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

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