4.5 Linearization

Linearization is the process of approximating a function near a given point using the tangent line at that point.

L(x) = f(a) + f'(a)(x - a)

The derivative of a function measures the rate at which the function's value changes as its input changes.

f'(a) represents the slope of the tangent line to the graph of f(x) at the point x=a.

L(x) = f(2) + f'(2)(x - 2) = 8 + 12(x - 2) = 12x - 16.

Linearization is used to approximate complex functions with simpler linear functions for easier calculations.

To find the tangent line, calculate the derivative at the point to get the slope, then use the point-slope form of the line.

The differential dy is given by dy = f'(x)dx, where dx is a small change in x.

dy = (8x + 1)dx.

The point of tangency is where the linear approximation is most accurate, as it is the point where the function and the tangent line touch.