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Section 4.1 - Antiderivative Explorations

An antiderivative of a function is just a second function whose derivative is the first one. For example, $g(x) = \sin(x)$ is an antiderivative of $f(x) = \cos(x)$ because $$g'(x) = \cos(x) = f(x)$$

How did I get there? There weren't any calculations or formulas involved. I just liked at the original function and said to myself, "Hmmm . . . can I think of a function whose derivative is $\cos(x)$?" This one was pretty straightforward. Sometimes you have to tinker a little - adding a negative sign or maybe multipling by a constant but, in the end, a lot of antiderivative problems come down to this: you fiddle with the function, rearrange it, factor it, rewrite it until you get to something that you can look at and say, ". . . and I know a function whose derivative is that."

With that in mind, see if you can come up with functions whose derivatives the following functions.

Function Antiderivative
$$f(x) = 3x^2$$
$$g(x) = x^3$$
$$f(x) = x^2$$
$$g(x) = \frac{1}{3}x^3$$
$$f(x) = \sin x$$
$$g(x) = -\cos x$$
$$f(x) = \frac{1}{x}$$
$$g(x) = \ln x$$

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