Ask anybody “what is a derivative?” and you’ll quickly find out that nobody can tell you exactly what it is in no uncertain terms without any question. If fact, most people you ask who have taken calculus can’t even come close. Think I’m wrong, try it and you’ll find out what I did, it took me eight years after college to find out and even then not exactly. That’s pathetic and unacceptable in my opinion. Of course I was programming the TI Calculators in calculus at the time so I had some interest to even ask the question. I mean, who cares right? No one out of college will ever calculate a derivative or integral again in any job outside of re-teaching it as a professor, so I/we understand that.
I had some interest because of programming step by step calculus into the calculators and while teaching tennis to this guy named Mike, I find out – he at one time worked at NASA. At the time of me teaching him tennis he had left NASA was a professional black jack player. Went all over the world making money at black jack in the casinos. I was also fooling around with on line poker at the time and asked him how come black jack and not poker? He said poker was too much gambling and black jack is relatively sure. He said he used differential equations to help him count cards and change the odds in his favor. Anyway I asked him what was a derivative? He said immediately that it was the slope of a tangent line to a curve. I said “but when I graph the derivative there is a line – but no tangent line and the slope is off.” He said the graph of the tangent line is meaningless, “of no value”. He said when you compute the derivative of a function at an “x” value you come up with a number and that is the slope of the tangent line at that point on the function. I thought even that was fabulous and an eye opener, but we had finished picking up tennis balls and so I let it go and started to again teach him tennis.
After that moment, I kept thinking and thinking and thinking of what he said and then it dawned on me. The number you get when computing the derivative at the chosen point “x”, no matter how deep the original function is the numerator of a fraction with the denominator = to 1. That was rise over run. If the numerator is 12, that is 12/1. You go 12 spaces up the “y” axis and one space over on the “x” axis. Draw a line from that point through the origin of a graph (0,0) and that is the exact angle or slope of the line of the derivative. For that line to me tangent, it must touch the curve at only one minute point, so that line has to be transposed to do that, however it will still be the exact slope. To transpose that line you have to compute the equation of that line to the desired point (point slope form) and then graph that function and you have the perfect picture of what a derivative is. That’s fabulous. (Equation of a line at a point on a curve is in my calc1 programs). Now at the next party you know all there is of “what’s a derivative” and can look like a nurd, I mean nerd. lol, Tom
p.s. (The exact angle of that slope by the way is tan^(-1)(12,1), Fabulous!!
You can use derivatives in alomst every branch of science. For example In physics:If s = f(t) is the position of a particle as a function of time, then the velocity, v, is the derivative of the function, s’ or f'(t).Let’s say you have the height equation of an object, h(t) = 20t 5tb2. The velocity would just be the derivative of this equation at a certain time. So if t = 1second, then v(t) = 20 10t and v(1) = 20 10(1). Then you can find that the velocity equals 10 meters/second. Another fact that I think is interesting is that the double derivative of f(t) would equal the acceleration because it would give you the units of acceleration, or m/sb2Another example would be in economics:Suppose the cost of producing x items isC(x) = 5000 + 25x -bdxb2 + ⅓xb3 (units are in dollars)C'(x) is the marginal cost, or the dollars per an item.An example: you can find out how much it would cost to produce the seventh item by finding out how much it cost to produce 6 items then 7 items and find the difference.C(6) = $ 5204 and C(7) = $ 5265 so the seventh item cost $ 61The marginal cost of 6 items would be:C'(x) = 25 -x + xb2C'(6) = 25 6 + 36 = 55
, I will admit that it’s fun for me when I get to the Pavlovian point, that is, I’ve mastered the use of a copncet and now I don’t have to think about it.But I have to get there myself, in this case, by seeing the chain rule and then doing a bunch of exercises. If you try to beat it in, you’re going to lose me.