{
 "source": "https://peabody.kids/flashcards/math-calculus-concepts",
 "id": "math-calculus-concepts",
 "title": "Calculus Concepts and Limits",
 "category": "math",
 "subject": "Math",
 "levels": [
  "college",
  "9-12"
 ],
 "description": "Core calculus vocabulary and limit values, from continuity to series tests.",
 "tags": [
  "calculus",
  "limits",
  "series"
 ],
 "frontLabel": "Term",
 "backLabel": "Meaning",
 "cards": [
  [
   "Limit",
   "The value a function approaches as the input approaches a number"
  ],
  [
   "One-sided limit",
   "A limit approached only from the left or only from the right"
  ],
  [
   "Limit at infinity",
   "The value a function approaches as x grows without bound"
  ],
  [
   "Continuous at a point",
   "The limit equals the function value there"
  ],
  [
   "Removable discontinuity",
   "A hole in the graph that could be filled in"
  ],
  [
   "Jump discontinuity",
   "The left and right limits exist but differ"
  ],
  [
   "Infinite discontinuity",
   "The function grows without bound near that point (vertical asymptote)"
  ],
  [
   "Intermediate value theorem",
   "A continuous function takes every value between f(a) and f(b)"
  ],
  [
   "Extreme value theorem",
   "A continuous function on a closed interval has a maximum and a minimum"
  ],
  [
   "Differentiable",
   "Having a derivative at a point; the graph is smooth there"
  ],
  [
   "Differentiable implies",
   "Continuous"
  ],
  [
   "Local maximum",
   "A point higher than nearby points"
  ],
  [
   "Local minimum",
   "A point lower than nearby points"
  ],
  [
   "Absolute (global) extremum",
   "The highest or lowest point over the whole domain"
  ],
  [
   "Optimization",
   "Using derivatives to find the best (largest or smallest) value"
  ],
  [
   "Linear approximation",
   "Using the tangent line to estimate nearby values"
  ],
  [
   "Differential",
   "dy = f'(x) dx; a small change in y"
  ],
  [
   "Antiderivative",
   "A function whose derivative is the given function"
  ],
  [
   "Riemann sum",
   "A sum of rectangle areas that approximates an area"
  ],
  [
   "Definite integral",
   "A number: signed area under a curve between limits"
  ],
  [
   "Net change",
   "The integral of a rate of change"
  ],
  [
   "Slope field",
   "A picture of tangent slopes for a differential equation"
  ],
  [
   "Differential equation",
   "An equation involving a function and its derivatives"
  ],
  [
   "Separable differential equation",
   "One where the variables can be moved to opposite sides"
  ],
  [
   "Exponential growth equation",
   "dy/dt = ky has solution y = Ce^(kt)"
  ],
  [
   "Sequence limit",
   "The value the terms approach as n grows"
  ],
  [
   "Geometric series test",
   "Converges if |r| < 1, with sum a/(1 − r)"
  ],
  [
   "p-series",
   "Σ 1/nᵖ converges if p > 1 and diverges if p ≤ 1"
  ],
  [
   "Ratio test",
   "Compare consecutive terms; convergence if the limit of |aₙ₊₁/aₙ| is less than 1"
  ],
  [
   "Alternating series test",
   "Converges if terms alternate sign, decrease in size and go to 0"
  ],
  [
   "Taylor series",
   "A power series that matches a function's derivatives at a point"
  ],
  [
   "Maclaurin series",
   "A Taylor series centered at 0"
  ],
  [
   "Maclaurin series of eˣ",
   "1 + x + x²/2! + x³/3! + ..."
  ],
  [
   "Maclaurin series of sin x",
   "x − x³/3! + x⁵/5! − ..."
  ],
  [
   "Maclaurin series of cos x",
   "1 − x²/2! + x⁴/4! − ..."
  ],
  [
   "Maclaurin series of 1/(1 − x)",
   "1 + x + x² + x³ + ... for |x| < 1"
  ],
  [
   "Radius of convergence",
   "How far from the center a power series converges"
  ],
  [
   "lim (x→0) sin x / x",
   "1"
  ],
  [
   "lim (x→0) (1 − cos x) / x",
   "0"
  ],
  [
   "lim (x→0) (eˣ − 1) / x",
   "1"
  ],
  [
   "lim (x→∞) (1 + 1/x)ˣ",
   "e"
  ],
  [
   "lim (x→∞) 1/x",
   "0"
  ],
  [
   "lim (x→1) (x² − 1)/(x − 1)",
   "2"
  ],
  [
   "lim (x→∞) (3x² + 1)/(x² + 5)",
   "3"
  ],
  [
   "lim (x→0) ln(1 + x)/x",
   "1"
  ],
  [
   "Squeeze theorem",
   "If a function is trapped between two functions with the same limit, it has that limit"
  ],
  [
   "Vertical asymptote",
   "A vertical line the graph approaches where values blow up"
  ],
  [
   "Horizontal asymptote",
   "A horizontal line the graph approaches as x goes to ±∞"
  ],
  [
   "Indeterminate form",
   "A limit form such as 0/0 or ∞/∞ that needs more work"
  ],
  [
   "Partial derivative",
   "A derivative with respect to one variable, others held constant"
  ],
  [
   "Gradient",
   "A vector of partial derivatives pointing in the direction of steepest increase"
  ],
  [
   "Double integral",
   "Integral over a 2D region, finding volume under a surface"
  ],
  [
   "Multivariable calculus",
   "Calculus of functions with more than one input"
  ],
  [
   "Differential vs. derivative",
   "The derivative is a rate; the differential is a small change"
  ],
  [
   "Leibniz notation",
   "dy/dx"
  ],
  [
   "Lagrange notation",
   "f'(x)"
  ],
  [
   "Newton's notation",
   "ẏ, a dot over the variable"
  ],
  [
   "Who developed calculus independently?",
   "Isaac Newton and Gottfried Leibniz"
  ]
 ]
}