Calculus for Business – Lecture 12: Antiderivatives & Substitution

Vocabulary flashcards covering the main terms and rules from Lecture 12 on antiderivatives, indefinite integrals, IVPs, and u-substitution.

Antiderivative

A function F(x) whose derivative equals a given function f(x), i.e., F′(x)=f(x).

Indefinite Integral

The family of all antiderivatives of f(x), written ∫f(x)dx = F(x)+C.

Integrand

The function f(x) being integrated in an integral expression.

Constant of Integration (C)

An arbitrary constant added to an antiderivative to represent all possible vertical shifts.

Power Rule for Integration

∫xⁿdx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.

Constant Multiple Rule (Integration)

∫k·f(x)dx = k∫f(x)dx, where k is a constant.

Sum–Difference Rule (Integration)

∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx.

Initial Value Problem (IVP)

A differential equation together with a condition specifying the value of the unknown function at a given point.

General Solution

The indefinite‐integral form of a differential equation containing the constant of integration C.

Particular Solution

The specific function obtained from the general solution after applying the initial condition(s) to determine C.

Integration by Substitution (u-Substitution)

A technique that simplifies an integral by substituting u = g(x) where g′(x) appears in the integrand, transforming ∫f(g(x))g′(x)dx into ∫f(u)du.

General Power Rule (Substitution Form)

If u = g(x) and du = g′(x)dx, then ∫uⁿdu = uⁿ⁺¹/(n+1)+C, enabling ∫g′(x)[g(x)]ⁿdx integration.

Marginal Cost

The derivative dC/dx representing the instantaneous rate of change of total cost with respect to the number of units produced.

Particular Cost Function

Total cost C(x) found by integrating marginal cost dC/dx and using a known cost at a specific production level to solve for C.

Indefinite Integral Notation

The symbol ∫ followed by f(x)dx, indicating the operation of finding all antiderivatives of f(x).