If you know the value of a function and its derivative at a point you can approximate values of the function near the point with[br][math]f(x_0+h)\approx f(x_0)+hf'(x_0)[/math][br]This approximation will be on the tangent line at [math] x= x_0[/math].[br]The applet below shows the approximation and the exact value for a function defined through several small black dots.[br]The dots can be moved to change the function.[br][math]x_0[/math] can be changed by moving the orange plus on the [math]x[/math]axis.[br]How far away the approximation is can be adjusted with the "h" slider.[br]The error shown is the absolute value of the difference between [math]f(x_0 + h)[/math] and [math]f(x_0) + h f'(x_0) [/math]

Some questions to answer[br][list][br][*] How does the error behave when "h" is varied?[br][*] How does the error behave at different [math]x_0[/math] locations?[br][*] How does the error behave when the function is smoother?[br][/list]