Question

Linear Approximation

Function is y=x2 .

Take three points, x=2, x=3, x=4.

Approximate this function at these three points for a deviation Δx =0.1. Which of the three points does the approximation works best? Which point does it works worst?

Answer 1

Without even knowing any calculus, it's easy to see that a linear approximation works best where the graph's curvature is less. Since a parabola is most curvy at its vertex, the best approximations are those farthest away. In this case, at x=4.

So, since the tangent line at x=h is

y-h^2 = 2h(x-h)
y = 2hx - h^2

so plug in Δx to find y at (x+Δx)

You will see that it is closer for larger x values.