Need help on calculus question!

Question

Find the local linearization
(a) Find the local linearization of f(x) = ln(1 + x3) at x = 0.
(b) Use the local linearization to estimate sin(0.2).

Answer 1

Local linealization refers to the rate of change of a function in a small neighborhood of a point. For example, for (a), where you have a function f(x), you want to examine the rate of change of f(x) with respect to a small change in x, as it occurs near the point x=0 and f(x=0). So you want to take the derivative of f(x) with respect to x, and then set x= 0. This rate of change of the function can then be used to approximate values of the funciton for small variations near a point of interest.

 

If f(x) = ln(1 + x^3), then

 

df/dx=  2x^2 / (1 + x^3)

 

so df/dx= 0 near x= 0, and the value of the function in the  neighborhood of x=0 will be 0. You can verify this by making a graph of the function.

 

 

For part (b), you do the same thing, with the function

f(x)= sin(x)

df/dx = cos(x) = 1 at x=0.

 

so, sin(0.2)= sin(0) + df/dx (at x=0) * (0.2) = 0.2.

 

This is the famous small anlge approximation for the sin function where the sin Θ = Θ (approximately).

 

- Ben