Definition of the minimum of f(x) on an interval: is the minimum of on an interval if for all on that interval.

Definition of the maximum of f(x) on an interval:

is the maximum of on an interval if for all on that interval.

The relative minimum and relative maximum of a function f(x):

If there is an interval over which is the minimum of on an interval if for all x on that interval, then it sis a relative minimum.

If there is an interval over which is the maximum of on an interval if for all x on that interval, then it is a relative maximum.

Critical number of a function:

Let be defined at a point, If or if is not differentiable at then is a critical number of

Extreme Value Theorem:

If a function is continuous on then over the interval there must exist at least one maximum and one minimum.

This tutorial describes application of differentiation such as related rated and extrema. By completing this tutorial you will create and analyze graphs of functions and their first derivatives. Also, examples showing the relative and absolute maxima and minima for a function are introduced with the help of examples.

Next, learning how to take derivatives implicitly is shown with the aid of example problems. Related rates problems are given with the introduction of example problems. The use of the KUDOS problem solving method is introduced to emphasize the concept of related rates and their applications. Graphs of the derivatives of functions are shown to point out the maximum and minimum values of a typical function.

Specific Tutorial Features:

• Graphic representation of derivatives.
  
• Example problems showing related rates problems.
  
• Implicit differentiation is shown with the use of derivatives and tangent lines.

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.

Graphs of functions and their derivatives
Going to (and Finding) Extremes
   Related maximum and minimum defined
   The definition of a critical number
   Finding extremes
Implicit differentiation
   Definition of implicit differentiation
Related rate problems