2-B Linearization & Differentials
Right: Investigate the secant and tangent lines at a point P on the graph of a function.
- 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.
- Use the input box for h or the button "" to move the point Q around P.
- The "Secant" and "Tangent" checkboxes will show/hide the respective lines.
- The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0.
- Use the "dx" checkbox to show/hide the horizontal change from x=c to x=c+h.
- Check the "Secant" box to also reveal a checkbox for . This will show/hide the vertical change along the graph of f from x = c to x=c+h.
- Check the "Tangent" box to reveal a checkbox for . This will show/hide the vertical change along the tangent line from x=c to x=c+h.
- Check the "Error" box to highlight the difference between and .
2-B Linearization and Local Linearity
As mentioned in the previous applet (on Differentiability), the existence of a derivative at a point is closely related to a concept called local linearity. When a function is differentiable it also has local linearity, and vice versa. If you zoom in close enough to a point on the graph of a differentiable function, the graph will look like a straight line. In fact, it will nearly be identical to its tangent line if you zoom in far enough.
Implications: Differentiable functions can be approximated with linear functions, called the linearization (i.e., tangent line), as long as you keep your inputs relatively close together. The point where the linearization is generated is called the center of the linearization. As long as the input is "close enough" to the center, the linearization should give good estimates of nearby function values.