local linear approximation

Question

Find the local linear approximation of f(x)=1/x at x= 1

Answer 1

The point-slope form of a line is y - y₁ = m(x - x₁) where the point is (x₁, y₁) and "m" is the slope.

So if x = 1, then f(x) = 1/1 or 1 making our point (1, 1). To find the slope, use the derivative.

To calculate the derivative, write the function as f(x) = x^-1 and use the power rule making f '(x) = -x^-2 or:

f '(x) = -1/x²

Since x = 1, f '(1) = -1/1² = -1

So, (x₁, y₁) = (1, 1) and m = -1. Plug them into y - y₁ = m(x - x₁)