# Find the position s as a function of time t from the given velocity v = ds/dt. Then evaluate the...

## Question:

Find the position s as a function of time t from the given velocity {eq}v =\frac{ds}{dt} {/eq}. Then evaluate the constant of integration so that {eq}s = s_0 {/eq} when, {eq}t = 0 {/eq}. {eq}v = 3t^2 , s_0 = 4 {/eq}.

## Antiderivatives

The opposite of a derivative is an antiderivative. Therefore, if we know that a function given is a derivative of another quantity, we can find that quantity by finding the antiderivative of the function that we've been given. There are an infinite amount of antiderivatives for every function.

Velocity is the derivative of position, so if we want to find the position function that corresponds to this problem, we need to find the antiderivative of the given velocity function. Our velocity function is a monomial, so we can reverse the power rule in order to achieve this goal.

{eq}\frac{d}{dx} x^n = n x ^{n-1}\\ \frac{d}{dx} \frac{1}{n+1}x^{n+1} = x^n {/eq}

This gives the following antiderivative.

{eq}s(t) = t^3 + c {/eq}

As with every antiderivative, we need to add an unknown constant term to the end. We can use the initial condition given, that the initial position is 4, to find the value of this constant term.

{eq}s(0) = (0)^3 + c = 4\\ c = 4 {/eq}

Therefore, the position function is {eq}s = t^3 + 4 {/eq}.