2-C Derivative Function
Instructions
You will construct the graph of a function whose values are the rates of change of the function on the left.
- Use the input box (or click-and-drag P) to adjust c to different input values.
- The "Estimate Slope" checkbox will show/hide a line segment at the point P. Click and drag the left endpoint (gray point) to adjust the line segment so that it is approximately tangent to the graph. The input box for h will adjust the length of the line segment.
- Click the "Mark Point" button to leave a trace of the tangent line segment (on the left) and a point whose y-coordinate represents the slope of the line segment (on the right).
- Click the "Tangent" checkbox to show/hid the actual tangent line (segment) and its slope (in green). If you move P this will leave a trace on the right graph.
- Click the "Derivative Function" checkbox to show/hide the graph of the derivative function.
2-C The Derivative Function
We previously learned about the derivative of a function at a point as the instantaneous rate of change of the function at that point. But, a function can have an instantaneous rate of change at many different points in its domain. So, we can also think of the derivative as a function, denoted by , whose output at is the derivative , i.e., the instantaneous rate of at . Whereas is visualized as the height of the graph of at , the derivative is visualized as the slope of at .
Because the derivative is a function itself, we will be concerned with being able to find a formula for the derivative function so that we can work with it algebraically (i.e., evaluate and solve). This will be a major component of Modules 3-4 as we work on differentiation techniques.