determine the local linearization near x=0

Question

Determine the local linearization of (1+x)^k near x=0. Use the linearization to estimat the square root of 1.32.

 

The steps I think I should take are fnd the y vale for the function, find the derivative, use the derivative to find the slope, get a tangent line, and evaluate square root of 1.32. However I am a little confused on how to start because is k a constant in this case or a part of the function. The problem doesnt explicitly state what two variables are being analyzed so I am a little stumped. Please help direct me on what I should do in terms of d/dx.

Answer 1

Let f(x) = √(1+x)  Then f'(x) = 1/[2√(1+x)]   Let a = 0 and let x =0.32

 

Then, f(x) ≈ L(x) = f(a) + f'(a)(x-a)

 

Substituting, we have f(0.32) = √1.32 ≈ L(0.32) = f(0) + f'(0)(0.32-0)

 

                                                                       = 1 + (1/2)(0.32) = 1.16  

Answer 2

Hello Olivia,

 

I think the problem is asking you to use k = 1/2. Then, the function becomes:

 

     y = √(1 + x)

 

Finding the square root of 1.32 would just be evaluating y(0.32).

 

The rest of the problem is exactly how you've guessed it. :)