Math 2.0

Real functions Definition

Antiderivative

Builds on Derivative

Definition

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R}. A function F ⁣:I→ℝF \colon I \to \mathbb{R} is an antiderivative of ff on II if FF is differentiable on II and F′(x)=f(x)F'(x) = f(x) for all x∈Ix \in I.

Remarks

If FF is an antiderivative of ff, then so is F+CF + C for each constant CC. On an interval, these are all the antiderivatives, because a function with zero derivative is constant. For example, x3/3x^3/3 is an antiderivative of x2x^2 on ℝ\mathbb{R}. A function with a jump has no antiderivative on an interval, by Darboux's theorem.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionAntiderivative, definition, no evidence levelAntiderivativedefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition
  4. Antiderivative definition

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Antiderivative, lead section
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