Antiderivatives

Business Calculus · Axiom Academy

Finding functions from their derivatives - the reverse of differentiation An antiderivative is a function whose derivative gives you the original function. It's the reverse of taking a derivative. F(x) is an antiderivative of f(x) if and only if F'(x) = f(x) 2 The +C: Infinitely Many Antiderivatives Here's a key insight: if F(x) is an antiderivative of f(x), then so is F(x) + C for any constant C. The derivative of any constant is 0. So x^2 , x^2 + 5 , and x^2 - 100 all have the same derivative: 2x . The +C represents this family of functions that all have the same derivative. The indefinite integral notation - "antiderivative of f(x)" Each derivative rule has a corresponding antiderivative rule. Here are the essential ones: Power Rule for Integration (the most important rule) The power rule doesn't work when n = -1. The antiderivative of x is |x| + C , not 0 (which is undefined). 4 Sum Rule and Constant Multiple Rule Just like with derivatives, we can integrate term by term: Sum Rule: Integrate each term separately Constant Multiple Rule: Constants factor out If we know the marginal cost MC(x), the antiderivative gives us the total cost function C(x): Marginal cost: MC(x) = 20 + 0.4x In cost functions, C often represents fixed costs - the costs that exist even at zero production. We can find C using initial conditions: "Fixed costs are 5,000" means C(0) = 5000.

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