Refining Linear Approximation

[IdForgotten]

Hello! This is a problem on some "advanced" homework I have and it's absolutely killing me.

"An approximation of the value of f(x) can be given by the formula f(x) = f(x0)+f'(x0)(TriangleX), where x = x0+triangle(x).

Improve this method of approximation by using the average value of f' between x and x0 rather than f'(x0)."

Since we haven't covered integration, there must be a reasonable way to solve and show this without using integrals?

 

[Deleted member 4993]

Hello! This is a problem on some "advanced" homework I have and it's absolutely killing me.

"An approximation of the value of f(x) can be given by the formula f(x) = f(x0)+f'(x0)(TriangleX), where x = x0+triangle(x).

Improve this method of approximation by using the average value of f' between x and x0 rather than f'(x0)."

Since we haven't covered integration, there must be a reasonable way to solve and show this without using integrals?

There is no integration involved here.

These are the first two terms of Taylor's series expansion of f(x) around f(x₀).

Work with that....