Calculus is useful for solving non-linear equations. For example, say you were looking for the area inside a flat rectangle. A flat rectangle is linear, and finding the area would just be a simple matter of multiplying one side to one of the perpendicular sides, or 3 inch times 5 inch equals 15 square inches which is the area. What if it was a rectangular box? Same difference, still linear - The length times width times height may equal 3 times 5 times 4 equals 60 inches cubed. These are "linear equations".
But what if it isn't linear? What if it's flat and curves around like a sine wave? Looping back and forth and never ending? And say you want to find the area of this flat sine wave from 0 to pi? This isn't linear at all. If you look at this curvy line from a distance, it's rounded. But what if you pin-pointed on one spot on that curvy line? And you keep getting closer and closer to that spot? Eventually, you would have a gnat's view of that one spot and it isn't curved at all. It would look like a straight line. Kind of like looking across the ocean. The earth is curving, but it looks dead straight for miles from here, cause you are so close to it.
From this point, you can take a measurement that is pretty much linear, do the 3 inch times 5 inch routine and record it. Now move across by 3 inches and do it again, line still looks straight here also. Now add it to the prior one. Keep doing this until you reach your destination of pi, add them all together and you get your area. You've made your curve into very tiny little rectangles. It's not absolutely exact, cause the itty bitty curve part at each interval wasn't added in, but it's really close. If you took and just drew a straight line across the entire curve, the dome part you miss is huge, but since you made a hole bunch of little domes, it's getting closer to the true area.
To make it closer to exact, keep making each small interval a little smaller. That makes the curve part more insignificant, making the answer more exact. The smallest interval would be zero inches, or whatever. But we can't do that. Can't divide by zero. But we can "approach" zero and get extremely close, say 1 zillionth of an inch maybe, then add up all zillion of these extremely thin rectangles. If you try to add up a zillion different rectangle areas, it may take a couple of hours, or so... Yep... Maybe even 3 hours considering you had to find the areas of a zillion rectangles in the first place..... (Bad joke).
By the way, doesn't have to be an area. Look at the squiggly line for a road on a map. The smaller you make the intervals, the more accurate you come to the distance of that interval, and you just add all of them up to come to the answer- How far is it to Denver? Draw a straight line from your house to Denver, and you would be wayyyy off.. Unless you live in Denver. That, by the way, is why Calculus' integral sign looks like a wavy "S". It's a summation of little intervals as their distance approach zero.
This is where calculus comes in. It allows you a quick mathematical procedure to find these irregular areas as their length "approaches" zero. It's impossible to master. There are so many different formulas to keep up with, so many different scenarios, it becomes endless. Nobody masters calculus, they just get used to the parts they need to know, and forget all the rest they learned in Calculus getting "up" to what they needed to know. In Calculus, you have a single integral, giving you length, a double integral giving you area, and a triple integral giving you depth area. Position is the derivative of velocity, and velocity is the derivative of acceleration. An integral goes the opposite direction where acceleration is the integral of velocity which is the integral of position. Everything we do on earth can be broken into areas, positions, accelerations, and velocities. Everything.
So if you have a hankering to know the exact force a 2,127ton meteorite with a 26degree angular velocity of 63,000Kph, spinning at 3,800kph will cause when it strikes a planet moving at -64 degree angular velocity of 40,300Kph, striking it at an angle of 42 degrees on a surface spinning at 4,600kph at a 12 degree angle, I suggest you break out a calculus book.
Or simplified... What is the charge in a 47uF capacitor in a particular circuit at 3.5uS? This is NOT linear. Calculus is the quickest and most accurate way to measure this and makes all these HDTV's, PDA's, and the occasionally successful Mars landings possible. Once you learn up to the part you need, it takes a couple of minutes to pull off the answers, instead of the rest of your life.
Comments
Well written answer to a tough question!
by Jodie44 on April 14th, 2006