Find the linearization of the natural log

Question

a) Find the linearization of the natural log, ln(x), at x=1

L(x) =

b) Use part (a) to estimate

ln(1.42)

c) Find the linearization of the cube root function, 3squarex =x^1/3 at x = 1

L(x) =

d) Use part (c) to estimate

3squre 1.42

e) Which estimate is more accurate? Explain

Answer 1

(a) The first thing we need to do is identify the method to do linear approximations. We know that the equation of a tangent line, L(x) for a function f(x) at some point x=a is:

L(x) ≈ f(a) + f^'(a)(x-a)

So the problem gives our f(x)=ln(x) and our a=1. We know that the derivative of ln(x) is 1/x. So our L(x) would be

L(x) ≈ ln(1) + (1/1)(x-1) = ln(1) + (x-1) = 0 + (x-1)

L(x) ≈ x-1

(b) So want to use L(x) to approximate ln(1.42). We would have

ln(1.42) ≈ 1.42 - 1 = 0.42

using a calculator we see that ln(1.42) = 0.35 so our approximation is ok-ish. Not great.

(c) I am not quite sure whether you mean 3x^1/2 or whether you mean x^1/3, but either way the method above can be followed using the derivative to approximate L(x).