How to find differentiation and integration of curves in general?

Question

Graph of function $f(x)$

How do I go about finding integration and differentiation of curves like these which yield other curves?

Answer 1

You can use a little bit of instinct for these things. Sure, you can just go and check the slope. But there are other signs as well. First of all, an integral of an odd function is even and vice-versa (if you disregard the constant offset). So, (A) and (B) are immediately out. Then, recall that integration smooths things. So integrating a discontinuous function makes a continuous one, a continuous piecewise linear becomes a smooth one with continuous derivative (no sharp angles) and so on. That rules out (D) and (B). Even more... you know that integrating a linear function gives a quadratic one. So the segments that are linear in the original function are parabolic in the result and constant parts become linear in the result. This makes it easy to sketch the curve by hand.

You can see that going forward (integrating) instead of trying backward (differentiating) also has a lot of clues and tricks, I'd say it's even more convenient than differentiation, because it's easier to see (integration makes for global properties, differentiation is local, so you must "probe" the function at different places). Of course, everything you know about differential properties of curves also applies (zero derivative is an extremum,...).

Answer 2

As $f$ is odd, $F$ must be even. This excludes (A) and (B). And $(D)$ is excluded because its derivative is zero in all the interval except in a finite number of points (where doesn't exists).