Conceptual Question Relating to Calculus?

Question

I know that the derivative of a function represents the slope of the function at specific points. So I suppose I was wondering what, if any, information can be gained about the original function by looking at the second (or greater) derivative. Or, if the second derivative doesn't provide any insights into the original function, does knowing the slope of the slope of the function tell you anything?

This isn't for class or anything; we were doing higher derivatives in class, and I was just wondering if there was any point/reason for them.

Answer 1

Once you get into higher derivatives they can at any point tell you where the local maximums and minimums (the dips) of the function are. Also you can find inflection points of the function, which means you can find out at what point the concavity of the curve changes. This is all in Grade 12 Calculus class in Ontario.

Answer 2

let's look at a common physical problem: displacement, s(t).

s'(t) is usually called velocity

s"(t) is then called acceleration.

Note positive acceleration means the velocity is increasing, right? Negative acceleration means it is decreasing, so zero acceleration means the velocity has somehow reached a maximum, minimum, or the displacement function has an an inflection point, ie, the displacement (not the velocity function) changed its sense of concavity.

I see this as a good example, simply because the derivatives have common names.

Answer 3

it will give you the first derivative of the function modulo a constant.
if you integrate that you will get the original function modulo a constant times x + another constant.

higher ( second ) derivatives are usefull to find max/min of the original function.