ln|x| + C
• ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C
• ∫e^x dx = e^x + C
• Linear inside: ∫e^ax dx = (1/a)e^ax + C; ∫cos ax dx = (1/a) sin ax + C; ∫sin ax dx = −(1/a) cos ax + C
• ∫1/(ax + b) dx = (1/a) ln|ax + b| + C
• ∫g′(x)/g(x) dx = ln|g(x)| + C; ∫g′(x)e^g(x) dx = e^g(x) + C

Definite integrals and areas

• ∫_a^b f(x) dx = F(b) − F(a), where F′(x) = f(x)
• ∫_a^b k f(x) dx = k∫_a^b f(x) dx, and the integral of a sum is the sum of the integrals
• ∫_a^b k dx = k(b − a) for a constant k
• Area between a curve and the x-axis = ∫_a^b |f(x)| dx
• Below the axis the integral is negative: take its size as the area
• Area between two curves = ∫_a^b (upper − lower) dx, with a and b where the curves meet
• An area is never negative

Kinematics

• v = ds/dt and a = dv/dt = d²s/dt²
• s = ∫v dt and v = ∫a dt, each with a constant found from the given values
• At rest: v = 0
• Changes direction: v = 0 and v changes sign
• Displacement from t₁ to t₂ = ∫v dt between those times
• Total distance from t₁ to t₂ = ∫|v| dt between those times
• Speed = |v|; units are m, m s⁻¹ and m s⁻²