01
Compare one nearby value with a farther value
At x = 2.1, the curve gives 4.41 while the line gives 4.4. The line is lower by 0.01. At x = 3, the curve gives 9 while the line gives 8, leaving an error of 1. Both comparisons use the same tangent line. They show why a good nearby approximation does not imply equal accuracy farther away. Record the input, curve value, line value and their difference in separate columns so the approximation cannot be mistaken for an exact calculation from the original curve.
02
Derive the error before choosing a tolerance
Subtracting the line from the curve gives x² − (4x − 4) = (x − 2)². This squared difference is never negative, so the tangent underestimates this particular curve except at x = 2, where they agree. If the allowed absolute vertical error is at most 0.04, then (x − 2)² must be at most 0.04. The permitted interval is 1.8 through 2.2, including its endpoints. This is an absolute error in the plotted vertical units, not a percentage or a universal rule for other curves.
03
Keep the approximation attached to its assumptions
A reusable comparison sheet should record the curve equation, chosen point, derivative, tangent equation, sample inputs and allowed error. If the chosen point changes, calculate a new tangent before reusing the sheet. If the curve changes, derive its error separately rather than carrying over the squared expression from this example. For a plotted illustration, state the axis range and mark the tolerance interval on the horizontal axis. The worksheet then connects the visual impression with checkable arithmetic and makes clear which conclusions belong only to the stated exercise.