Education Standards

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

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

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

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

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