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