IB Mathematics: Analysis and Approaches HL
Higher Level flashcards for IB Diploma Mathematics: Analysis and Approaches (first exams 2021). Covers the full SL core plus all AHL extensions across number & algebra, functions, geometry & trigonometry, statistics & probability, and calculus.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 471 kort
Innehåll
- An arithmetic sequence has the form uₙ = 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 Sₙ = (n/2)(2u₁ + (n−1)d) = (n/2)(u₁ + uₙ).
- A geometric sequence has the form uₙ = u₁ rⁿ⁻¹, where r is the common ratio (r ≠ 0).
- The sum of the first n terms of a geometric series is Sₙ = u₁(rⁿ − 1)/(r − 1) = u₁(1 − rⁿ)/(1 − r), for r ≠ 1.
- The sum to infinity of a geometric series is S∞ = u₁/(1 − r), and it exists only when |r| < 1.
- Sigma notation: Σ (from k=1 to n) of aₖ means the sum a₁ + a₂ + ··· + aₙ. The index variable below the Σ is the lower limit; the value above is the upper limit.
- Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁰ = 1 (a ≠ 0).
- A negative exponent gives a reciprocal: a⁻ⁿ = 1/aⁿ. A fractional exponent is a root: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ).
- The logarithm is the inverse of the exponential: a^x = b ⇔ x = logₐ b, for a > 0, a ≠ 1, b > 0.
- Laws of logarithms: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, and logₐ(xⁿ) = n logₐ x.
- The change-of-base formula: logₐ x = (logₘ x)/(logₘ a). This lets any logarithm be computed using a common base such as 10 or e.
- The natural logarithm ln x = logₑ x uses base e ≈ 2.71828. Key identities: ln(e^x) = x and e^(ln x) = x.
- Standard (scientific) form writes a number as a × 10ᵏ, where 1 ≤ |a| < 10 and k is an integer.
- The binomial theorem: (a + b)ⁿ = Σ (from r=0 to n) of C(n,r) aⁿ⁻ʳ bʳ, where C(n,r) = n!/(r!(n−r)!) are the binomial coefficients.
- Pascal's triangle gives binomial coefficients: each entry is the sum of the two above it. Row n (starting n=0) lists C(n,0), C(n,1), ..., C(n,n).
- The general term (r+1)th term in the expansion of (a + b)ⁿ is C(n,r) aⁿ⁻ʳ bʳ, used to find a specific term without expanding fully.
- [HL] The extended binomial theorem applies to fractional or negative n: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + ..., valid for |x| < 1 as an infinite series.
- [HL] Proof by mathematical induction: prove the base case (n=1), assume true for n=k, then prove it for n=k+1. This establishes the statement for all positive integers n.
- [HL] Proof by contradiction: assume the negation of the statement is true, then derive a logical contradiction. The classic example proves √2 is irrational.
- [HL] Disproof by counterexample: a single example for which a statement fails proves the statement false. One counterexample is enough to disprove a universal claim.
- [HL] A permutation counts ordered arrangements: nPr = n!/(n−r)!. The order of selection matters.
- [HL] A combination counts unordered selections: nCr = C(n,r) = n!/(r!(n−r)!). The order of selection does not matter.
- [HL] Partial fractions decompose a rational function into simpler terms, e.g. (5x−2)/((x−1)(x+2)) = A/(x−1) + B/(x+2). Used to simplify integration.
- [HL] The imaginary unit i is defined by i² = −1, so √(−1) = i. A complex number is written z = a + bi, where a is the real part and b is the imaginary part.
- [HL] The complex conjugate of z = a + bi is z* = a − bi. It satisfies z z* = a² + b² = |z|², a real non-negative number.
- [HL] The modulus of z = a + bi is |z| = √(a² + b²), the distance from the origin on the Argand diagram. The argument arg(z) is the angle from the positive real axis.
- [HL] Polar (modulus-argument) form: z = r(cos θ + i sin θ) = r cis θ, where r = |z| and θ = arg(z).
- [HL] Euler form: z = r e^(iθ). Euler's identity is the special case e^(iπ) + 1 = 0, linking e, i, π, 1 and 0.
- [HL] In polar form, multiplication multiplies moduli and adds arguments: z₁z₂ = r₁r₂ cis(θ₁ + θ₂). Division divides moduli and subtracts arguments.
- [HL] De Moivre's theorem: (r cis θ)ⁿ = rⁿ cis(nθ), i.e. (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Used for powers and roots of complex numbers.
- [HL] The n distinct nth roots of a complex number lie equally spaced on a circle of radius r^(1/n) in the Argand diagram, separated by angle 2π/n.
- [HL] A system of linear equations can have a unique solution, infinitely many solutions, or no solution, found by elimination (row reduction) or by using a GDC.
- [HL] A system of three linear equations in three unknowns represents three planes; geometrically a unique solution is a single point, no solution means the planes do not share a common point.
- Compound interest with n compounding periods per year: FV = PV(1 + r/n)^(nt), where r is the annual rate (as a decimal) and t is the time in years.
- The number e ≈ 2.71828 is the base of natural exponential growth and is the limit of (1 + 1/n)ⁿ as n → ∞.
- An arithmetic series diverges (sum grows without bound) for d ≠ 0, whereas a geometric series converges to a finite sum only when |r| < 1.
- [HL] Complex roots of real polynomials occur in conjugate pairs: if a + bi is a root and the coefficients are real, then a − bi is also a root.
- The factorial n! = n × (n−1) × ··· × 2 × 1, with 0! = 1 by definition. It counts the number of orderings of n distinct objects.
- A function f maps each element of its domain to exactly one element of its range. The domain is the set of allowed inputs; the range is the set of resulting outputs.
- The composite function (f ∘ g)(x) = f(g(x)) means apply g first, then f. In general f ∘ g ≠ g ∘ f.
- The inverse function f⁻¹ reverses f: f(f⁻¹(x)) = x. Its graph is the reflection of y = f(x) in the line y = x. A function has an inverse only if it is one-to-one.
- A quadratic function f(x) = ax² + bx + c has a parabolic graph. Its vertex (axis of symmetry) is at x = −b/(2a).
- The quadratic formula gives the roots of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac))/(2a).
- The discriminant Δ = b² − 4ac determines the roots of a quadratic: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
- Completed-square (vertex) form f(x) = a(x − h)² + k shows the vertex at (h, k) directly. Useful for finding maximum or minimum values.
- Graph transformations of y = f(x): y = f(x) + k shifts vertically by k; y = f(x − h) shifts horizontally by h; y = a f(x) stretches vertically by factor a; y = f(bx) stretches horizontally by factor 1/b.
- An exponential model f(x) = a·bˣ + c grows when b > 1 and decays when 0 < b < 1. The line y = c is a horizontal asymptote.
- A logarithmic model f(x) = a ln(x) + b is the inverse shape of an exponential. It has a vertical asymptote where its argument equals zero.
- For a quadratic ax² + bx + c = 0, the sum of the roots is −b/a and the product of the roots is c/a.
- [HL] The factor theorem: (x − a) is a factor of polynomial P(x) if and only if P(a) = 0. So a is a root precisely when (x − a) divides P(x).