Why You Need Calculus
Suppose you are driving your car at 60 miles per hour. After 2 hours, how far have you gone? Ok, that is an easy problem. Now suppose you are flying your spaceship, and your speed is now given by the equation speed = time2. Now how far will you go in the next two hours? Ok, you’ll probably never need to know the answer to the second question. But engineers who want to land spaceships on the moon do. Physicists and other scientists also need to know this kind of stuff. And that is where calculus comes in.
Contents
- The Secret of Riemann Sums. Hardcore math geeks learn calculus by studying integration first using the technique of Riemann sums. But it is also the best way to develop intuition, so that is how we’ll do it too. Riemann sums are annoying, but I’ve chosen examples that make the math nice and simple.
- Using Riemann sums to Solve Hard Problems The math only gets slightly messier, but now we can solve problems that used to be beyond our reach.
- Derivatives and the Power Rule Finding derivatives is much easier than finding integrals.
- The Fundamental Theorem of Calculus We struggled finding integrals of functions with exponents larger than 3. Not any more. Now we’ll rip off our knowledge of derivatives and apply it to integrals. Nice shortcut.
- The Product Rule Why the product rule makes perfect intuitive sense. Then once you’ve got the product rule you get the quotient rule and fractional exponents almost for free.
- The Taylor Series Single term polynomials of the form axn are easy to integrate. And as luck would have it, you can write complicated formulas such x2sin x as a power series of these terms.
