Question

use a tangent line at (27,3) to the fxn y=x^(1/3) to estimate (25)^(1/3)

Answer 1

First find the slope using \(y'=(1/3)x^{-2/3}\). At \(x=27\) we have \(y'=1/27\). Therefore the tangent line is of the form \(y=\frac{1}{27}x+b\). Plug in the point (27,3) we conclude that b=2. Therefore the tangent line is \(y=\frac{x}{27}+2\). Now the estimate of of \(25^{1/3}\) become \(\frac{25}{27}+2=\frac{79}{27}\)