Meaning of DerivativeDate: 10/12/96 at 18:6:19 From: Dominic Tsang Subject: Meaning of Derivative Can you give me a plain English meaning of the idea of a derivative? Thanks! Date: 10/13/96 at 15:54:4 From: Doctor Scott Subject: Re: Meaning of Derivative Dominic, The derivative does, in fact, have a nice "plain English" meaning. You probably learned that the derivative is the slope of the tangent line to the curve at a point. The derivative describes the rate of change of the function at that point. Actually, it is the *instantaneous* rate of change of the function at the point. For example, if you are blowing up a balloon, the volume of the balloon depends on the radius of the balloon. That is, V = (4/3)*pi*r^3. The derivative of V (with respect to r) would tell you how fast the volume is changing as the radius changes. Hope this helps! -Doctor Scott, The Math Forum Check out our web site! Date: 10/13/96 at 16:2:48 From: Doctor Ken Subject: Re: Derivative Hi Dominic - Here's an example of the derivative and what it means. Let's say you've got a function, call it f(x). The derivative of f, f'(x), tells you how fast f is changing. If the derivative is positive, f is increasing, and if the derivative is negative, f is decreasing. If the derivative is 0, then at that point f is neither increasing nor decreasing. Let's say f(x) represents the position of a car on a straight road at time x. Then the derivative f'(x) tells you the velocity (that's like speed) of the car at time x. If the car is going forward the velocity will be positive, and if it's going backward the velocity will be negative. If you take another derivative, you'll call it f''(x), and that tells you the acceleration of the car at time x. If the car is speeding up the acceleration will be positive, and if it's slowing down it will be negative. Here's a little-known fact I'll let you in on: if you take another derivative, that has a name too. It's called "jerk," and we write f'''(x). Can you figure out what the physical interpretation of jerk is? The derivative is used all the time in physics, in exactly this kind of way. -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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