Heh, *“is hard”* kind of says it all!

I still remember my seventh-grade science teacher telling us how, in physics, “work” requires moving weight against gravity. Just pushing weight around on a level surface, technically, wasn’t “work”.

Science is weird! 😆

]]>I hope you did, too.

Gotta say, I love living in the boonies. Walked in, stood behind one person for less than a minute, checked in, got my ballot, marked it, put it in the machine, and got my sticker, all in just a few painless minutes. Small Cities Rule!

]]>We want to calculate the roots of the derivatives — points where the value is zero. So we will.

Given the requirement that ** a** and

As ** x** grows larger,

Since the exponential never vanishes, for the derivatives to vanish requires their first parts, the polynomial, vanish. So, we just need to see if and where they do.

The first one is easy. Rewritten slightly, it’s obvious that for this to be true…

…the left part is a constant, so the only way for this function to vanish is for ** x** itself to be zero. And indeed, the first derivative has just the one root,

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The second derivative is only slightly less obvious:

We again factor out the constants, ignore them (they’ll never be zero), and focus on what remains:

So, the second derivative has two roots, ** x=±c/v2**. We can see that

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The third derivative is more of the same. Absent the exponential, we have:

We again ignore the factored-out constants focus on what remains:

So, the third derivative has the two roots **±v3/2× c** as well as one at

And that’s it. **QED**?

This appendix shows step-by-step how to calculate the derivatives of exponential functions.

We’ll start with the base equation:

We use a special instance of the chain rule to derive an exponential function. Recall that the basic chain rule is:

For exponential functions we have:

Because the derivative of the exponential function is the exponential function. That’s kind of its whole deal. It’s its own derivative. Few functions can say that.

So, for the temperature equation, we get the first derivative like this:

Which, rearranged a little is the first derivative shown above:

To derive this one, we’ll need the product rule since there are now two factors that are functions of ** x**. The product rule is:

In the case of the first derivative, the two factors are:

So, we start with:

We factor out the exponential and do the derivative (** x** vanishes), which gives us:

Then we do the multiplication on the left of the plus and multiply the right side by c²/c² to make the denominators match:

And again, with a little rearranging, we have the second derivative shown above:

This one also has two factors that are functions of ** x**, so deriving it is similar to the previous one. Here the two factors are:

The second one is the same, of course. The exponential function keeps reappearing in its derivatives! Given these functions, we derive them:

As before, we factor out the exponential and do the derivative (this time we still get a function of ** x**):

Again, we do the multiplication on the left and multiply the right side of the plus sign by c²/c² to make the denominators match:

The second and third terms inside the parentheses match, so we can reduce this to the third derivative shown in the post:

We won’t derive it further (although we could; endlessly). We just wanted this to calculate its roots. Those tell us where the minima and maxima of the second derivative are, and *those* indicate the maximum curvature parts of the base function (which, admittedly, falls into the *Nice to Know* category).

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The temperature equation isn’t parameterized for time (yet), so there’s no way to calculate the time derivative at this, um, *time*.