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
This approximation will be on the tangent line at .
The applet below shows the approximation and the exact value for a function defined through several small black dots.
The dots can be moved to change the function.
can be changed by moving the orange plus on the axis.
How far away the approximation is can be adjusted with the "h" slider.
The error shown is the absolute value of the difference between and

Some questions to answer

How does the error behave when "h" is varied?

How does the error behave at different locations?

How does the error behave when the function is smoother?