# Higher order derivatives and graphs modules

## Overview

After completing this section, students should be able to do the following.

- Use the first derivative to determine whether a function is increasing or decreasing.
- Define higher order derivatives.
- Compare differing notations for higher order derivatives.
- Identify the relationships between the function and its first and second derivatives.
- Sketch a graph of the second derivative, given the original function.
- Sketch a graph of the original function, given the graph of its first or second derivative.
- Sketch a graph of a function satisfying certain constraints on its higher-order derivatives.
- State the relationship between concavity and the second derivative.
- Interpret the second derivative of a position function as acceleration.
- Calculate higher order derivatives.
- Given the graph of a position function, find time intervals where the velocity is positive/negative
- Given the graph of a position function, find time intervals where the velocity is increasing/decreasing

# Rates of rates

## Ximera Module

Two young mathematicians look at graph of a function, its first derivative, and its second derivative.

# Higher order derivatives and graphs

## Ximera Module

Here we make a connection between a graph of a function and its derivative and higher order derivatives.

# Concavity

## Ximera Module

Here we examine what the second derivative tells us about the geometry of functions.

# Poison, velocity, and acceleration

## Ximera Module

Here we discuss how position, velocity, and acceleration relate to higher derivatives.