In algebra, you learn about the slope of a line which is equal to the ratio of the rise to the run. In other words, slope equals rise over run. Here you see that the rise is about half as long as the run so the line has a slope of about ½. The slope on a curve is constantly changing so to figure it you need to switch to calculus.
Just like the line you saw before, this line has a slope of about ½ and the slope is the same at every point between A and B. But you can see that unlike the line, the steepness of the curve is changing between A and B. At A the curve is less steep than the line and at B, the curve is steeper than the line.
What do you do if you want the exact slope at say point C, zoom in. When you zoom in far enough, infinitely far the little piece of the curve becomes straight and you can figure the slope the old fashion way. That is how differentiation works. Because of the derivative of the curve is the slope which is equals rise over run or rise per run, the derivative is also a rate of this per that like miles per hour or gallons per minute.
The name of the particular rate simply depends on the units used on the x and y axis. The two graphs here show a relationship between distance and time. They could represent a trip in your car. Here is a regular algebra problem, if you know where A and B are you can determine the slope between A and B. In this problem that slope gives you the average in miles per hour for the interval from A to B.
In this problem you can use the derivative of the curve to determine the exact slope or steepness at point C. While both problems determines slope the second one gives the single point on the curve that is the instantaneous rate in miles per hour at the point of C. For that you need to go beyond algebra to calculus to get the answer.
Transcription by:
Scribe4you Transcription Services