(11-17-2013, 04:24 PM)Peter_C Wrote:To the non-mathematician here's what you need to know about the BIG ideas of calculus:(11-15-2013, 04:42 PM)justMongo Wrote:
Since my calculus text is in the Smithsonian, what text would you recommend for a refresher course?
Since I am not actually sure what 'calculus' is - I assume I should just stop thinking about this post, right?
Calculus comes in two parts. Differential calculus studies something called the derivative, which is nothing more than the rate something is changing. Do you remember the idea of the slope of a straight line?
Slope = rise/run = (change in y)/(change in x)=rate y changes with respect to x
In differential calculus, you learn how to calculate the slope of curves---functions that are not lines. And the slope of the function is just the rate the function is changing. In other words, differential calculus studies the Rate problem: How do functions change? For example: Velocity is the rate function for position, so velocity is the derivative of position. That's really main BIG idea of differential calculus.
Integral calculus studies the area trapped underneath a curve and over a particular interval of [i]x[i]-values. In K-12 mathematics we learn all about finding areas of squares, rectangles, triangles, circles, and shapes based directly on these things like trapezoids. In integral calculus we learn how to find the area trapped under a parabola. Or under one hump of a sine wave. This is the Area problem, and it's the main thing integral calculus is concerned with.
The first way to solve the Area problem is by approximating the area under the curve with a whole lot of very skinny rectangles and adding them up. As the widths of the rectangles gets smaller and the number of rectangles in the approximation gets larger, the approximation gets better and better. So Area is defined in terms of a nasty limit. It's not very pretty and it's not very easy to work with, but it's an important idea. This limit definition of Area is the first BIG idea of integral calculus.
The connection between the Area problem and Rate problem is something called The Fundamental Theorem of Calculus. The Fundamental Theorem says "the rate the area under y=f[x] is changing at x=a is given by f[a]" This basically says, the Area problem (from integral calculus) and the Rate problem (from differential calculus) are inverse problems. And the Fundamental Theorem of Calculus provides the mathematician and scientist with important short cuts for solving the Area problem without going through the nasty limit definition first used to describe the Area problem. This is the second BIG idea from integral calculus.