2-A The Derivative at a Point
Instructions
The graph of a function is shown in the applet.
- Use the input box for c or click and drag the point on the graph to change the point where you want to investigate the instantaneous rate of change.
- Adjust the slider tool for h to move the point Q around P.
- The "Secant" checkbox will show/hide the secant line between P and Q and the slope of the secant line (i.e., average rate of change).
- The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0.
- The "Tangent" checkbox will show/hide the tangent line at P and its slope.
- Use the "" and observe the relationship between the secant and tangent lines.
2-A The Derivative (at a Point)
As we have seen, the instantaneous rate of change of a quantity can be estimated using the average rate of change over a small interval. And we define the instantaneous rate of change to be the limit of those average rates of change as the width of the interval shrinks to 0.
The derivative of a function at a point in its domain is the instantaneous rate of change of at . Because we previously introduced instantaneous rates of change as limits of average rates of change, we have a formula for calculating the derivative at , denoted by .
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