AP Calculus AB
Advanced Placement Calculus AB covering the College Board CED Units 1-8: limits and continuity, differentiation rules, applications of derivatives, integration, differential equations, and applications of integrals.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 512 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).
- Limit notation lim x→c⁻ f(x) means the limit from the left (values less than c); lim x→c⁺ f(x) means the limit from the right.
- A function f is continuous at x = c if: (1) f(c) is defined, (2) lim x→c f(x) exists, and (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 but the limit exists.
- A jump discontinuity occurs when the left- and right-hand limits both exist but are not equal — the function 'jumps' to a different value.
- An infinite (essential) discontinuity occurs when one or both one-sided limits at x = c are ±∞. This produces a vertical asymptote.
- Limit laws: lim of sum = sum of limits; lim of product = product of limits; lim of quotient = quotient of limits (if denominator limit ≠ 0). These require that each individual limit exists.
- The 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. Used for limits like lim x→0 x²·sin(1/x) = 0.
- Special trigonometric limit: lim x→0 sin(x)/x = 1. This is the foundation for derivative formulas of trig functions.
- Special trigonometric limit: lim x→0 (1 - cos(x))/x = 0. Useful when simplifying limits with trig in the numerator.
- A function f has a horizontal asymptote y = L if lim x→∞ f(x) = L or lim x→−∞ f(x) = L.
- Rational function end-behavior: if degree of numerator < degree of denominator → horizontal asymptote y = 0; if degrees are equal → horizontal asymptote = ratio of leading coefficients; if numerator degree > denominator → no horizontal asymptote.
- Indeterminate forms include 0/0, ∞/∞, 0·∞, ∞ − ∞, 0⁰, ∞⁰, and 1^∞. They require algebraic manipulation or L'Hôpital's Rule to evaluate.
- The 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.
- IVT corollary: if f is continuous on [a, b] and f(a) and f(b) have opposite signs, then f has at least one root in (a, b).
- Polynomial functions are continuous everywhere on ℝ. Their limits at any point c equal f(c) by direct substitution.
- Rational functions are continuous on their domain — everywhere except where the denominator equals zero.
- Removable discontinuity at x = c can be 'patched' by redefining f(c) to equal lim x→c f(x). The function becomes continuous after the patch.
- If f and g are continuous at c, then f ± g, f·g, and f/g (provided g(c) ≠ 0) are all continuous at c. The composition f∘g is also continuous when g is continuous at c and f is continuous at g(c).
- Average rate of change of f on [a, b] = (f(b) - f(a))/(b - a). Geometrically, this is the slope of the secant line through (a, f(a)) and (b, f(b)).
- Derivative definition (limit form): f'(x) = lim h→0 (f(x+h) - f(x))/h. This measures the instantaneous rate of change.
- Alternate derivative definition: f'(c) = lim x→c (f(x) - f(c))/(x - c). Useful when you want the derivative at a specific point c.
- Geometric meaning of the derivative: f'(c) equals the slope of the tangent line to the graph of f at x = c.
- Differentiability implies continuity: if f is differentiable at c, then f is continuous at c. The converse is FALSE — a continuous function need not be differentiable (e.g., |x| at x = 0).
- Common reasons a function is not differentiable at x = c: (1) discontinuity at c, (2) a corner or cusp, (3) a vertical tangent (slope is undefined/infinite).
- Constant rule: d/dx[c] = 0. The derivative of any constant is zero.
- Power rule: d/dx[xⁿ] = n·x^(n−1), valid for any real n. Example: d/dx[x⁵] = 5x⁴; d/dx[√x] = (1/2)x^(−1/2).
- Constant multiple rule: d/dx[c·f(x)] = c·f'(x). Sum/difference rule: d/dx[f ± g] = f' ± g'.
- Product rule: d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x). Mnemonic: 'first times derivative of second, plus second times derivative of first.'
- 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, over the square of what's below.'
- Derivatives of basic trig: d/dx[sin x] = cos x; d/dx[cos x] = -sin x; d/dx[tan x] = sec² x.
- Reciprocal trig derivatives: d/dx[cot x] = -csc² x; d/dx[sec x] = sec x · tan x; d/dx[csc x] = -csc x · cot x.
- Exponential derivative: d/dx[eˣ] = eˣ. The natural exponential is its own derivative — a defining property of e.
- General exponential derivative: d/dx[aˣ] = aˣ · ln(a), where a > 0. Reduces to eˣ when a = e since ln(e) = 1.
- Natural log derivative: d/dx[ln x] = 1/x for x > 0. More generally, d/dx[log_a(x)] = 1/(x · ln a).
- Notation for derivatives: f'(x), df/dx, y', dy/dx, and D_x[f]. The Leibniz form dy/dx emphasizes 'derivative of y with respect to x'.
- Higher-order derivatives: f''(x) is the derivative of f'(x); f'''(x) is the third derivative. Notation: f^(n)(x) for the n-th derivative or dⁿy/dxⁿ.
- Linear equation of the tangent line at (a, f(a)): y - f(a) = f'(a)·(x - a). The normal line has slope -1/f'(a) (perpendicular).
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x). Mnemonic: 'derivative of the outer (with the inner left alone) times the derivative of the inner.'
- Leibniz form of chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx). The 'du's cancel formally, but the equation is rigorous.
- Chain rule for exponentials: d/dx[e^u] = e^u · u', where u = u(x). Example: d/dx[e^(x²)] = e^(x²) · 2x.
- Chain rule for logs: d/dx[ln(u)] = u'/u. Example: d/dx[ln(x² + 1)] = 2x/(x² + 1).
- Implicit differentiation: when y is defined implicitly by an equation in x and y, differentiate both sides with respect to x and treat y as a function of x. Apply chain rule whenever y appears: d/dx[y²] = 2y · dy/dx.
- Derivative of an inverse: if g = f⁻¹, then g'(x) = 1/f'(g(x)). At a specific point: (f⁻¹)'(b) = 1/f'(a) when f(a) = b.
- Inverse trig derivatives: d/dx[arcsin x] = 1/√(1 - x²); d/dx[arccos x] = -1/√(1 - x²); d/dx[arctan x] = 1/(1 + x²).
- More inverse trig derivatives: d/dx[arccot x] = -1/(1 + x²); d/dx[arcsec x] = 1/(|x|·√(x² - 1)); d/dx[arccsc x] = -1/(|x|·√(x² - 1)).
- Second derivative via implicit differentiation: differentiate dy/dx with respect to x, again using chain rule whenever y appears. The result usually contains dy/dx, which you substitute from the first step.
- Logarithmic differentiation: take ln of both sides before differentiating. Powerful for products, quotients, and functions of the form f(x)^g(x). Example: y = xˣ → ln y = x ln x → y'/y = ln x + 1 → y' = xˣ(ln x + 1).
- If s(t) is position, then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s''(t). Speed is |v(t)| — always non-negative.