Learning Resources
Outcomes
In this lesson you will learn
- what is meant by the derivative of a function
- how to find the instantaneous velocity at any time, t
By the end of this section students will be able to:
- demonstrate an understanding for slope functions and their connection to differentiation
- determine and apply the derivative of a function
- demonstrate an understanding for the conceptual foundations of limit, the area under a curve, the rate of change, and the slope of the tangent line, and their applications
- derive and apply the power rule
- evaluate and apply limits
- analyze relations, functions, and their graphs
- use tables and graphs as tools to interpret expressions
Introduction
Several problems involving the calculation of an instantaneous rate of change have been solved in the previous section. No doubt some, if not all, of them took a great deal of time to solve. The purpose of this section then, is to offer other ways of finding the instantaneous rate of change; namely, the derivative. As the title of this section suggests, the derivative is the general formula for the slope of the tangent at any point on a graph.
The latter part of the section will involve the investigation of the graph of the slope function of the original function. This graph can provide key information about the shape of the original graph. Thus, by the end of this section, you should have a greater understanding of how slope is connected to the shape of a graph.
It should take 6 to 7 hours to complete this section.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- slope of a tangent
- slope of a secant
- instantaneous velocity
- right and left hand limits and their proper notations
- the formula for volume of a sphere (for question #8)
- the formula for volume of a cylinder (for question #9)