I am tutoring a friend in calculus. Right now, she is working on finding relative maxima and minima as well as Rolle's theorem. While she gets how to find relative maxima and minima she does not get why it works. I have repeatedly explained to her the notion of the derivative. She can tell me its the rate of change or its the limit of the slope as it approaches zero, but can't explain "it" to me. I really want her to understand the concept of the derivative and more generally of Calculus. Do any of you have advice in how to explain these basic notions of Calculus to her or where to find such explanations? I really want her to get this, but also to understand the beauty and elegance of the results. Do you have any advice?

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For local (relative) maxima, the geometric intuition is reasonably clear: on top of a mountain, unless there is a sharp peak, things flatten out. – André Nicolas Nov 11 '11 at 21:25
Draw a picture of a graph that moves up and down. Have her pick out the relative mins and maxes. Ask her what the slope of the tangent line is at that point. Make sure there are some points where the slope is 0 and some where it is undefined (such as a sharp point). – Graphth Nov 11 '11 at 21:26
@AndréNicolas I did both of those and she was able to point. I am just wondering about some ways to reinforce and develop her understanding of the derivative to where she really understands it. – analysisj Nov 11 '11 at 21:30
Maybe no one ever fully understands. Teaching the course many times helps! At his stage, the best test is ability to solve problems, including some conceptual ones for which the amount of calculation needed is small. Also, ability to transfer from geometric to kinematic and back is a good test. – André Nicolas Nov 11 '11 at 21:42
@AndréNicolas Yes, I agree. Especially if it's some undergrad student that isn't majoring in math, they probably won't fully understand a lot of the stuff. The more they are exposed to it, the more they will understand it. If they can get the simpler concepts now, the more difficult concepts will make more sense later. – Graphth Nov 11 '11 at 21:54

Although the derivative, even in elementary calculus, is a pointwise notion, virtually all of the applications only make use of interval properties. Here's how I initially explain the calculus approach to max/min problems.

I draw a typical looking polynomial graph that has several turning points. Then I explain how we don't need to bother with the points within an interval where the derivative is positive. This is because the function will be strictly increasing on any such interval, and so no such point can be a local max (since points just to the right of it will correspond to higher points on the graph) and no such point can be a local min (since points just to the left of it correspond to lower points on the graph). Also, we don't need to bother with the points within an interval where the derivative is negative. This is because the function will be strictly decreasing on any such interval, and so no such point can be a local max (since points just to the left of it correspond to higher points on the graph) and no such point can be a local min (since points just to the right of it correspond to lower points on the graph).

The remaining possibilities are what we need to investigate, namely the points where the derivative is zero and the points where the derivative does not exist. We call these the critical points of the function. Then I discuss the first derivative test. Yes, I realize that the set of points where the derivative is positive does not have to be a union of open intervals (indeed, there exist everywhere differentiable functions $f$ such that each of the sets $\{x: \; f'(x) > 0\},$ $\{x: \; f'(x) = 0\},$ and $\{x: \; f'(x) < 0\}$ is dense in $\mathbb R$), but you don't need to get into these issues now (and probably not ever in beginning calculus).

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I am not sure why you want her to know more about Calculus, but to answer your question in general, there are some tutorials on Calculus that use animation. They vary in quality.

Here are some examples of what you may find (these are of basic level):

Video-Def. of the derivative

Video-Calculus Haiku/Animation

Plotting Derivatives

The fundamental Theorem of Calculus

I hope this helps.

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