Objectives: In this tutorial, we determine "how good" does the tangent line approximate a function near some point. After working through these materials, the student should be able

• to visualize the tangent line as an approximation to the graph of the function; and
• to determine on what interval is the tangent line e-close to the function.

Modules:

Problem. Let f be a function, let a be in the domain of f, g(x) = m (x - a) + f(a) be the tangent line to the graph of f at x = a. We know that the tangent line is an approximation to f near x = a. The problem that we want to consider is how close to a is g a good approximation to f. More precisely, given e > 0, the problem is to determine for what x near a is |f(x) - g(x)| < e?

• Discussion [Using Flash]

• A LiveMath notebook that illustrates graphically a solution for this problem.

• A JavaScript module that illustrates a numerical investigation of this problem.

• Using a graphing calculator to find a solution for this problem:
  • TI-86 graphing calculator [Flash version]
  • TI-85 Graphing Calculator