What is the Purpose of Determining a Derivative?
Date: 12 Jan 1995 12:57:41 -0500 From: Jenny Lay Subject: Help! Dear Dr. Math, Earlier this semester we learned about derivatives in Calculus. I know how to determine the derivative of something, but what is the purpose? Thanks. Jenny
Date: 12 Jan 1995 14:11:43 -0500 From: Dr. Ken Subject: Re: Help! Hello there! There are lots of reasons we'd want to take the derivative of something. First of all, let's say you're riding in your shiny new sports car and you have the best odometer in the world. It will tell you to the nearest thousandth of a mile (or something like that) how far you've gone. If you graphed what the odometer tells you as a function of time, so that time is on the x-axis and distance is on the y-axis, you could take the derivative of this function and figure out your speed for every point in your journey. So all the information about your speed and acceleration and everything can be gotten from the odometer, as long as you know how to take derivatives. Here's a question my calculus teacher once asked me: in cars, there's both an odometer and a speedometer. Essentially, the speedometer takes the derivative of the odometer information (before it gets to the odometer though; it's straight from the wheels). How does it do that? It's been doing that since way before on-board computers happened to cars. So essentially, they've found a purely mechanical way to take derivatives. Neat stuff, worth researching. The derivative is also quite an intuitive concept, I think. Let's say you have a growth chart on your wall. If you're a human (which I believe you are) you'll probably have a couple of periods when you grew faster than at other times in your life. If the marks were made at regular intervals, they'd be more spread out in certain periods and more clustered together in others. So it's not hard to figure out from this chart that you grew faster in those growth spurt times than in the lull times. Well, how fast you grew is just the derivative with respect to time of how tall you were. So the derivative will be big sometimes, small sometimes, and once you hit 40 years old, it will be negative (some people say). So these are a couple of real-life examples. Other examples that are based on integration (the inverse of differentiation) would include finding the volume of some objects, finding the area of some regions in a plane, and stuff like that. And trust me, if you go on and do some more in math, taking the derivative of functions will be SHEER BLISS compared with some of the more nasty stuff (which is more rewarding. Stick with math!). So that's how I feel about derivatives.
Date: 12 Jan 1995 15:23:35 -0500 From: Dr. Elizabeth Subject: Re: Help! Hi Jenny! One of the nicest things you can do with derivatives is to find out where the maximum value of a function is - which can be a very useful thing to know. The derivative of a function is its slope at any given point (actually it's the slope of the tangent line to the point, but even though a point can't have a slope, I always found it easier just to think of the derivative this way). At the highest point of a function (its maximum value), it's gone as far up as it's going to, and it's about to start heading back down. Its slope at this point, then, will be zero, since it isn't going up, and it isn't going down. Let's say you had a function and you wanted to know where, exactly, it would reach a maximum (I'll give you an example of why you might want to know this in a minute). Take the derivative of the function and set it equal to zero. Then solve for x or a or the number of feet of fence or the number of frogs or whatever it is that your independant variable happens to be. Here's my example: I throw a ball straight up at a speed of 6 meters per second somewhere where there's absolutely no air (no air resistance). When will it be at its highest point? I can write the ball's height in an equation: the height at any time t will be the ball's upward velocity times the amount of time it's been going up (6m/sec times time, t) minus its gravitational acceleration downward times the amount of time it's been up there squared (10 m/sec/sec *that's the acceleration of gravity* times time squared, t^2). Leaving out the units so that the math is easier to see here: Height = 6t-10t^2. If we take the derivative of this function, we have an equation for the slope of this function at any given time, t. The value of t for which this new equation is equal to zero is the same t at which the height of the ball will be a maximum. Derivative of height = 6-20t = 0 6 = 20t t = 6/20 second The ball will reach its maximum height in 6/20 of a second. Hope this helps! Elizabeth, a math doctor
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