AP Calculus BC
Advanced Placement Calculus BC covering the College Board CED Units 1-10: all AB topics (limits, derivatives, integration, differential equations) plus BC-only material — parametric/polar/vector functions and infinite sequences and series.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 403 kort
Innehåll
- A limit describes the value that a function f(x) approaches as x approaches a particular value c. Notation: lim x→c f(x) = L.
- A two-sided limit lim x→c f(x) exists if and only if both one-sided limits exist and are equal: lim x→c⁻ f(x) = lim x→c⁺ f(x).
- A function f is continuous at x = c if all three hold: (1) f(c) is defined, (2) lim x→c f(x) exists, (3) lim x→c f(x) = f(c).
- A removable discontinuity (hole) occurs when lim x→c f(x) exists but does not equal f(c), or when f(c) is undefined while the limit exists.
- A jump discontinuity occurs when both one-sided limits exist but are not equal — the function jumps to a different value at x = c.
- An infinite (essential) discontinuity occurs when one or both one-sided limits at x = c are ±∞. This produces a vertical asymptote.
- Limit laws: limits of sums, differences, products, and quotients are the sum/difference/product/quotient of the individual limits, provided each limit exists (and denominator ≠ 0 for quotients).
- Indeterminate forms include 0/0, ∞/∞, 0·∞, ∞ − ∞, 0⁰, ∞⁰, and 1^∞. These require algebraic manipulation, L'Hôpital's rule, or special techniques to evaluate.
- Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) near c, and lim x→c g(x) = lim x→c h(x) = L, then lim x→c f(x) = L also.
- Intermediate Value Theorem (IVT): if f is continuous on [a, b] and N is any value between f(a) and f(b), then there exists c in (a, b) such that f(c) = N.
- Two special trig limits: lim x→0 (sin x)/x = 1 and lim x→0 (1 − cos x)/x = 0. Both are foundational for proving d/dx[sin x] = cos x.
- Limits at infinity: for a rational function p(x)/q(x), if deg(p) < deg(q) the limit is 0; if deg(p) = deg(q) the limit is the ratio of leading coefficients; if deg(p) > deg(q) the limit is ±∞.
- Horizontal asymptote y = L exists if lim x→∞ f(x) = L or lim x→−∞ f(x) = L. A function can have at most two horizontal asymptotes (one on each side).
- Polynomial functions, exponential functions, sine, cosine, and absolute value are continuous on all of ℝ. Rational and root functions are continuous on their domains.
- Informal ε–δ definition: lim x→c f(x) = L means for every ε > 0, we can find δ > 0 such that 0 < |x − c| < δ implies |f(x) − L| < ε.
- Derivative as a limit: f′(x) = lim h→0 [f(x+h) − f(x)]/h. This is the slope of the tangent line and the instantaneous rate of change at x.
- Alternate derivative definition: f′(c) = lim x→c [f(x) − f(c)]/(x − c). Useful when computing the derivative at a specific point.
- Differentiability implies continuity: if f is differentiable at x = c, then f is continuous at c. The converse is FALSE — continuity does not imply differentiability (e.g. |x| at x = 0).
- Non-differentiable points: a function fails to be differentiable at corners, cusps, vertical tangents, and discontinuities. The left and right derivatives differ at these points.
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹ for any real n. The most fundamental differentiation rule.
- Derivative of a constant: d/dx[c] = 0. Constant multiple rule: d/dx[c·f(x)] = c·f′(x).
- Sum/difference rule: d/dx[f(x) ± g(x)] = f′(x) ± g′(x). The derivative of a sum equals the sum of the derivatives.
- Product rule: d/dx[f(x)g(x)] = f′(x)g(x) + f(x)g′(x). Read as 'derivative of first times second, plus first times derivative of second'.
- Quotient rule: d/dx[f(x)/g(x)] = [f′(x)g(x) − f(x)g′(x)] / [g(x)]². Mnemonic: 'low d-high minus high d-low, square the bottom and away we go'.
- Derivatives of exponentials: d/dx[eˣ] = eˣ and d/dx[aˣ] = aˣ·ln a. eˣ is unique — its derivative equals itself.
