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AP Calculus Terms and Rules
Limits, derivatives, integrals, and the rules and theorems that connect them.
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- LimitTermThe value a function approaches as the input approaches a point
- Continuity at a pointTermThe limit exists, the function is defined, and they are equal
- Removable discontinuityTermA hole in a graph
- Jump discontinuityTermOne-sided limits exist but differ
- Vertical asymptoteTermA line the function grows without bound near
- Horizontal asymptoteTermA value the function approaches at infinity
- Intermediate Value TheoremTermA continuous function takes every value between f(a) and f(b)
- DerivativeTermInstantaneous rate of change; slope of the tangent line
- Limit definition of derivativeTermlim as h to 0 of [f(x+h) - f(x)] / h
- Differentiability implies continuityTermA function differentiable at a point is continuous there
- Power ruleTermd/dx x^n = n x^(n-1)
- Product ruleTerm(fg)' = f'g + fg'
- Quotient ruleTerm(f/g)' = (f'g - fg') / g^2
- Chain ruleTermd/dx f(g(x)) = f'(g(x)) times g'(x)
- Derivative of sin xTermcos x
- Derivative of cos xTerm-sin x
- Derivative of tan xTermsec^2 x
- Derivative of e^xTerme^x
- Derivative of ln xTerm1/x
- Derivative of a^xTerma^x ln a
- Derivative of arctan xTerm1 / (1 + x^2)
- Derivative of arcsin xTerm1 / sqrt(1 - x^2)
- Implicit differentiationTermDifferentiate both sides with respect to x, treating y as a function of x
- Related ratesTermFinding rates of change of connected quantities over time
- Mean Value TheoremTermSome point has a tangent slope equal to the average slope over the interval
- Rolle's TheoremTermIf f(a) = f(b), the derivative is zero somewhere between
- Extreme Value TheoremTermA continuous function on a closed interval has a max and a min
- Critical pointTermA point where f' is zero or undefined
- First derivative testTermSign change of f' identifies local maxima and minima
- Second derivative testTermf'' positive means local min; negative means local max at a critical point
- Concave upTermf'' is positive
- Inflection pointTermWhere concavity changes
- OptimizationTermUsing derivatives to find the best value of a quantity
- L'Hopital's RuleTermFor 0/0 or infinity/infinity, differentiate the top and bottom
- Linear approximationTermUsing the tangent line to estimate nearby values
- Position, velocity, accelerationTermFirst and second derivatives of position
- SpeedTermAbsolute value of velocity
- Riemann sumTermApproximating area by adding areas of rectangles
- Left Riemann sumTermUses left endpoints for rectangle heights
- Trapezoidal ruleTermApproximates area with trapezoids
- Definite integralTermNet signed area between a curve and the x-axis
- AntiderivativeTermA function whose derivative is the given function
- Indefinite integralTermThe family of antiderivatives, plus C
- Power rule for integralsTermIntegral of x^n = x^(n+1)/(n+1) + C, for n not equal to -1
- Integral of 1/xTermln |x| + C
- Integral of e^xTerme^x + C
- Integral of cos xTermsin x + C
- Integral of sin xTerm-cos x + C
- Fundamental Theorem of Calculus, part 1Termd/dx of integral from a to x of f(t) dt equals f(x)
- Fundamental Theorem of Calculus, part 2TermIntegral from a to b of f = F(b) - F(a)
- u-substitutionTermIntegration technique that reverses the chain rule
- Integration by partsTermIntegral of u dv = uv - integral of v du
- Average value of a functionTerm(1/(b - a)) times the integral of f from a to b
- Area between curvesTermIntegral of top function minus bottom function
- Disk methodTermVolume = pi times integral of radius squared
- Washer methodTermVolume using outer radius squared minus inner radius squared
- Differential equationTermAn equation involving a function and its derivatives
- Separation of variablesTermRewrite so each variable is on one side, then integrate
- Slope fieldTermGraph showing tangent slopes at many points
- Exponential growth modelTermdy/dt = ky gives y = Ce^(kt)
- Logistic growthTermGrowth that levels off at a carrying capacity
- Improper integralTermIntegral with an infinite limit or an infinite discontinuity
- SequenceTermAn ordered list of numbers
- SeriesTermA sum of the terms of a sequence
- Geometric seriesTermConverges to a/(1 - r) when |r| is less than 1
- Harmonic seriesTermSum of 1/n; diverges
- p-seriesTermSum of 1/n^p converges when p is greater than 1
- Ratio testTermSeries converges if the limit of |a(n+1)/a(n)| is less than 1
- Alternating series testTermTerms alternate, decrease, and approach zero imply convergence
- Absolute convergenceTermThe series of absolute values converges
- Taylor seriesTermInfinite polynomial built from a function's derivatives at a point
- Maclaurin seriesTermTaylor series centered at zero
- Maclaurin series for e^xTerm1 + x + x^2/2! + x^3/3! + ...
- Radius of convergenceTermDistance from the center within which a power series converges
- Lagrange error boundTermLimit on the error of a Taylor polynomial approximation
- Parametric derivativeTermdy/dx = (dy/dt) / (dx/dt)
- Polar areaTermArea = 1/2 times integral of r squared d theta
- Arc lengthTermIntegral of sqrt(1 + (dy/dx)^2) dx
- Euler's methodTermStep-by-step numerical approximation of a differential equation
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