IB Maths: Applications and Interpretation formulas

Number and algebra

Standard form, approximation and errors

• Standard form: a × 10^k with 1 ≤ a < 10 and k an integer
• Multiply: multiply the a parts and add the powers. Divide: divide the a parts and subtract the powers
• Significant figures are counted from the first non-zero digit, so 0.004 52 has 3 significant figures
• Bounds: stated value ± half of the rounding unit, e.g. 8.4 to 1 decimal place lies from 8.35 to 8.45
• Largest possible product or sum uses the upper bounds; smallest uses the lower bounds
• percentage error = |(approximate − exact) ÷ exact| × 100% (ε = |(v_A − v_E)/v_E| × 100%)
• To estimate, round each number to 1 significant figure first, then calculate

Arithmetic sequences and series

• common difference = any term − the term before it (d = u_n+1 − u_n)
• nth term: u_n = u₁ + (n − 1)d
• Sum of the first n terms: S_n = (n/2)(2u₁ + (n − 1)d)
• Sum, if you know the last term: S_n = (n/2)(u₁ + u_n)
• Given two terms, subtract them to find d: u₈ − u₃ = 5d
• Simple interest: total value = P + P × (r/100) × n, with r in percent per year and n in years
• An arithmetic model fits when the differences between consecutive values are roughly constant

Geometric sequences and series

• common ratio = any term ÷ the term before it (r = u_n+1 ÷ u_n)
• nth term: u_n = u₁ × r^{n − 1}
• Sum of the first n terms: S_n = u₁(rⁿ − 1)/(r − 1), for r ≠ 1
• Same sum, easier when r < 1: S_n = u₁(1 − rⁿ)/(1 − r)
• Increase of p% per period: r = 1 + p/100. Decrease of p%: r = 1 − p/100
• Given two terms, divide them to find r: u₅ ÷ u₂ = r³
• Value after n periods of growth = starting value × rⁿ, because the starting value is at time 0

Financial applications: compound interest, depreciation, loans and annuities

• Compound interest: FV = PV × (1 + r/(100k))^kn, with r = nominal annual rate in %, k = compounding periods per year, n = years
• k = 1 yearly, 2 half-yearly, 4 quarterly, 12 monthly
• Depreciation at r% per year: value = PV × (1 − r/100)ⁿ
• TVM solver: N = number of payments, I% = nominal annual rate, PV, PMT, FV, P/Y and C/Y = payments and compounding periods per year
• Loan fully repaid: FV = 0. Savings annuity starting from nothing: PV = 0
• total interest on a loan = payment × number of payments − amount borrowed
• Real value: use real rate ≈ interest rate − inflation rate, then apply the compound interest formula

Exponents and logarithms

• a^m × aⁿ = a^{m + n} and a^m ÷ aⁿ = a^{m − n}
• (a^m)ⁿ = a^mn and (ab)ⁿ = aⁿbⁿ
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ, for a ≠ 0
• a^x = b is equivalent to log_a b = x (a > 0, a ≠ 1, b > 0)
• 10^x = b gives x = log₁₀ b. e^x = b gives x = ln b
• To solve a^x = b: x = (log b) ÷ (log a)
• If log₁₀ y = c then y = 10^c

Systems of linear equations and polynomial equations (technology)

• Define each unknown clearly, e.g. a = number of adult tickets, c = number of child tickets
• Write one equation for each fact: usually one for the total number and one for the total cost
• Line up the equations as ax + by + cz = d before entering the coefficients, with 0 for a missing variable
• Rearrange a polynomial equation to the form ... = 0 before using the solver
• A polynomial of degree n has at most n real solutions: a quadratic up to 2, a cubic up to 3
• Reject solutions that make no sense in context or lie outside the given domain
• Check by substituting the answers back into the original equations

