Question

estimate cuberoot 30 using local linear approximation

Answer 1

Okay, so \[
30 = x^2
\]We want to find the roots of: \[
f(x) = x^2-30
\]

Answer 2

no its cube root not square root

Answer 3

Okay, well \(3^3 = 27\) and \(4^3 = 64\) so clearly \(27\) is closest cube.\[
f(x) = x^{1/3}\implies df =\frac 13 x^{-2/3}dx
\]We use use the fact that when \(dx=\Delta x = x-a\): \[
\Delta f \approx df\implies f(x)-f(a)\approx \frac{1}{3}(a)^{-2/3}(x-a)
\]

Answer 4

In this case \(x=30\) and \(a=27\).

Answer 5

\[
f(30)-f(27)\approx \frac{1}{3}(27)^{-2/3}(30-27)
\]