IB Mathematics: Applications and Interpretation SL
Standard Level common-core flashcards for IB Diploma Programme Mathematics: Applications and Interpretation (first exams 2021). Covers number and algebra, functions and modelling, geometry and trigonometry, statistics and probability, and introductory calculus, with emphasis on real-world modelling, technology (GDC) and interpreting results in context.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 419 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.
- In standard form a × 10^k, a positive exponent k means a large number (≥ 10) and a negative exponent means a small number (between 0 and 1).
- SI prefixes scale by powers of ten: kilo (k) = 10³, mega (M) = 10⁶, giga (G) = 10⁹, tera (T) = 10¹².
- Small-value SI prefixes: milli (m) = 10⁻³, micro (µ) = 10⁻⁶, nano (n) = 10⁻⁹, pico (p) = 10⁻¹².
- To multiply numbers in standard form, multiply the coefficients and add the exponents: (a × 10^m)(b × 10^n) = ab × 10^(m+n), then adjust so 1 ≤ coefficient < 10.
- Percentage error = |v_A − v_E| / |v_E| × 100%, where v_E is the exact (accepted) value and v_A is the approximate (measured) value.
- Rounding to a number of significant figures: count significant digits from the first non-zero digit. E.g. 0.04057 to 3 s.f. is 0.0406.
- An upper bound is the largest value a rounded measurement could be; a lower bound is the smallest. A length given as 24 cm to the nearest cm has bounds 23.5 cm ≤ length < 24.5 cm.
- Percentage error is always taken as a positive value (the absolute value ensures it), since it measures the size of the discrepancy, not its direction.
- The nth term of an arithmetic sequence is u_n = u₁ + (n − 1)d, where u₁ is the first term and d is the common difference.
- The sum of the first n terms of an arithmetic series is S_n = (n/2)(2u₁ + (n − 1)d) = (n/2)(u₁ + u_n).
- In an arithmetic sequence the common difference d is found by subtracting any term from the next: d = u_(n+1) − u_n.
- The nth term of a geometric sequence is u_n = u₁ × r^(n−1), where u₁ is the first term and r is the common ratio.
- The sum of the first n terms of a geometric series is S_n = u₁(r^n − 1)/(r − 1) = u₁(1 − r^n)/(1 − r), valid for r ≠ 1.
- In a geometric sequence the common ratio r is found by dividing any term by the previous one: r = u_(n+1) / u_n.
- Simple interest is interest paid only on the original principal: I = Prt, where P is principal, r the interest rate per period (as a decimal) and t the number of periods.
- The IB compound interest formula is FV = PV × (1 + r/(100k))^(kn), where FV is future value, PV present value, r the annual interest rate (%), k the number of compounding periods per year and n the number of years.
- Compound interest grows as a geometric sequence: each period multiplies the balance by (1 + i), where i is the interest rate per compounding period.
- Compounding more frequently (e.g. monthly vs annually) at the same nominal annual rate produces a slightly larger future value, because interest is earned on interest sooner.
- Real value (purchasing power) accounts for inflation: a nominal return must exceed the inflation rate for an investment to gain real value.
- An annuity is a sequence of equal payments made at regular intervals; its value is found with the GDC's finance solver (TVM Solver) rather than by hand at SL.
- Amortization is the process of paying off a loan with regular equal payments; each payment covers interest on the outstanding balance plus a portion of the principal.
- On a GDC's TVM (Time Value of Money) solver, cash inflows and outflows have opposite signs: money paid out (e.g. a deposit) is negative and money received is positive (or vice versa, kept consistent).
- Depreciation is the loss in value of an asset over time. Reducing-balance (geometric) depreciation multiplies the value by a fixed factor (1 − r) each year.
- Laws of exponents: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), and (a^m)^n = a^(mn).
- A negative exponent gives a reciprocal: a^(−n) = 1/a^n. A zero exponent gives 1: a⁰ = 1 (for a ≠ 0).
- The logarithm is the inverse of exponentiation: if a^x = b then x = log_a b. So log_a b answers 'to what power must a be raised to get b?'
- log₁₀ x (written log x) has base 10; ln x is the natural logarithm with base e ≈ 2.718. Both are available directly on the GDC.
- Systems of linear equations and polynomial equations can be solved on a GDC using its equation solver or by graphing both sides and finding intersection points.
- A linear equation in two unknowns represents a line; a system of two such equations has a unique solution where the lines intersect, no solution if parallel, or infinitely many if identical.
- A linear function has the form f(x) = mx + c, where m is the gradient (slope) and c is the y-intercept.
- The gradient of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁), the change in y divided by the change in x.
- The x-intercept of a graph is where y = 0 (it crosses the x-axis); the y-intercept is where x = 0 (it crosses the y-axis).
- Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines have gradients whose product is −1 (m₁ × m₂ = −1).
- The domain of a function is the set of all allowed input (x) values; the range is the set of all resulting output (y) values.
- A quadratic function f(x) = ax² + bx + c graphs as a parabola, opening upward if a > 0 and downward if a < 0.
- The axis of symmetry of the parabola y = ax² + bx + c is the vertical line x = −b/(2a), which also gives the x-coordinate of the vertex.
- The vertex of a parabola is its turning point: a minimum if it opens upward, a maximum if it opens downward.
- An exponential model has the form f(x) = k a^x + c (a > 0). It represents growth if a > 1 and decay if 0 < a < 1.
- A horizontal asymptote is a horizontal line that a graph approaches but never reaches. For f(x) = k a^x + c, the horizontal asymptote is y = c.
- A cubic model has the form f(x) = ax³ + bx² + cx + d. Its graph can have up to two turning points and crosses the x-axis up to three times.
- A sinusoidal model has the form f(x) = a sin(bx) + d (or with cosine), used for periodic phenomena such as tides, temperature cycles and daylight hours.
- For f(x) = a sin(bx) + d: the amplitude is |a| (half the distance between max and min), the principal axis is y = d, and the period is 360°/b (in degrees).
- Direct variation (proportion) is modelled by y = kx: y is directly proportional to x, so doubling x doubles y. The graph is a straight line through the origin.
- Inverse variation is modelled by y = k/x: y is inversely proportional to x, so doubling x halves y. The graph is a hyperbola.
- A function is a relation in which each input value maps to exactly one output value. A graph represents a function if it passes the vertical line test.
- When modelling in context, always interpret parameters meaningfully: e.g. in a cost model C(n) = 5n + 200, the 5 is the cost per item and 200 is the fixed cost.
- A model is only valid over an appropriate domain. Extrapolating far outside the data used to build it can give unrealistic predictions (e.g. negative populations).
- The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D is √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).
- The midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D is ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2), the average of each coordinate.