Motion

Motion is a large topic of calculus and physics. In calculus, motion questions are used to evaluate a student's understanding of the difference quotient, derivatives, and integrals. For now we will focus on the first two, but we will look at integrals in time.

The whole show begins with a position function, s(t). If you want to find something's instantaneous velocity [v(t)], you just take the derivative of the position function. If you want to find something's instantaneous acceleration [a(t)], you just take the derivative of the velocity function. I equate these ideas to finding the tangent line, so I always remember to take the derivative.

We also talked about displacement. Displacement is your change in position. I always think about it as the distance from my starting point after I'm all done with the movement. "How far is it from where I began?"

We also talked about average values. Average velocity is the change in position divided by your change in time. Average acceleration is the change in velocity divided by your change in time. I just equate these ideas to the secant line, so I always remember what it is that I am doing.

I also asked that you look at example 6 on page 129. This talks about how derivatives are used in economics. In general, profit = revenue - cost or money in - money out. If you want to find the marginal revenue, the amount of money gained by producing one more unit or how the revenue is changing at that instant, you simply take the derivative or the revenue function. If you want to find the marginal cost, the extra cost of producing one more unit or how the cost is changing at that instant, you simply take the derivative or the cost function. If you want to find the marginal profit, you subtract the marginal cost from the marginal revenue.