Finding Antiderivatives -- Lucifer's Rules
I hope you enjoyed the previous activities about applications of integrals. I designed these activities to help you understand integration without having to do any tedious calculations.
You probably noticed however that the antiderivatives always seemed to appear "out of nowhere." For instance, in the previous activity, when we integrated
v(x)=-9.8x+v_0, there was almost no fanfare regarding the appearance of the antiderivative s(x)=-4.9x^2+v_0*x+s_0. Similarly, when we studied the intuitive proof of the FTC by taking a close look at the integral of the traffic on Route 15, the antiderivative just appeared. The reason for this was that I didn't want you to take your eye off the prize: building conceptual knowledge about the integral.
It's time however to talk about a process to find antiderivatives. This process amounts to undoing the Monkey Rules from earlier in the textbook.
The good news is that conceptually, this is a good bit easier than what we've done so far in this chapter on integrals.
The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives, you need to think a bit like a forensic analyst who studies the site of an explosion to see what sort of bomb was used. We'll discuss this analogy more later when we practice finding antiderivatives.
Because of the new level of difficulty of attempting to do the Monkey Rules backwards, I call the rules for finding antiderivatives Lucifer's Rules.
In general, your ability to calculate antiderivatives doesn't really impact your conceptual understanding of calculus. However, if you plan to continue your study of calculus, you do need to have at least a passing understanding of the fundamentals of finding antiderivatives. In future seasons we'll need a little bit of skill finding antiderivatives.
One last word before we get started: As we temporarily move away from studying integrals as an applied topic, we need some notation for antiderivative. After all, when an integral is calculated, you end up with a number, but an antiderivative is a function. The standard notation for "the antiderivative of f(x)" is:
The only difference between this notation for "the antiderivative of f(x)" and the notation for "the integral of f(x) from x=a to x=b