Functions

Equations of straight lines

• gradient = change in y ÷ change in x (m = (y₂ − y₁)/(x₂ − x₁))
• Gradient-intercept form: y = mx + c, with gradient m and y-intercept c
• Point-gradient form: y − y₁ = m(x − x₁)
• General form ax + by + d = 0: rearrange to y = −(a/b)x − d/b, so the gradient is −a/b
• Parallel lines have equal gradients: m₁ = m₂
• Perpendicular lines: m₁ × m₂ = −1, so the second gradient is −1/m₁
• The x-intercept is found by setting y = 0

Functions, graphs and their key features

• f(a) means put x = a into the rule. Use brackets for negatives: (−2)² = 4
• To find f⁻¹(x): write y = f(x), swap x and y, then make y the subject
• The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f
• Zeros (x-intercepts): solve f(x) = 0. y-intercept: work out f(0)
• Quadratic y = ax² + bx + c: axis of symmetry and vertex at x = −b/(2a)
• y = a + b/(x − c) has vertical asymptote x = c and horizontal asymptote y = a
• For the range on a restricted domain, check the end points and any turning point inside the domain

Linear, quadratic, cubic and variation models

• Linear model: f(x) = mx + c, with rate of change m and starting value c
• Piecewise model: first decide which piece the x value belongs to, then use only that rule
• Quadratic f(x) = ax² + bx + c: axis of symmetry x = −b/(2a); the vertex is a maximum if a < 0 and a minimum if a > 0
• The axis of symmetry is halfway between the two zeros, or between any two points with the same y value
• y varies directly as xⁿ: y = kxⁿ
• y varies inversely as xⁿ: y = k/xⁿ, so y × xⁿ is constant
• The domain of a model is limited by the context, e.g. lengths and times cannot be negative

Exponential and sinusoidal models; the modelling process

• Exponential model: y = k × a^x + c. With k > 0 it is growth when a > 1 and decay when 0 < a < 1.
• Horizontal asymptote of y = k × a^x + c is y = c; the starting value (x = 0) is k + c.
• Constant ratio between y values for equal steps in x means exponential; constant difference means linear.
• For y = a sin(bx) + d or y = a cos(bx) + d: amplitude = |a| = (maximum − minimum) ÷ 2.
• Principal axis: y = d, where d = (maximum + minimum) ÷ 2.
• Period = 360° ÷ b, with x in degrees.
• Maximum value = d + |a|, minimum value = d − |a|.

Geometry and trigonometry

Three-dimensional geometry: distance, solids and angles

• Distance: d = √((x₁ − x₂)² + (y₁ − y₂)² + (z₁ − z₂)²)
• Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
• Volume of a right pyramid = (1/3) × base area × height
• Right cone: volume = (1/3)πr²h; curved surface area = πrl, where slant height l = √(r² + h²)
• Sphere: volume = (4/3)πr³; surface area = 4πr²
• Hemisphere: volume = (2/3)πr³; curved surface = 2πr²; solid hemisphere total surface = 3πr²
• Space diagonal of a cuboid = √(length² + width² + height²)

Right-angled and non-right-angled trigonometry

• 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; the angles add up to 180°.

Applications of trigonometry: elevation, bearings, arcs and sectors

• Angle of depression from the top = angle of elevation from the bottom (alternate angles between horizontal lines).
• Bearings: clockwise from north, three figures (e.g. 070°, 250°).
• Back bearing: add 180° if the bearing is less than 180°, subtract 180° if it is more.
• Arc length = (θ/360) × 2πr, with θ in degrees
• Area of a sector = (θ/360) × πr²
• Perimeter of a sector = arc length + 2r
• Area of a segment = area of sector − area of triangle = (θ/360) × πr² − ½r² sin θ

Perpendicular bisectors and Voronoi diagrams

• Perpendicular bisector: passes through the midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) with gradient −1 ÷ (gradient of the line joining the points).
• Edge: every point on it is equidistant from the two sites on either side.
• Vertex: equidistant from the three (or more) nearest sites, where their cells meet.
• Adding a site creates one new cell; only the cells next to it get smaller and the rest of the diagram does not change.
• Nearest neighbour interpolation: a point takes the value of the site whose cell it lies in.
• Toxic waste dump problem: the point furthest from all sites is at a vertex (inside the region); compare the vertices and choose the one with the greatest distance to its nearest sites.

