On the Left: [br][list][*]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 [math]\Delta y[/math] 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. [/*][/list][br]On the Right: [list][*]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 "[math]h\to0[/math]" to bring the point Q closer to P. [/*][*]Use the Secant and Tangent checkboxes to show/hide the secant and tangent lines. [/*][/list]

For certain types of functions, if you [b]zoom in [i]close enough[/i][/b] the graph of the function looks like a straight line. This line is called the [b]tangent line[/b] or the [b]linearization[/b]. Think of the linearization as the "best" linear function that can estimate the function better than any other linear function. [br][br]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:[br][br][math]y-f(c)=f'(c)(x-c)[/math][br][br]Solving this for y gives us a function formula for the linearization:[br][br][math]y=L(x)=f(c)+f'(c)(x-c)[/math]