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Calculus
Explore change, accumulation and approximation through single-variable calculus.
Functions and limits
Functions
A function assigns exactly one output to each input in its domain; its domain matters when comparing expressions.
No reviewed resource yetLimits
A limit describes how function values approach a value as the input approaches a point, without requiring the function to be defined at that point.
No reviewed resource yetContinuity
Continuity at a point means that the function value equals its limit there; continuity on an interval requires this at each relevant point.
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Differentiation
Derivatives
A derivative is the limit of a difference quotient, measuring local rate of change when that limit exists.
1 reviewed resourcesChain rule
The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.
No reviewed resource yetImplicit differentiation
Differentiate an equation relating variables and use the chain rule, while checking when the resulting relation determines a derivative.
No reviewed resource yetMean value theorem
For a function continuous on a closed interval and differentiable inside it, some interior derivative equals the average rate of change.
No reviewed resource yetOptimization
Derivatives help identify candidate extrema; endpoints, domain restrictions and the distinction between local and global extrema must also be checked.
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Integration
Antiderivatives
An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.
No reviewed resource yetDefinite integrals
A Riemann integral describes signed accumulation as a limit of sums, for functions satisfying the integrability conditions.
No reviewed resource yetFundamental theorem of calculus
For a continuous integrand, differentiation of its accumulated integral recovers the integrand and antiderivatives evaluate definite integrals.
No reviewed resource yetIntegration by parts
The product rule yields an integration identity that trades one integral for another, with suitable differentiability and integrability.
No reviewed resource yetImproper integrals
Integrals over unbounded intervals or near singularities are defined through limits; convergence must be established.
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Infinite processes
Series
An infinite series converges when its sequence of partial sums has a finite limit; terms tending to zero alone do not suffice.
No reviewed resource yetTaylor series
Taylor coefficients come from derivatives at a point; equality of a function to its series requires additional convergence and remainder conditions.
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