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Calculus Concepts and Limits
Core calculus vocabulary and limit values, from continuity to series tests.
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- LimitTermThe value a function approaches as the input approaches a number
- One-sided limitTermA limit approached only from the left or only from the right
- Limit at infinityTermThe value a function approaches as x grows without bound
- Continuous at a pointTermThe limit equals the function value there
- Removable discontinuityTermA hole in the graph that could be filled in
- Jump discontinuityTermThe left and right limits exist but differ
- Infinite discontinuityTermThe function grows without bound near that point (vertical asymptote)
- Intermediate value theoremTermA continuous function takes every value between f(a) and f(b)
- Extreme value theoremTermA continuous function on a closed interval has a maximum and a minimum
- DifferentiableTermHaving a derivative at a point; the graph is smooth there
- Differentiable impliesTermContinuous
- Local maximumTermA point higher than nearby points
- Local minimumTermA point lower than nearby points
- Absolute (global) extremumTermThe highest or lowest point over the whole domain
- OptimizationTermUsing derivatives to find the best (largest or smallest) value
- Linear approximationTermUsing the tangent line to estimate nearby values
- DifferentialTermdy = f'(x) dx; a small change in y
- AntiderivativeTermA function whose derivative is the given function
- Riemann sumTermA sum of rectangle areas that approximates an area
- Definite integralTermA number: signed area under a curve between limits
- Net changeTermThe integral of a rate of change
- Slope fieldTermA picture of tangent slopes for a differential equation
- Differential equationTermAn equation involving a function and its derivatives
- Separable differential equationTermOne where the variables can be moved to opposite sides
- Exponential growth equationTermdy/dt = ky has solution y = Ce^(kt)
- Sequence limitTermThe value the terms approach as n grows
- Geometric series testTermConverges if |r| < 1, with sum a/(1 − r)
- p-seriesTermΣ 1/nᵖ converges if p > 1 and diverges if p ≤ 1
- Ratio testTermCompare consecutive terms; convergence if the limit of |aₙ₊₁/aₙ| is less than 1
- Alternating series testTermConverges if terms alternate sign, decrease in size and go to 0
- Taylor seriesTermA power series that matches a function's derivatives at a point
- Maclaurin seriesTermA Taylor series centered at 0
- Maclaurin series of eˣTerm1 + x + x²/2! + x³/3! + ...
- Maclaurin series of sin xTermx − x³/3! + x⁵/5! − ...
- Maclaurin series of cos xTerm1 − x²/2! + x⁴/4! − ...
- Maclaurin series of 1/(1 − x)Term1 + x + x² + x³ + ... for |x| < 1
- Radius of convergenceTermHow far from the center a power series converges
- lim (x→0) sin x / xTerm1
- lim (x→0) (1 − cos x) / xTerm0
- lim (x→0) (eˣ − 1) / xTerm1
- lim (x→∞) (1 + 1/x)ˣTerme
- lim (x→∞) 1/xTerm0
- lim (x→1) (x² − 1)/(x − 1)Term2
- lim (x→∞) (3x² + 1)/(x² + 5)Term3
- lim (x→0) ln(1 + x)/xTerm1
- Squeeze theoremTermIf a function is trapped between two functions with the same limit, it has that limit
- Vertical asymptoteTermA vertical line the graph approaches where values blow up
- Horizontal asymptoteTermA horizontal line the graph approaches as x goes to ±∞
- Indeterminate formTermA limit form such as 0/0 or ∞/∞ that needs more work
- Partial derivativeTermA derivative with respect to one variable, others held constant
- GradientTermA vector of partial derivatives pointing in the direction of steepest increase
- Double integralTermIntegral over a 2D region, finding volume under a surface
- Multivariable calculusTermCalculus of functions with more than one input
- Differential vs. derivativeTermThe derivative is a rate; the differential is a small change
- Leibniz notationTermdy/dx
- Lagrange notationTermf'(x)
- Newton's notationTermẏ, a dot over the variable
- Who developed calculus independently?TermIsaac Newton and Gottfried Leibniz
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