{
 "source": "https://peabody.kids/flashcards/tp-ap-calculus",
 "id": "tp-ap-calculus",
 "title": "AP Calculus Terms and Rules",
 "category": "test-prep",
 "subject": "AP Calculus",
 "levels": [
  "9-12",
  "college"
 ],
 "description": "Limits, derivatives, integrals, and the rules and theorems that connect them.",
 "tags": [
  "ap",
  "calculus",
  "math"
 ],
 "frontLabel": "Term",
 "backLabel": "Meaning",
 "cards": [
  [
   "Limit",
   "The value a function approaches as the input approaches a point"
  ],
  [
   "Continuity at a point",
   "The limit exists, the function is defined, and they are equal"
  ],
  [
   "Removable discontinuity",
   "A hole in a graph"
  ],
  [
   "Jump discontinuity",
   "One-sided limits exist but differ"
  ],
  [
   "Vertical asymptote",
   "A line the function grows without bound near"
  ],
  [
   "Horizontal asymptote",
   "A value the function approaches at infinity"
  ],
  [
   "Intermediate Value Theorem",
   "A continuous function takes every value between f(a) and f(b)"
  ],
  [
   "Derivative",
   "Instantaneous rate of change; slope of the tangent line"
  ],
  [
   "Limit definition of derivative",
   "lim as h to 0 of [f(x+h) - f(x)] / h"
  ],
  [
   "Differentiability implies continuity",
   "A function differentiable at a point is continuous there"
  ],
  [
   "Power rule",
   "d/dx x^n = n x^(n-1)"
  ],
  [
   "Product rule",
   "(fg)' = f'g + fg'"
  ],
  [
   "Quotient rule",
   "(f/g)' = (f'g - fg') / g^2"
  ],
  [
   "Chain rule",
   "d/dx f(g(x)) = f'(g(x)) times g'(x)"
  ],
  [
   "Derivative of sin x",
   "cos x"
  ],
  [
   "Derivative of cos x",
   "-sin x"
  ],
  [
   "Derivative of tan x",
   "sec^2 x"
  ],
  [
   "Derivative of e^x",
   "e^x"
  ],
  [
   "Derivative of ln x",
   "1/x"
  ],
  [
   "Derivative of a^x",
   "a^x ln a"
  ],
  [
   "Derivative of arctan x",
   "1 / (1 + x^2)"
  ],
  [
   "Derivative of arcsin x",
   "1 / sqrt(1 - x^2)"
  ],
  [
   "Implicit differentiation",
   "Differentiate both sides with respect to x, treating y as a function of x"
  ],
  [
   "Related rates",
   "Finding rates of change of connected quantities over time"
  ],
  [
   "Mean Value Theorem",
   "Some point has a tangent slope equal to the average slope over the interval"
  ],
  [
   "Rolle's Theorem",
   "If f(a) = f(b), the derivative is zero somewhere between"
  ],
  [
   "Extreme Value Theorem",
   "A continuous function on a closed interval has a max and a min"
  ],
  [
   "Critical point",
   "A point where f' is zero or undefined"
  ],
  [
   "First derivative test",
   "Sign change of f' identifies local maxima and minima"
  ],
  [
   "Second derivative test",
   "f'' positive means local min; negative means local max at a critical point"
  ],
  [
   "Concave up",
   "f'' is positive"
  ],
  [
   "Inflection point",
   "Where concavity changes"
  ],
  [
   "Optimization",
   "Using derivatives to find the best value of a quantity"
  ],
  [
   "L'Hopital's Rule",
   "For 0/0 or infinity/infinity, differentiate the top and bottom"
  ],
  [
   "Linear approximation",
   "Using the tangent line to estimate nearby values"
  ],
  [
   "Position, velocity, acceleration",
   "First and second derivatives of position"
  ],
  [
   "Speed",
   "Absolute value of velocity"
  ],
  [
   "Riemann sum",
   "Approximating area by adding areas of rectangles"