Statistics and probability

Collecting data and sampling

• Interquartile range: IQR = Q₃ − Q₁
• Outlier: a value less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR.
• Simple random: every member, and every possible sample of size n, has an equal chance of being chosen.
• Systematic: a random start, then every kth member of a list.
• Stratified: the population is split into groups (strata) and each group is sampled at random in proportion to its size: number from a group = (group size ÷ population size) × sample size.
• Quota: a fixed number is wanted from each group, but the interviewer chooses who, so it is not random.
• Convenience: the people easiest to reach are used; quick but likely to be biased.

Presenting data: histograms, cumulative frequency and box plots

• Cumulative frequency is plotted against the upper boundary of each class.
• For n values on a cumulative frequency graph: median at n/2, Q₁ at n/4, Q₃ at 3n/4.
• The pth percentile is read at a cumulative frequency of (p/100) × n.
• Number of values greater than x = n − (cumulative frequency at x).
• IQR = Q₃ − Q₁ = length of the box; it holds the middle 50% of the data.
• Compare distributions with the median (average) and the IQR (spread): a smaller IQR means more consistent values.
• Longer right whisker and median nearer Q₁: positive (right) skew. The reverse: negative (left) skew.

Measures of central tendency and dispersion

• Mean = sum of values ÷ number of values; from a frequency table, mean = Σfx ÷ n, where n = Σf.
• Median: the ((n + 1)/2)th value in order; for an even n, the mean of the two middle values.
• Grouped data: use mid-interval values for x; the modal class is the class with the highest frequency (equal widths).
• Range = maximum − minimum; IQR = Q₃ − Q₁
• Variance = (standard deviation)² = σ²
• Add a constant k to every value: the mean increases by k; the standard deviation and variance do not change.
• Multiply every value by k: the mean and standard deviation are multiplied by k (|k| for the standard deviation); the variance is multiplied by k².

Correlation and regression

• −1 ≤ r ≤ 1: r near 1 is strong positive, r near −1 is strong negative, r near 0 is little or no linear correlation.
• A line of best fit drawn by eye passes through the mean point (x̄, ȳ); so does the regression line.
• Regression line y = ax + b: a is the change in y for each increase of 1 in x; b is the value of y when x = 0.
• Use the y on x line only to predict y from x, and only for x inside the range of the data (interpolation).
• Extrapolation, predicting outside the range of the data, is unreliable because the pattern may not continue.
• Spearman's r_s: rank each variable, then find r for the ranks; equal values share the average of their ranks.
• r_s = 1 means y always increases when x increases; the points need not lie on a straight line.

Probability: Venn diagrams, tree diagrams and conditional probability

• P(A′) = 1 − P(A), where A′ means A does not happen
• Expected number of occurrences = number of trials × probability
• Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)
• Conditional: P(A | B) = P(A ∩ B) ÷ P(B)
• Independent: P(A ∩ B) = P(A) × P(B), which is the same as P(A | B) = P(A)
• Tree diagram: multiply along the branches, then add the routes you want

Discrete random variables and the binomial distribution

• The probabilities in a distribution add up to 1: ΣP(X = x) = 1
• Expected value: E(X) = Σ x × P(X = x), multiply each value by its probability and add
• Fair game: expected gain = 0, where gain = winnings − cost
• X ~ B(n, p): P(X = r) from binomial pdf; P(X ≤ r) from binomial cdf
• P(X ≥ r) = 1 − P(X ≤ r − 1)
• Binomial mean: E(X) = np
• Binomial variance: Var(X) = np(1 − p); standard deviation is its square root

