Quandaries and Queries


Hello, my name is a jathiyah, i am a freshman in college. I wanted to know how would you tell (on a graph diplaying two funtions), which funtion is the derivative of the other?


Hi Jathiyah,

Assuming you have to images (graphs) and they do have the relationship that one is the derivative of the other, then a few simple local features should be enough to tell the difference.

For example, if you look at the maxima of the original function, they should be zero values of the other. That alone might tell you.
[The minima of the original will also be zeros.]

At the next level, a maximum will have (left to right) postive slope, zero slope, negative slope- and the derivative will be coming down from the positive side, to zero, and on into the negative values.

Visually, the concavity of the original (shaped like a cup down imples the derivative is decreasing, shaped like a cup upwards imples the derivative is increasing) give some other matches.

A final step is that the maxima and minima of the derivative will be the points of inflection of the original function.

When you have studied a bit more calculus, you will learn the fundamental theorem of calculus. That will give the connection back in the other direction. Given the derivative, find the function. This can be worked out as expressions of 'the area under the curve' gives a function with your starting point as the derivative. With that image (probably in you text, but later) you can work both backwards and forwards - a nice check on all your reasoning.

Walter Whiteley


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