Calculus Problem of the Week October 21, 2011

Sometimes easy and sometimes hard, our calculus problem of the week could come from any calculus topic. If you really want to get better at calculus, following these problems is a great way to make yourself practice! Past calculus problems of the week.

This week’s problem:
(click “see the solution” at the bottom of post to, well, see the solution. If it doesn’t work, please click the title of this post and then try (I’m working on this!))

Let f(x) be a differentiable function at x=0 and \lim_{x->0}\dfrac{f(x)}{x}=4. Find f(0).

See the solution.

Think about this for a second: If f(0)=5, f(0)=100, or f(x) as anything other than 0, we wouldn’t be saying the limit is 4.

No, instead, we would be saying the limit doesn’t exist since “plugging in” the zero would give us \dfrac{0}{0} (it is reasonable to assume \dfrac{f(x)}{x} is as simplified as it can be[/latex]).

Therefore, f(0)=0. In this case, finding the limit would involve some kind of algebraic manipulation perhaps that we just can’t see since we don’t know what f(x) is exactly! You can probably think of quite a few functions that would work here like f(x)=4x for example.