Linear approximation

Question

Using equation Δf≈f'(a)Δx , estimate Δf=f(3.02)-f(3) for the function f(x)= x⁴

 

Hi everyone, I need steps on how to do this. First, I dont undstand this equation " Δf≈f'(a)Δx " and what to plug into there. Second, what is delta x for? 

 

So far in my calculus class Ive been grasping every topic easily, but we just started linear approximation and these question was given and I just dont understand. 

Answer 1

The way I interpret " Δf≈f'(a)Δx " is:   'a small change in the function value (f) is approximately equal to the derivative of that function evaluated at some point a (f'(a)) times a small change in x (Δx).' 

We know the derivative of f(x), i.e. f'(x), evaluated at a point a is nothing more than the slope of the tangent line of f(x) at the point where x=a.  So that slope times a small change in x  approximates the change in the function.

 

f(x) = x⁴

f'(x) = 4x³

 

f'(3) = 4*(3)³ = 108

Δx = 3.02-3 = 0.02

 

So Δf ≈ 108*(0.02) = 2.16

 

How good is this approximation?

f(3.02) = 3.02⁴ = 83.1816

f(3) = 3⁴ = 81

f(3.02)-f(3) = 83.1816-81 = 2.1816   (so its off by about 1%)