Definition of the Derivative Part 1

In this lesson, we learn how to approximate the derivative of a function by making the interval h smaller and smaller. The derivative describes the rate of change of the function at a given point and is represented by the slope of the tangent line to the function at that point. With the help of an applet, we can see how the slope of the secant line changes as we move the interval closer to the point of interest, eventually obtaining an approximation of the slope of the tangent line, which is the derivative.

Be able to explain the definition of the derivative (THE fundamental notion of Calculus) in terms of a limit of the difference quotient and more importantly the Cartesian plot of secant lines vs tangent line to a function.

• What is the formal definition of a derivative?
• What is the limit notation for a derivative?
• How do you approximate a derivative?
• How do you find the average rate of change or slope of the secant line between two points that are very close together on a curve?
• How do you find the slope of a function at a point?
• What happens as h gets smaller and smaller as you try to get two points on a secant line as close together as possible to find the slope at a point of a function?