Objectives: In this tutorial, we define the tangent line at a point of the graph of a function. The tangent line is represented as the limit of secant lines. The approximation of the function by the tangent line is investigated graphically. After working through these materials, the student should be able

• to visualize the tangent line as the limit of secant lines;
• to visualize the tangent line as an approximation to the graph; and
• to approximate the slope of the tangent line both graphically and numerically.

Modules:

 Definition. Let y = f(x) be a function and suppose that a is in the domain of f. A line L containing (a, f(a)) is a tangent line to f at x = a if the slope of L is the limit • Discussion [Using Flash]

• A LiveMath notebook illustrating the tangent line as the limit of secant lines.

• A JavaScript exploration of the numerical estimates of the slope of the tangent line to the graph of various functions.

• Comment: In the following two modules, the graphs of a function f and a tangent line to f at some x = a are drawn. It is shown that by zooming in towards the point (a, f(a)) the two graphs almost coincide. Hence, the slope of the tangent line can be estimated from the graph of the function.

• An animation demonstrating the estimation of the slope of the tangent by zooming in.

• A LiveMath notebook which compares graphically a function with a tangent line.

• Using a graphing calculator to illustrate the tangent line as the limit of secant lines.

• Using a graphing calculator to plot the graph of a function together with a tangent line at a given point.