Linear Approximation?

Question

Let f(x)=sqrt(x-12) and use the Linear Approximation to this function at a=37 with Δx=0.4 to estimate f(37.4)−f(37)=Δf≈df=

Answer 1

Given

f(x) = √(x-12)

Δx = x-x₀ = 0.4

f(37) = √(37-12) = √25 = 5

f(37.4) = √(37.4-12) = √25.4

f(x) = √(x-12)

Apply diffenceation.

f '(x) = 1/[2√(x-12)]

Substitute x = 37

f '(37) = 1/[2√(37-12)]

f '(37) = 1/[2√(25)]

f '(37) = 1/(2*5)

f '(37) = 1/10

Formula for Linear Approximation : f(x) = f(x₀) + (x-x₀)f '(x₀)

Consider x = 37.4 and  x₀ = 37

f(37.4) = f(37) + (0.4)f '(37)

Substitute f '(37) = 1/10

f(37.4) = f(37) + (0.4)(1/10)

f(37.4) - f(37) = 0.04

The solution is f(37.4) - f(37) = 0.04