IB Mathematics: Applications and Interpretation HL
Comprehensive flashcards for IB Diploma Mathematics: Applications and Interpretation Higher Level (first exams 2021). Covers all five topics — Number & algebra, Functions, Geometry & trigonometry, Statistics & probability, and Calculus — at full HL depth (SL core plus all AHL extensions), emphasising real-world modelling, statistics, finance and GDC/technology. HL-only content is marked [HL].
Ämne: Matematik · Nivå: Gymnasium (16–19) · 473 kort
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- Standard form (scientific notation) writes a number as a × 10^k, where 1 ≤ a < 10 and k is an integer. Example: 0.00043 = 4.3 × 10⁻⁴.
- Common SI prefixes: tera (T) = 10¹², giga (G) = 10⁹, mega (M) = 10⁶, kilo (k) = 10³, milli (m) = 10⁻³, micro (μ) = 10⁻⁶, nano (n) = 10⁻⁹.
- Percentage error = |v_A − v_E| / |v_E| × 100%, where v_A is the approximate (measured) value and v_E is the exact value.
- Rounding: to round to n significant figures, keep the first n non-zero-leading digits. Decimal places (d.p.) count digits after the decimal point. Always round the final answer, not intermediate values.
- Upper and lower bounds: a value rounded to a given accuracy lies within ± half the rounding unit. E.g. a length given as 24 cm (nearest cm) lies in [23.5, 24.5) cm.
- An arithmetic sequence has a constant common difference d. The nth term: u_n = u₁ + (n − 1)d.
- Sum of the first n terms of an arithmetic series: S_n = (n/2)(2u₁ + (n − 1)d) = (n/2)(u₁ + u_n).
- A geometric sequence has a constant common ratio r. The nth term: u_n = u₁ · r^(n−1).
- Sum of the first n terms of a geometric series: S_n = u₁(rⁿ − 1)/(r − 1) = u₁(1 − rⁿ)/(1 − r), for r ≠ 1.
- Sum to infinity of a geometric series: S_∞ = u₁/(1 − r), valid only when |r| < 1 (the series converges).
- Simple interest: I = Prt, where P = principal, r = interest rate per period (as a decimal), t = number of periods. The amount grows linearly.
- Compound interest: FV = PV(1 + r/k)^(kn), where PV = present value, r = annual nominal rate, k = compounding periods per year, n = number of years. Growth is geometric (exponential).
- On a GDC's finance solver (TVM), the variables are: N (number of payments), I% (annual rate), PV (present value), PMT (payment), FV (future value), P/Y and C/Y (payments/compounding per year). Cash inflows are positive, outflows negative.
- An annuity is a series of equal regular payments. Amortization is repaying a loan by regular payments that cover both interest and principal; early payments are mostly interest, later payments mostly principal.
- Depreciation reduces an asset's value over time. Reducing-balance (declining) depreciation uses a compound formula with a negative rate: value after n years = PV(1 − r)ⁿ.
- Inflation is a sustained rise in the general price level, reducing money's purchasing power. Real interest rate ≈ nominal interest rate − inflation rate. Inflation is modelled with compound growth.
- Laws of exponents: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a⁰ = 1; a^(−n) = 1/aⁿ; a^(1/n) = ⁿ√a.
- A logarithm is the inverse of exponentiation: if a^x = b then log_a b = x (a > 0, a ≠ 1). log₁₀ is the common log; ln = log_e is the natural log (e ≈ 2.71828).
- [HL] Laws of logarithms: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(xⁿ) = n·log_a x; change of base: log_a x = (log_b x)/(log_b a).
- [HL] A complex number in Cartesian form: z = a + bi, where i² = −1, a = Re(z) is the real part, b = Im(z) is the imaginary part. Plotted on an Argand diagram (horizontal = real, vertical = imaginary).
- [HL] Modulus of z = a + bi is |z| = r = √(a² + b²); argument arg(z) = θ is the angle from the positive real axis. The complex conjugate z* = a − bi.
- [HL] Polar (modulus-argument) form: z = r(cosθ + i·sinθ) = r·cisθ. Euler form: z = r·e^(iθ). These are equivalent ways of writing the same complex number.
- [HL] Multiplying in polar form: multiply moduli, add arguments — r₁e^(iθ₁) · r₂e^(iθ₂) = r₁r₂·e^(i(θ₁+θ₂)). Dividing: divide moduli, subtract arguments.
- [HL] In AC-circuit analysis, complex numbers represent impedance Z = R + iX, where R is resistance and X is reactance. The modulus |Z| gives the total impedance magnitude and arg(Z) the phase angle between voltage and current.
