Calculus Problem of the Week October 7 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!))

This week, I am going with something other than a mere calculation. Give an example of a function which does not have a derivative of zero at any point in its domain and explain why your function “works” as an example. (There is one type of function that would be the easy example – if you think you figure that out, try to find a different example of another type of function)

See the solution.

As you start to think about this, a natural thought would be “what does it mean for the derivative to be zero at a point?”. There are several possible answers here but one is probably more useful: it means that the tangent line is horizontal.

In other words, this question is asking you to find a function that does not have a horizontal tangent line. The “easier” set of function for which this would be true is the set of linear functions y=mx+b. Unless the slope is zero, y'=m where m is a nonzero number.

What about other examples? If you can find a function where f'(x)=0 has no solutions then you are set! For instance, \dfrac{1}{x+5} which has a derivative of \dfrac{-1}{(x+5)^2}. This derivative is never equal to zero.