3B-1. Linearization & Differentials
Instructions:
On the Left:
- Use the check box for f(x) to show/hide the graph of f.
- Use the checkbox for dx to show/hide the horizontal change in x.
- Use the checkboxes for and dy to show/hide the vertical changes in y. Notice the difference between the two. Use the Error checkbox to show/hide the error between the two.
- Use the Zoom In / Zoom Out buttons as needed.
- Use the input box for f(x) to define the function.
- Use the input boxes for c and h to set the locations of P and Q. Use the button "" to bring the point Q closer to P.
- Use the Secant and Tangent checkboxes to show/hide the secant and tangent lines.
Linearization & Differentials
For certain types of functions, if you zoom in close enough the graph of the function looks like a straight line. This line is called the tangent line or the linearization. Think of the linearization as the "best" linear function that can estimate the function better than any other linear function.
This linear function goes through the point P which has coordinates (c, f(c)). By definition, the slope of this line is the derivative f'(c). Therefore, since we have the slope and a point on the line, we can write the equation of the line in point-slope form:
Solving this for y gives us a function formula for the linearization: