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

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