Problem:
For which x is the linear approximation to

y = f(x) = x3 - 4 x

at x = -1 accurate to within 0.5?

Visualization:
Use the following LiveMath notebook in which the graph of y = f(x) = x3 - 4 x is colored blue and the tangent line at x = -1 is colored orange. The two red graphs are the graphs of f translated vertically 0.5 and -0.5 units. By zooming in, you can determine the values of x for which the graph of the tangent line stays in the region between the latter two graphs.

You can change the value of the point of tangency a and the accuracy k desired. View an animation to see how this can be done. In addition, you can make changes to the function y = x3 - 4x. It appears that you must click twice on the graph to get the graphs correctly plotted.

If you get the message "Cannot draw this graph ..." in the notebook, just move the cursor onto the graphing area and click the left mouse button.