## To estimate the value of f(h) for small values of h using linear approximation, we can use the linearization formula. The linearization of f(h) at a point a is given by:

L(h) = f(a) + f'(a)(h - a)

In this case, we want to estimate f(h) for small values of h, so let's choose a = 0 (since small values of h means h is close to 0). Now we can calculate the linear approximation of f(h).

First, let's find the derivative of f(h) using the power rule and chain rule:

f'(h) = (1/3)(8 + 2h)^(-2/3) * 2

Now, substitute a = 0 into f(h) and f'(h):

f(0) = (8 + 2(0))^(1/3) = 8^(1/3) = 2

f'(0) = (1/3)(8 + 2(0))^(-2/3) * 2 = (1/3)(8^(-2/3)) * 2 = 2/3

Using the linearization formula, we can approximate f(h) for small values of h:

L(h) = f(a) + f'(a)(h - a) = 2 + (2/3)h

Therefore, the linear approximation of f(h) for small values of h is L(h) = 2 + (2/3)h.