A journey along the graph of a function - Increasing and Decreasing Functions

Click the Start button to start the journey of the Tortoise along the graph of the function. When the Tortoise stops, click on him to start the next section of the trip. Irina Boyadzhiev, Created with GeoGebra

This applet can be used for illustration of “increasing” and “decreasing” intervals for a function. The students with some knowledge of calculus can look at the graph of the derivative and how it relates to the intervals of increasing/decreasing.

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Tracing the Derivative

Consider the function $f(x) = x^2 - 3x$. The number m is the slope of line a tangent to f at Point A. Point B is the ordered pair (n,m) where n is the x coordinate of A and m the slope of line a. Now drag point A along the graph. What do you observe? What can you say about the traces of point B?

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The graph of the traces of point B is the derivative function of f. Can you explain why?

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Graphs and Their Properties

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Questions:
1.)   Which interval/s of the function f is/are increasing? is/are decreasing?
2.)   What is the relationship between the critical points of the graph (maximum, minimum, point of inflection) to the function to its derivative?
3.)   What is the relationship between critical points and the slope of the tangent line?

The Geometry of Derivative

Move point Q towards point P.


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As Q approaches P,

1.) What do you observe about the inclination of the secant line in relation to the inclination of the tangent line?
2.) What do observe about the value of h?
3.) What do you observe about the value of the slope of the secant line?
4.) What happens to the value of h and the slope of the tangent line if Q coincides with P?