Anti-derivatives

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Anti-derivatives

Antiderivatives
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Antiderivatives are not Integrals

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Antiderivatives

Definition
If $F(x)$ is a function with $F'(x)=f(x)$, then we say that $F(x)$ is an antiderivative of $f(x)$.

Example:
$F(x)=x^3$ is an antiderivative of $f(x)=3x^2$. Also, $x^3+7$ is an anti-derivative of $3x^2$, since $$\frac{d(x^3)}{dx} = 3x^2 \text{ and }\frac{d(x^3+7)}{dx}=3x^2.$$ The most general antiderivative of $f$ is $F(x)=x^3+C$, where $c$ is an arbitrary constant.

As you will begin to see,

Every continuous function has an antiderivative, and in fact has infinitely many antiderivatives.
Two antiderivatives for the same function $f(x)$ differ by a constant.
To find all antiderivatives of $f(x)$, find one anti-derivative and write "+ C" for the arbitrary constant. We call this the most general antiderivative of $f$.
Graphically, any two antiderivatives have identical graphs, only vertically shifted, which is what happens as you vary the constant.

DO: Find 5 more antiderivatives of $f(x)=3x^2$.

DO: Find 3 antiderivatives of $g(x)=2x$.

DO: Find the most general antiderivatives of $f$ and $g$.

Antiderivatives come up frequently in physics.

Since velocity is the derivative of position, position is the antiderivative of velocity. If you know the velocity for all time, and if you know the starting position, you can figure out the position for all time.
Since acceleration is the derivative of velocity, velocity is the antiderivative of acceleration. If you know the acceleration for all time, and if you know the starting velocity, you can figure out the velocity for all time.
Much of physics involves Newton's law: Force = mass $\times$ acceleration. If you can figure out the force, you can figure out the acceleration. From there, you can often get the velocity and position by antidifferentiation.