## Summary

- The kinematic quantities of displacement, velocity, and acceleration can be described by all sorts of functions.
- The function describing one quantity can be transformed into functions describing the other two quantities.
- The procedure for doing so is either…
- differentiation (finding the derivative) or…
- The derivative of displacement with time is velocity.
- The derivative of velocity with time is acceleration.

- integration
(finding the integral).
- The integral of acceleration over time is velocity.
- The integral of velocity over time is displacement.

- The techniques of calculus can also be used to analyze functions — including those that describe motion.
- The
*first* derivative of a function…
- is the instantaneous rate of change of the function.
- determines the slope of a line tangent to a graph of the function.
- equals
*zero* at a local extrema (*maximum* or *minimum*) or a *saddle* point of the function.

- The
*second* derivative of a function…
- is used to determine the direction of concavity of the graph of a function.
- The graph of a function is
*concave up* if its second derivative is *positive*.
- The graph of a function is
*concave down* if its second derivative is *negative*.
- An
*inflection* point (the transition between two different concavities) occurs where the second derivative is *zero*.

- is used to distinguish extrema.
- An extremum is a local
*maximum* if the second derivative of the function at that point is *negative*.
- An extremum is a local
*minimum* if the second derivative of the function at that point is *positive*.
- A
*saddle* point occurs at a location where both the first and second derivatives of a function are *zero*.