- [HL] A matrix is a rectangular array of numbers with dimensions m × n (rows × columns). Matrices are added/subtracted element-wise (same dimensions required) and multiplied by a scalar element-wise.
- [HL] Matrix multiplication AB is defined when columns of A = rows of B; an (m×n)(n×p) product is m×p. Multiplication is NOT commutative (AB ≠ BA in general). The identity matrix I satisfies AI = IA = A.
- [HL] For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. A matrix is singular (non-invertible) if and only if its determinant is 0.
- [HL] The inverse of A = [[a, b], [c, d]] is A⁻¹ = (1/det A)[[d, −b], [−c, a]], provided det A ≠ 0. It satisfies AA⁻¹ = A⁻¹A = I. For larger matrices use a GDC.
- [HL] A system of linear equations can be written as AX = B; the solution is X = A⁻¹B when A is invertible. This handles 2×2 and 3×3 systems efficiently on a GDC.
- [HL] An eigenvector v of a square matrix M satisfies Mv = λv for a scalar λ (the eigenvalue): M only scales v, it does not change its direction. Eigenvalues are found from det(M − λI) = 0.
- [HL] The characteristic equation det(M − λI) = 0 for a 2×2 matrix gives a quadratic in λ. Each eigenvalue λ is substituted back into (M − λI)v = 0 to find its eigenvector(s).
- [HL] Diagonalisation: a matrix M with distinct eigenvalues can be written M = PDP⁻¹, where D is diagonal (eigenvalues on the diagonal) and P has the corresponding eigenvectors as columns. Then Mⁿ = PDⁿP⁻¹, making powers easy to compute.
- A function f maps each input x to exactly one output f(x). The domain is the set of allowed inputs; the range is the set of resulting outputs.
- A linear model y = mx + c has gradient (slope) m = rise/run = (y₂ − y₁)/(x₂ − x₁) and y-intercept c (value of y when x = 0).
- The x-intercept (root/zero) of a function is where its graph crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
- A quadratic model y = ax² + bx + c graphs as a parabola. It opens upward if a > 0, downward if a < 0. The axis of symmetry is x = −b/(2a), which locates the vertex (maximum or minimum).
- Roots of a quadratic ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a). The discriminant Δ = b² − 4ac: Δ > 0 two real roots, Δ = 0 one repeated root, Δ < 0 no real roots.
- A cubic model y = ax³ + bx² + cx + d can have up to 3 real roots and up to 2 turning points (a local max and a local min). It is useful for data with an inflection.
- An exponential model y = k·a^x + c (a > 0) models growth (a > 1) or decay (0 < a < 1). The horizontal line y = c is a horizontal asymptote the curve approaches but never reaches.
- A sinusoidal model y = a·sin(b(x − c)) + d: |a| is the amplitude, the period is 360°/b (or 2π/b in radians), c is the horizontal (phase) shift, and d is the vertical shift (principal axis y = d).
- Direct variation: y = kx (y is proportional to x); the graph is a straight line through the origin. Inverse variation: y = k/x (y is inversely proportional to x); the graph is a hyperbola.
- When fitting a model to data, you choose the function family from the shape of the scatter plot, find parameters (often via regression on a GDC), then interpret parameters in context and judge the fit.
- Interpreting a model: the gradient/rate parameter gives the change in output per unit input (e.g. cost per item), and the intercept gives the output when the input is zero (e.g. fixed cost). Always state units.
- An asymptote is a line a curve approaches arbitrarily closely. Exponential models have horizontal asymptotes; y = k/x has both a horizontal asymptote (y = 0) and a vertical asymptote (x = 0).
- [HL] A logistic model y = L/(1 + Ce^(−kx)) describes growth that starts roughly exponentially then levels off at a carrying capacity L (the horizontal asymptote). Used for populations and the spread of disease.
- [HL] A piecewise function is defined by different expressions on different intervals of the domain, e.g. a tax model where the rate changes above income thresholds. Each piece applies only on its stated interval.
- [HL] Function transformations of y = f(x): y = f(x) + k shifts up k; y = f(x − h) shifts right h; y = a·f(x) stretches vertically by factor a; y = f(bx) stretches horizontally by factor 1/b; y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.
- Distance between two 3D points (x₁,y₁,z₁) and (x₂,y₂,z₂): d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). The midpoint is the average of each coordinate.
- Volume of common 3D solids: cuboid = lwh; cylinder = πr²h; cone = (1/3)πr²h; sphere = (4/3)πr³; pyramid = (1/3) × base area × height.
- Surface area of common solids: cylinder = 2πr² + 2πrh; sphere = 4πr²; cone (curved part) = πrl, where l is the slant height; total cone area = πr² + πrl.