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.
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.
illustrates a numerical investigation of this problem.
- Using a graphing calculator to find a solution for this problem: