Lucifer's Rules Practice
There are a few more Lucifer's Rules, but we won't see those in this season; check out Season 2 of Calculus For the People (Fall 2020), or take Calculus 2 at your local college or university.
Let's take a moment to be sure we can use those of Lucifer's Rules we do have before moving to wrap things up.
Let's take a look at the function from the first applet from the Definition of the Integral. The function is f(x)=x^3-6x^2+9x+2. Looking back over Lucifer's Rules, we see the antiderivative of this function is not too challenging:
The key is using Lucifer's 3rd Rule to realize you can tackle each term one at a time, and then using Lucifer's 1st Rule to find the antiderivative of each term.
You might struggle with something like
6x^2, but I encourage you to pause and just think it through. It's tempting to "linearize" these types of calculations (and many calculus textbooks give elaborate processes) and give you exact step by step instructions, but you'll learn it better by just stopping and thinking about the integral with your problem solving hat on.
For instance, maybe try out a few possibilities and don't be afraid of being wrong. You might try 6x^3 as a possible antiderivative for 6x^2. But then when you use the Monkey Rules to check your guess, you'll see the derivative of 6x^3 is 18x^2. You're off by a third. And that's how you end up with 6*1/3x^3.
In Season 2 we will make this a bit more methodical, but for now, I really encourage you to use Lucifer's 1st Rule. It's not only easier, it also prepares you for future antiderivative Lucifer Rules in Season 2.Quick Check: Calculate
If you're chomping at the bit to learn more Lucifer's Rules: great! Be sure to check out Season 2 of Calculus for the People coming in Fall of 2020.