  ],
  [
   "Left Riemann sum",
   "Uses left endpoints for rectangle heights"
  ],
  [
   "Trapezoidal rule",
   "Approximates area with trapezoids"
  ],
  [
   "Definite integral",
   "Net signed area between a curve and the x-axis"
  ],
  [
   "Antiderivative",
   "A function whose derivative is the given function"
  ],
  [
   "Indefinite integral",
   "The family of antiderivatives, plus C"
  ],
  [
   "Power rule for integrals",
   "Integral of x^n = x^(n+1)/(n+1) + C, for n not equal to -1"
  ],
  [
   "Integral of 1/x",
   "ln |x| + C"
  ],
  [
   "Integral of e^x",
   "e^x + C"
  ],
  [
   "Integral of cos x",
   "sin x + C"
  ],
  [
   "Integral of sin x",
   "-cos x + C"
  ],
  [
   "Fundamental Theorem of Calculus, part 1",
   "d/dx of integral from a to x of f(t) dt equals f(x)"
  ],
  [
   "Fundamental Theorem of Calculus, part 2",
   "Integral from a to b of f = F(b) - F(a)"
  ],
  [
   "u-substitution",
   "Integration technique that reverses the chain rule"
  ],
  [
   "Integration by parts",
   "Integral of u dv = uv - integral of v du"
  ],
  [
   "Average value of a function",
   "(1/(b - a)) times the integral of f from a to b"
  ],
  [
   "Area between curves",
   "Integral of top function minus bottom function"
  ],
  [
   "Disk method",
   "Volume = pi times integral of radius squared"
  ],
  [
   "Washer method",
   "Volume using outer radius squared minus inner radius squared"
  ],
  [
   "Differential equation",
   "An equation involving a function and its derivatives"
  ],
  [
   "Separation of variables",
   "Rewrite so each variable is on one side, then integrate"
  ],
  [
   "Slope field",
   "Graph showing tangent slopes at many points"
  ],
  [
   "Exponential growth model",
   "dy/dt = ky gives y = Ce^(kt)"
  ],
  [
   "Logistic growth",
   "Growth that levels off at a carrying capacity"
  ],
  [
   "Improper integral",
   "Integral with an infinite limit or an infinite discontinuity"
  ],
  [
   "Sequence",
   "An ordered list of numbers"
  ],
  [
   "Series",
   "A sum of the terms of a sequence"
  ],
  [
   "Geometric series",
   "Converges to a/(1 - r) when |r| is less than 1"
  ],
  [
   "Harmonic series",
   "Sum of 1/n; diverges"
  ],
  [
   "p-series",
   "Sum of 1/n^p converges when p is greater than 1"
  ],
  [
   "Ratio test",
   "Series converges if the limit of |a(n+1)/a(n)| is less than 1"
  ],
  [
   "Alternating series test",
   "Terms alternate, decrease, and approach zero imply convergence"
  ],
  [
   "Absolute convergence",
   "The series of absolute values converges"
  ],
  [
   "Taylor series",
   "Infinite polynomial built from a function's derivatives at a point"
  ],
  [
   "Maclaurin series",
   "Taylor series centered at zero"
  ],
  [
   "Maclaurin series for e^x",
   "1 + x + x^2/2! + x^3/3! + ..."
  ],
  [
   "Radius of convergence",
   "Distance from the center within which a power series converges"
  ],
  [
   "Lagrange error bound",
   "Limit on the error of a Taylor polynomial approximation"
  ],
  [
   "Parametric derivative",
   "dy/dx = (dy/dt) / (dx/dt)"
  ],
  [
   "Polar area",
   "Area = 1/2 times integral of r squared d theta"
  ],
  [
   "Arc length",
   "Integral of sqrt(1 + (dy/dx)^2) dx"
  ],
  [
   "Euler's method",
   "Step-by-step numerical approximation of a differential equation"
  ]
 ]
}