An Introduction to Derivatives
Date: 05/17/98 at 14:15:43 From: Erika Subject: Third derivative? Hello, I have to research derivatives for a project. It asks about a third derivative. I haven't seen a second mentioned so I was wondering, is there a third derivative? Thank you for your time.
Date: 05/20/98 at 22:44:11 From: Doctor Pat Subject: Re: Third derivative? Erika, Yes, there is a second derivative, and a third, and a fourth... at least as long as the function continues to be differentiable. With polynomial functions you can take derivatives forever, but after long enough they all get kind of boring. I'll let you figure out why. A derivative is just the rate of change of one thing as measured by the change in another. Wow, I find that hard to understand and I wrote it. How about we look at a real world example? If you go from here to there, you change your position. Let's pretend there is a number line along which you are walking, and a big clock on the wall. As you walk, your position and the clock are both changing. Your velocity (speed with a direction attached, but you knew that already, I suspect), which is the First derivative of position with respect to time, dP/dt, is how far you go divided by how long it takes. You may recognize this as the sixth grade formula for rate: distance rate = -------- time In calculus, we make the change in time smaller and smaller and use limits to come up with an "instantaneous" velocity. If you are moving with a constant speed the average speed (sixth grade method) and the dP/dt (really cool calculus method) are the same. So what is a second derivative? Well if you were walking along and decided to walk a little faster, the first derivative (your velocity) would also increase. The second derivative measures the change in the first derivative per unit of "whatever" (in our example, time). The common language for the change in velocity per unit of time is acceleration. When you step on the gas, you "change" your speed (velocity). How fast it is changing is the idea of a second derivative. And if your second derivative is changing, guess what we call that? We don't have names for all the different derivatives, or at least I don't know the names if they do exist, but you can see how each one is related to the one before it. Graphically you should also try to understand that if you graph y versus the nth derivative of x, then the slope of the curve at any point is the (n+1)th derivative of y with respect to x. Let's walk through the example with some computations of derivatives. If you do not know how to calculate derivatives and are not expected to, ignore all of this. If the position of some object is given as x(t) = t^3 + 2t + 1, then we can figure its position at any time by evaluating for the value of t. At t = 0, the position is x = 1 at t = 1 the position is x = 4. We don't know how it got there, or where else it has been, but we know that between t = 0 and t = 1 it moved 3 units to the right. Its AVERAGE velocity was +3 units per time unit. If we take the derivative of x(t) with respect to time, we get the velocity. This is the FIRST derivative: x'(t) = 3t^2 + 2 We can use this function the same way we used the position function, only we get the velocity at each moment. For example at time t = 0 the velocity was 2, and at time t = 1 the velocity was 5. From this we can see that not only is it moving to the right, but it is speeding up. We can expect that it will move even farther to the right in the next unit of time than it did before. As before, now that we know two values of a function, we can figure the average acceleration (change in velocity/change in time) and get 3 units per time unit squared. If we want the instanteous acceleration at any moment, we take the derivative of velocity, which is the second derivative of position: x''(t) = v'(t) = a(t) = 6t This is a function which gives us the acceleration at any time. By now I think you get the picture. By the way, the third derivative of position with respect to time, which is the change in acceleration with respect to time, is named for what happens when you experience a sudden change of acceleration: you feel a jerk. Hope that helps. There are lots of ideas and lots of language involved in the ideas of calculus, but they really help to explain ideas that involve change and motion in a way earlier math could not. That is why calculus was so necessary for Kepler and Newton when they worked with the orbits of planets and the laws of gravity. If you need help on the mechanics of calculating a derivative, drop us another note. -Doctor Pat, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 05/22/98 at 08:19:42 From: Goodeffort Subject: Re: Third derivative? Thank you for helping me. The information you sent was just what I needed! :-)
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