Question

Use an appropriate local linear approximation to estimate the value of (122)^2/3

Answer 1

$$f(x)=x^{\frac {2}{3}}$$

$$f(x_0+\varDelta x)\approx f(x_0)+d[f(x_0)]$$

$$x_0=125, \varDelta x=-3$$

$$d[f(x_0)]=f^/(x_0)\times \varDelta x$$

$$f^/(x)=\frac {2}{3x^{\frac{1}{3}}}$$

$$d[f(x_0)]=f^/(x_0)\times \varDelta x=\frac {2}{15} \times(-3)=-0.4$$

$$f(x_0+\varDelta x)\approx f(x_0)+d[f(x_0)] \\ \approx 25-0.4=24.6$$