Summary of Antiderivatives | Calculus I

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Module 4: Applications of Derivatives

Summary of Antiderivatives

Essential Concepts

If $F$ is an antiderivative of $f$ , then every antiderivative of $f$ is of the form $F(x)+C$ for some constant $C$ .
Solving the initial-value problem
$\frac{dy}{dx}=f(x),y(x_0)=y_0$

requires us first to find the set of antiderivatives of $f$ and then to look for the particular antiderivative that also satisfies the initial condition.

Glossary

antiderivative: a function $F$ such that $F^{\prime}(x)=f(x)$ for all $x$ in the domain of $f$ is an antiderivative of $f$

indefinite integral: the most general antiderivative of $f(x)$ is the indefinite integral of $f$ ; we use the notation $\displaystyle\int f(x) dx$ to denote the indefinite integral of $f$

initial value problem: a problem that requires finding a function $y$ that satisfies the differential equation $\frac{dy}{dx}=f(x)$ together with the initial condition $y(x_0)=y_0$