A few days ago I pulled off a minor miracle.

First, demonstrate the power rule and a few of the trig derivatives (handy Desmos thing!). Then, as a side discussion, write the equations of motion they're probably familiar with from their algebra-based physics course [x(t) = x0+v0*t+1/2*a*t^2 etc.]

With x(t), v(t), and a(t) sitting up there, take the derivative of x(t). WHAT DID YOU DO. Take the derivative again. OMG NO WAY NO WAY.

Then, grab a tennis ball and mimic this flight path.

Questions to ask as you flail around with a tennis ball:

- generally, is the slope at the beginning positive or negative?
- what do they say about velocity at the peak of these flight paths?
- what slope would a tangent line have at the peak?
- generally, is the slope at the end positive or negative?

Confirm their findings a few times. I generalize it with something like "so, you're telling me velocity is plus, zero, then minus here?" Nods all around. However you want to word the conclusion is up to you, but make a connection between the position changes being caused by the sign changes in velocity. In the middle we start losing velocity, the ball can no longer maintain its height, etc.

Now the tricky leap. Have them visualize the balls velocity. Again, arms flailing, move the tennis ball in a negative sloping line, mirroring what they told you before with "plus, zero, minus." They'll stumble here a bit because (at least my groups anyway) aren't super comfortable with velocity graphs. They've seen them, but have they *seen* them?

The connection to the power rule is huge here. At some point you've established that power functions reduce by one power when taking a derivative. So somewhere in the back of their head should be "quadratics reduce to linear." A few repeats of that line and BAM, they make a small step towards curve sketching f'(x) based on knowledge of f(x).

If you jump that mental hurdle (take your time), the next move will be easier. Start talking about the velocity changes of the ball. Plus, zero, minus (lots of tennis ball flailing). Plus, zero minus (flail flail flail). What's the slope of the *velocity* curve? Negative? Ok...and what's the general sign convention for gravity? Nega---OMG YOU DID IT AGAIN.

In 20 minutes of tennis ball flailing you just discussed a majority of differential calculus. Refer to this metaphor as much as possible. It is the most helpful thing, especially when it's time to establish the role of integrals.