Find the linear approximation of f(x)=sqrt[3]{x} when x=8.

Question

Accepted Answer

Step:1 ：Find the derivative of the function (f(x) = x^{1/3}). The derivative is (f&#x27;(x) = frac{1}{3}x^{-2/3}).Step:2 ：Evaluate (f(8)) and (f&#x27;(8)). We find that (f(8) = 2) and (f&#x27;(8) = frac{1}{12}).Step:3 ：Write the equation of the tangent line using the formula (L(x) = f(a) + f&#x27;(a)(x - a)). Substituting the values we found, we get (L(x) = 2 + frac{1}{12}(x - 8)).Step:4 ：(boxed{L(x) = 2 + frac{1}{12}(x - 8)}) is the linear approximation of (f(x) = x^{1/3}) at (x = 8).

Answer

Step: 1 ：Find the derivative of the function (f(x) = x^{1/3}). The derivative is (f&#x27;(x) = frac{1}{3}x^{-2/3}).Step: 2 ：Evaluate (f(8)) and (f&#x27;(8)). We find that (f(8) = 2) and (f&#x27;(8) = frac{1}{12}).Step: 3 ：Write the equation of the tangent line using the formula (L(x) = f(a) + f&#x27;(a)(x - a)). Substituting the values we found, we get (L(x) = 2 + frac{1}{12}(x - 8)).Step: 4 ：(boxed{L(x) = 2 + frac{1}{12}(x - 8)}) is the linear approximation of (f(x) = x^{1/3}) at (x = 8).

Find the linear approximation of $f(x)=\sqrt[3]{x}$ when $x=8$.

Answer

Popular Problems