Is calculus practical? Will you ever see it again, after this semester?
Sure, every time it rains. For example, farmers operate their own rain gauges because a forecast for the entire county might not apply on, say, the east slope of their high hill. The farmer can't afford to trust averages for the county.
This is from a farm journal:
Last night you heard the weather reporter say “Rainfall measurements of 100 mm/ 3 inches were recorded in [your area] between 6am to 8pm.” Then you’re telling yourself that can’t be true. Yesterday it didn’t rain at all. You’re sure because you were waiting for the rain all day long.
What is the truth then? The local weather service uses sophisticated tools and a team of professionals monitoring the rainfall amount and other conditions. However, they don’t cover every acre of land. Their measurements are based on averages of a few chosen locations. In addition, rainfall is not distributed evenly.
The information they have may not be much use to you, especially when you’re into farming. You need exact rainfall data in your specific area if you want to be successful. You need useful information that will help you better forecast your yield and harvest time. That’s why a rain gauge is an invaluable tool for you to have.
Look at these two rain gauges. Your friend the farmer has two problems with rain. Which rain gauge does he need?
His first problem is not enough rain. His second problem is too much rain too fast. Too much too fast can wash out his sweetpeas.
The first rain gauge looks like a measuring cup on a stick. You check how full it is and then you empty it for next time. Like baking a cake.
How do you know how much rain fell between midnight and midnight? By checking at midnight. By taking a flashlight and a pencil and paper out there, maybe in the rain, and writing the amount, and emptying the cup before you head back to bed by way of the bathroom.
What if you don't read it until five AM? After 29 hours instead of 24? Well, you could divide the inches of rain by 29 instead of 24 and still get an amount per hour. Or you could empty the thing after 48 hours and divide the amount by 48, but your average is getting sloppier and less useful.
Say you got two inches of rain. Did it come in a rush, in twenty minutes? Or in a slow drizzle for two straight days? This rain gauge can't tell you.
The second rain gauge works like a teeter-totter (or see-saw, depending on where you grew up). Rain comes down the funnel onto the high side of the see-saw until it fills the cup. The weight of the water tips the see-saw and dumps the cup. The other cup goes up and starts to fill. The rain gauge keeps a record of all the times it tips. Maybe it tipped like crazy for twenty minutes and hardly tipped at all after that. You know your two inches of rain came in a twenty-minute rush, not a long slow drizzle.
What's the difference? Think of the two-inch accumulation as the integral, and the rush of two inches in twenty minutes as the derivative.
I get a gorgeous ten-day weather forecast that has both: Precip. Accum. Total (mm) and Hourly Liquid Precip. (mm).
If you know one you can figure the other. The Fundamental Theorem of Calculus? Big name for a simple idea. If you know the rate you can figure the accumulation. If you know the accumulation you can figure the rate. You've been doing it all your life, in a rough way. How long to grandma's house 200 miles away? Well, if you can do fifty it's four hours. At night in the rain you might drop that to forty and it's five hours. But that assumes you can hold a steady speed. Your speed will vary if you have great roads through the flats mixed with tricky back roads in the hills. Now what? You can't count on your forty or fifty average. You're just guessing. That's good enough for Sunday dinner with grandma but not good enough for a manned landing on the moon.
What you learned of geometry and algebra has been known for two and a half thousand years. Calculus, closer to two and a half centuries.
One more example. Suppose the odometer in your friend's car is broken but he has a mobile app that records his speed every ten seconds while he follows his Google map directions.
Even with the help of Google maps he got lost and went God knows where along the way. When he got lost he went faster, then slower, then stopped beside the road to think, then backed into a drive and turned around, and so on, over and over. He went many different speeds along the way. If you graphed his speed the line would go up and down and up and down, not straight and level like someone doing seventy on the Thruway.
He needs to know how far he went. Can you tell him?
Yep. Not where, exactly, but how far.
You have all those different speeds. Speed and velocity are different. If you speed up when you're lost you get more speed but less velocity. Velocity is speed in a selected direction.
You have his speed every ten seconds. Basically you just figure how far he went in ten seconds, over and over. Say for example he went two miles an hour for ten seconds. Two miles an hour is two miles in 3600 seconds (60 times 60). In those ten seconds he went 2/360 miles, or just over 29 feet. Now do that every ten seconds. Then add them all up. Sounds like a lot of arithmetic, eh? Fortunately someone found a shortcut, a way to skip most of the arithmetic. Calculus is the shortcut.