- Derivatives of logarithms: d/dx[ln x] = 1/x for x > 0, and d/dx[logₐ x] = 1/(x·ln a).
- Trig derivatives: d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[tan x] = sec²x, d/dx[cot x] = −csc²x, d/dx[sec x] = sec x tan x, d/dx[csc x] = −csc x cot x.
- Inverse trig derivatives: d/dx[arcsin x] = 1/√(1−x²), d/dx[arccos x] = −1/√(1−x²), d/dx[arctan x] = 1/(1+x²).
- Higher-order derivatives: f″(x) = derivative of f′(x), and so on. f″ represents acceleration (when f is position) or concavity. Notations: f″(x), d²y/dx², y″.
- Equation of the tangent line at x = a: y − f(a) = f′(a)(x − a). Normal line uses slope −1/f′(a) (perpendicular).
- Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x). Differentiate the outer function evaluated at the inner, then multiply by the derivative of the inner.
- Leibniz form of chain rule: dy/dx = (dy/du)(du/dx). Useful when y depends on u which depends on x.
- Implicit differentiation: differentiate both sides of an equation in x and y with respect to x, treating y as a function of x (so d/dx[y²] = 2y·dy/dx), then solve for dy/dx.
- Inverse function theorem: if g = f⁻¹, then g′(x) = 1/f′(g(x)). Equivalently, if y = f⁻¹(x), then dy/dx = 1/[f′(y)].
- Logarithmic differentiation: to differentiate y = f(x)^g(x), take ln of both sides to get ln y = g(x)·ln f(x), then differentiate implicitly.
- Chain rule with multiple compositions: for y = f(g(h(x))), dy/dx = f′(g(h(x)))·g′(h(x))·h′(x). Apply chain rule layer by layer from outside in.
- Second derivative via implicit differentiation: after solving for dy/dx, differentiate again — substitute the expression for dy/dx wherever it appears to express d²y/dx² in terms of x and y.
- Chain rule applied to e^u(x): d/dx[e^u(x)] = e^u(x) · u′(x). Applied to ln(u(x)): d/dx[ln u(x)] = u′(x)/u(x).
- Chain rule with trig: d/dx[sin u(x)] = cos u(x) · u′(x). For sin(3x²): derivative is cos(3x²)·6x.
- Symmetric inverse trig pairings: arccos and arcsin have derivatives differing only in sign, and arccot/arctan likewise. Each pair adds to a constant (π/2), so derivatives are negatives.
- Position, velocity, acceleration: if s(t) is position, then v(t) = s′(t) is velocity and a(t) = v′(t) = s″(t) is acceleration. Speed = |v(t)|.
- Particle is speeding up when velocity and acceleration have the same sign; slowing down when they have opposite signs. At rest when v(t) = 0.
- Related rates: if two quantities x and y are related by an equation and both change over time, differentiate the equation implicitly with respect to t to relate dx/dt and dy/dt.
- Linear approximation (tangent line approximation): for x near a, f(x) ≈ L(x) = f(a) + f′(a)(x − a). Used to estimate function values without a calculator.
- L'Hôpital's rule: if lim f(x)/g(x) is 0/0 or ±∞/±∞, then lim f(x)/g(x) = lim f′(x)/g′(x), provided the latter limit exists.
- L'Hôpital traps: rule applies ONLY to 0/0 and ±∞/±∞. Other indeterminate forms (0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰) must first be rewritten as 0/0 or ∞/∞.
- Rates of change in context: f′(x) interprets as the instantaneous rate of change of f with respect to x. Units of f′ = units of f ÷ units of x.
- Average rate of change of f on [a, b] = [f(b) − f(a)]/(b − a). This is the slope of the secant line. The MVT relates this to an instantaneous rate f′(c) somewhere in (a, b).
- Cone/sphere/cylinder formulas for related rates: V_cone = (1/3)πr²h, V_sphere = (4/3)πr³, V_cylinder = πr²h, A_sphere = 4πr², A_circle = πr².
- Total displacement vs total distance: displacement on [a, b] = ∫ₐᵇ v(t) dt (signed). Total distance = ∫ₐᵇ |v(t)| dt (always non-negative).