Introduction to Antiderivatives

Calculus 1 · Axiom Academy

LESSON Introduction to Antiderivatives Running differentiation in reverse: given a rate of change, recover the function it came from — and discover why the answer is never just one function. An antiderivative of a function f is a function F whose derivative gives f back. Differentiating F returns you to f — so F "undoes" the act of differentiation. At every x , the slope of F(x)=x^2 is exactly the height of f(x)=2x . Because F'(x)=2x=f(x) , the function F(x)=x^2 is an antiderivative of f(x)=2x . Here is the twist. If F is an antiderivative of f , then so is F(x)+C for any constant C — because the derivative of a constant is zero, adding C changes nothing about the slope. x^2+C is the curve x^2 slid straight up or down by C . The shape is untouched. Sliding a curve up or down never tilts it, so at each x all the shifted curves share one slope. , so differentiating x^2+C erases C and returns 2x no matter what C was. Every curve in this stack differentiates to the same f(x)=2x — they are all antiderivatives of it. 3. Writing the Whole Family at Once To capture every member of the family in one expression, we attach " " to a single antiderivative. This is the indefinite integral , and its notation is: The symbol means "all functions whose derivative is f ." Each value of C picks one curve out of the family; the moving tangent below shows that as C slides the curve up and down, the slope at every point stays locked.

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