Suppose that f(h)=(8+2h)^1/3
Use the linear approximation to estimate the value of f(h) for small values of h. Give your answer in terms of h.
Thanks for all the help.
(Sorry Bob for not stating the subject last time)
d f(h) = 1/2 ((8+2h)^-2/3 (2)
= (1/f(h))^2
so if f(h) = 10, then for small delta h, the d f(h)= 1/100
Neat.
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.