Question

For fun question

Answer 1

Consider the function $f(x)=x^{1/3}$, which has derivative $f'(x)=\dfrac13x^{-2/3}$.

The linear approximation of $f(x)$ for some value $x$ within a neighborhood of $x=c$ is given by

$f(x)\approx f'(c)(x-c)+f(c)$

Let $c=64$. Then $(63.97)^{1/3}$ can be estimated to be

$f(63.97)\approxf'(64)(63.97-64)+f(64)$
$\sqrt[3]{63.97}\approx4-\dfrac{0.03}{48}=3.999375$

Since $f'(x)>0$ for $x>0$, it follows that $f(x)$ must be strictly increasing over that part of its domain, which means the linear approximation lies strictly above the function $f(x)$. This means the estimated value is an overestimation.

Indeed, the actual value is closer to the number 3.999374902...