Question

How do you show that the linearization of `f(x) = (1+x)^k` at x=0 is `L(x) = 1+kx`?

Answer 1

Note that `f(0)=(1+0)^k=1`.

Assuming `k` is a constant, we see that `f'(x)=k(1+x)^(k-1)`. Thus `f'(0)=k(1+0)^(k-1)=k*1=k`.

Using the point `(0,1)` and slope of `k` we can write the linearization function: `L(x)=k(x+0)+1=1+kx`.