Rolle’s Theorem:

If is continuous over and differentiable over and if then there is at least one number in such that

The Mean Value Theorem:

If is continuous over and differentiable over , and if then there is at least one number in such that

Definition of the second derivative of a function:

The second derivative is the derivative of the derivative of a function.

First Derivative Test:

If is a function continuous on and differentiable on then:

Iffor all in then f is increasing on

If for all x in then f is decreasing on

If for all x in then f is constant on

Second Derivative Test:

If f is a function continuous on and differentiable on then:

If for all x in then is concave up on

If for all x in then is concave down on

If for all x in then is changing concavity on

Concavity:

Curvature of a line or object, represented by the focus of the points on that region.

Concave upward:

Curvature such that the “bowl” of the curvature is directed upwards, or the focus is above the curve.

Concave downward:

Curvature such that the “bowl” of the curvature is directed downwards, or the focus is below the curve.

This tutorial discusses Rolle’s Theorem and the Mean Value Theorem with the help of examples and graphical representation of the concepts on hand. Learning how to take second order derivatives is discussed in this tutorial. The use of second derivatives to find possible points of inflections is covered in this tutorial.

The knowledge attained so far about curve sketching and their properties is mentioned in this tutorial. The usage of the Mean Value and Rolle’s Theorem is covered in some sections of this tutorial. Taking higher order derivatives requires a deeper understanding of the second derivative and its generality. The first and second derivative tests are covered in this tutorial.  

Specific Tutorial Features:

• Animated diagrams showing the first and second derivative tests.
  
• Graphic organizers to illustrate the inflection points of a function and the different types of concavity covered in this tutorial.

Series Features:

• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
  
• Definition slides introduce terms as they are needed.
  
• Visual representation of concepts
  
• Animated examples—worked out step by step
  
• A concise summary is given at the conclusion of the tutorial.

Rolle’s Theorem and the Mean Value Theorem
Taking Higher Order Derivatives
Points of Inflection- An application of the second derivative
   Concavity and its different types
Curve Sketching