IB Mathematics: Analysis and Approaches SL
Comprehensive flashcards for IB Diploma Mathematics: Analysis and Approaches Standard Level (first exams 2021). Covers the SL common core across all five topics: Number & algebra, Functions, Geometry & trigonometry, Statistics & probability, and Calculus.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 421 kort
Innehåll
- Scientific notation expresses a number in the form a × 10ᵏ, where 1 ≤ a < 10 and k is an integer.
- An arithmetic sequence has a common difference d: each term is found by adding d to the previous term. The nth term is uₙ = u₁ + (n − 1)d.
- 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 a common ratio r: each term is the previous term multiplied by r. The nth term is uₙ = u₁ × rⁿ⁻¹.
- 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 (converges) only when |r| < 1.
- Sigma notation Σ is used to write sums compactly. Σ from r=1 to n of u_r means u₁ + u₂ + ... + uₙ.
- Compound interest: the future value is FV = PV × (1 + r/n)^(nt), where r is the annual rate (as a decimal), n is compounding periods per year, and t is years.
- Depreciation reduces an asset's value over time. Under reducing-balance depreciation the value follows a geometric sequence with a ratio less than 1.
- 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 ≠ 0).
- Negative and fractional exponents: a⁻ⁿ = 1/aⁿ, and a^(1/n) = the nth root of a (ⁿ√a). Also a^(m/n) = (ⁿ√a)^m.
- A logarithm answers "to what power must the base be raised?" If a^x = b then log_a b = x, for a > 0, a ≠ 1, b > 0.
- Laws of logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x (same base throughout).
- Change of base formula: log_a x = (log_b x)/(log_b a). This lets you evaluate a logarithm of any base using a calculator's log₁₀ or ln.
- The natural logarithm ln x is the logarithm to base e, where e ≈ 2.718. So ln x = log_e x, and ln e = 1.
- log_a 1 = 0 and log_a a = 1 for any valid base a. Also a^(log_a x) = x and log_a(a^x) = x (logs and exponentials are inverse operations).
- The binomial theorem (for positive integer n): (a + b)ⁿ = Σ from r=0 to n of (nCr) a^(n−r) b^r, where nCr are the binomial coefficients.
- The binomial coefficient nCr = n! / (r!(n − r)!) gives the number of ways to choose r items from n. It is also written C(n, r) or ⁿC_r.
- Pascal's triangle gives binomial coefficients: each entry is the sum of the two above it. Row n (starting at n=0) gives the coefficients of (a + b)ⁿ.
- To find the general term of the binomial expansion: the (r+1)th term is (nCr) a^(n−r) b^r. This lets you find a specific term without expanding fully.
- Approximation: rounding to a given number of decimal places or significant figures. Percentage error = |approx − exact| / |exact| × 100%.
- An annuity is a series of equal payments over time; its value can be computed using geometric-series sums, which underlie loan and savings calculations.
- The number e ≈ 2.71828 is an irrational constant that is the base of natural exponential growth; the function e^x is its own derivative.
- To solve an exponential equation like a^x = b, take logarithms of both sides: x = (log b)/(log a). This works for any base.
- In an arithmetic sequence the difference uₙ₊₁ − uₙ is constant; in a geometric sequence the ratio uₙ₊₁ / uₙ is constant. This test identifies which type a sequence is.
- A function f maps each input x to exactly one output f(x). The notation f(x) means "the value of f at x". A relation where one input gives two outputs is not a function.
- The domain of a function is the set of all permitted input values (x); the range is the set of all resulting output values (y or f(x)).
- The composite function (f ∘ g)(x) = f(g(x)) means apply g first, then apply f to the result. In general f ∘ g ≠ g ∘ f.
- The inverse function f⁻¹ reverses f: if f(a) = b then f⁻¹(b) = a. A function has an inverse only if it is one-to-one (passes the horizontal line test).
- The graph of f⁻¹ is the reflection of the graph of f in the line y = x. The domain and range of f swap to become the range and domain of f⁻¹.
- A quadratic function f(x) = ax² + bx + c has a parabolic graph: it opens upward if a > 0 and downward if a < 0.
- The axis of symmetry of f(x) = ax² + bx + c is the vertical line x = −b/(2a); the vertex lies on this line.
- The vertex (turning point) form of a quadratic is f(x) = a(x − h)² + k, where (h, k) is the vertex. Completing the square converts standard form to vertex form.
- The discriminant of ax² + bx + c = 0 is Δ = b² − 4ac. It determines the nature of the roots without solving the equation.
- Discriminant and roots: if Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated (double) real root; if Δ < 0 there are no real roots.
- The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a).
- The y-intercept of f(x) = ax² + bx + c is c (set x = 0). The x-intercepts (roots) are found by solving ax² + bx + c = 0.
- The reciprocal function f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Its graph is a hyperbola in the first and third quadrants.
- For a rational function f(x) = (ax + b)/(cx + d), the vertical asymptote is x = −d/c (denominator zero) and the horizontal asymptote is y = a/c.
- An exponential function f(x) = a^x (a > 0, a ≠ 1) has a horizontal asymptote at y = 0 and passes through (0, 1). It increases if a > 1 and decreases if 0 < a < 1.
- The logarithmic function f(x) = log_a x is the inverse of a^x. It has a vertical asymptote at x = 0, passes through (1, 0), and is defined only for x > 0.
- Translation of graphs: y = f(x) + k shifts the graph up by k; y = f(x − h) shifts it right by h (horizontal shifts are 'opposite' to the sign).
- Stretches: y = p·f(x) is a vertical stretch by factor p; y = f(qx) is a horizontal stretch by factor 1/q (scale factor 1/q on the x-axis).
- Reflections: y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis.
- For ax² + bx + c = 0 with roots α and β: the sum of roots is α + β = −b/a and the product is αβ = c/a.
- A function is even if f(−x) = f(x) (symmetric in the y-axis) and odd if f(−x) = −f(x) (symmetric about the origin).
- Self-inverse property: f ∘ f⁻¹(x) = x and f⁻¹ ∘ f(x) = x. Composing a function with its inverse returns the original input.
- Exponential models of the form y = a·b^x or y = a·e^(kx) describe growth (k > 0) or decay (k < 0), used for populations, radioactivity, and cooling.
- To find an inverse function algebraically: replace f(x) with y, swap x and y, then solve for y. The result is f⁻¹(x).
- The distance between points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D is √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).