We have built up a whole list of tools that we can use to actually find or calculate these derivatives. We know how to differentiate most functions (and combinations of functions) that we can think of!
Go back through each of the above derivatives, and find a different option for \(f(x)\) that still works. Make sure that it is something completely unique, and not just an equivalent function that is written differently.
We want to try to define and name these “backwards derivatives.” Instead of calling them the “negative first” derivative, we will name them as antiderivatives.
We won’t be undoing the Power Rule with either of these! We might try to think about functions whose derivatives are \(\frac{1}{x}\) and \(\frac{1}{1+x^2}\text{.}\)
We use absolute values in the logarithm because we want to find a function whose derivative is \(g(x)\) on the whole interval that \(g(x)\) is defined. The log function is only defined for positive inputs, but we would like to be able to put any non-zero input into our function (since that’s the domain of \(g\)).
A computer program is trying to sort a group of computer files based on their size. The program isn’t very efficient, and the time that it takes to sort the files increases when it tries to sort more files.
The time that it takes, measured in seconds, based on the total, cumulative size of the files \(g\text{,}\) measured in gigabytes, is modeled by a function \(T(g)\text{.}\) We don’t know the function, but we do know that the time increases at an instantaneous rate of \(0.0001g\) seconds when the total size, \(g\) increases slightly. That is, the rate of change is a function of the file size, \(g\text{,}\) itself.
Find all of the possibilities for the function modeling the time, \(T\text{,}\) that it takes the computer program to sort files with a total size of \(g\text{.}\)
What does your constant \(C\) represent, here? You can interpret it graphically, interpret it by thinking about derivatives, but you should also interpret it in terms of the time that it takes this program to sort these files by size.
We call this type of problem an “initial value problem.” Here, we ended up solving for a family of antiderivatives, but then using some more information about that antiderivative (in this case, information about file size and time) to find the specific antiderivative function that was relevant.
For some function \(f(x)\text{,}\) if we want to find an antiderivative function \(F(x)\) and we know some “initial value,” \(F(a)\text{,}\) then we can find the exact antiderivative by:
Finding the family of antiderivatives: \(F(x)+C\text{.}\)
To finish this out, we’ll just build some notation that represents these families of antiderivatives. We can use words to describe them, but it will be helpful to introduce some quick way of writing this using notation.
\(dx\) is a differential, and represents both the “end” of the integral as well as an indicator of what the input variable of the integrand should be (or what variable we antidifferentiate “with regard to”).
If we are following the path set out by us already when we learned about derivatives, then at some point we will need to think about how to interpret these antiderivatives. What does \(F(x)\) tell us about \(f(x)\text{?}\)
What does \(f(x)\) tell us about \(f'(x)\text{?}\) We’re probably so used to thinking about what \(f'(x)\) tells us about \(f(x)\) that it might be hard to reverse the interpretation. And that’s ok!
Instead of worrying about uncovering this connection naturally, let’s just take a different approach, by stating the relationship first: Over the next few sections, we’ll discover that antiderivatives of \(f(x)\) are deeply connected to areas carved out by the graph of \(f(x)\text{.}\)
Our definition of an antiderivative is that \(F(x)\) is an antiderivative of \(f(x)\) if \(F'(x) = f(x)\text{.}\) Why does this not mean that \(F(x)\) is the antiderivative of \(f(x)\text{?}\)