The normal distribution

• About 68% of values lie within 1 standard deviation of the mean (μ ± σ)
• About 95% lie within 2 standard deviations (μ ± 2σ)
• About 99.7% lie within 3 standard deviations (μ ± 3σ)
• P(X < μ) = P(X > μ) = 0.5, and P(X > a) = 1 − P(X < a)
• Normal cdf needs: lower bound, upper bound, μ, σ
• Inverse normal needs the area to the LEFT of the value; for the top 10% enter 0.90
• Unknown μ or σ: find z with inverse normal (mean 0, standard deviation 1), then use z = (x − μ) ÷ σ

Hypothesis testing: χ² and t-tests

• If p-value < significance level, reject H₀; otherwise do not reject H₀
• If χ²_calc > critical value, reject H₀; otherwise do not reject H₀
• Expected frequency in a contingency table = (row total × column total) ÷ grand total
• Degrees of freedom, test for independence: (rows − 1) × (columns − 1)
• Degrees of freedom, goodness of fit: number of categories − 1
• χ²_calc = sum of (observed − expected)² ÷ expected
• t-test: H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ or μ₁ < μ₂ (one-tailed); the pooled test assumes normal populations with equal variances

Calculus

The derivative as a rate of change; increasing and decreasing functions

• dy/dx = f′(x) = gradient of the curve = gradient of the tangent at that point
• A limit is the value an expression gets closer to as x approaches a number; x does not have to reach it
• f′(x) > 0 on an interval: f is increasing there
• f′(x) < 0 on an interval: f is decreasing there
• f′(x) = 0: the tangent is horizontal (a stationary point)
• Units of a rate: units of y ÷ units of x, e.g. dV/dt in litres per minute
• A negative rate means the quantity is falling

Differentiating axⁿ; tangents and normals

• If y = axⁿ, then dy/dx = anx^{n − 1}
• The derivative of ax is a; the derivative of a constant is 0
• Negative powers follow the same rule: x⁻² differentiates to −2x⁻³
• Gradient at a point: put the x-value into dy/dx
• Tangent at (x₁, y₁) with gradient m: y − y₁ = m(x − x₁)
• Gradient of the normal = −1 ÷ m, because the two gradients multiply to −1
• Find y₁ from the ORIGINAL equation of the curve, not from dy/dx

Stationary points and optimisation

• Stationary point: dy/dx = 0
• Gradient + then − : local maximum
• Gradient − then + : local minimum
• Quadratic y = ax² + bx + c: stationary point at x = −b ÷ (2a); maximum if a < 0, minimum if a > 0
• Optimisation: use the given condition to remove one variable, differentiate, set the derivative equal to 0, solve
• Answer what is asked: the x-value, or the maximum or minimum value found by putting x back in

Integration and areas

• ∫ axⁿ dx = ax^{n + 1} ÷ (n + 1) + C, for n ≠ −1
• ∫ a dx = ax + C (a constant integrates to ax)
• Integrate a sum one term at a time
• To find C: substitute the given point (x, y) into the integrated function
• Definite integral from a to b: [F(x)] = F(b) − F(a), with no + C
• Area under y = f(x) from x = a to x = b, where f(x) > 0, is the definite integral of f(x) from a to b
• Integrating a rate gives the total change, e.g. ∫ dV/dt dt gives volume

The trapezoidal rule

• Interval width: h = (b − a) ÷ n, where n is the number of intervals (strips)
• Area ≈ (h ÷ 2) × [first height + last height + 2 × (sum of the heights in between)]
• n intervals use n + 1 heights (y-values)
• Area of one trapezoid = (h ÷ 2) × (sum of its two parallel sides)
• Concave up curve: overestimate; concave down curve: underestimate
• More intervals of smaller width give a more accurate estimate
• Under a speed-time graph the area is distance; across a river the depths give the cross-sectional area