Question

Let $S$ be a square with sides of length $x$. If we approximate the change in size of the area of $S$ by $\left.h \cdot \frac{d A}{d x}\right|_{x=x_0}$, when the sides are changed from $x_0$ to $x_0+h$, then the absolute value of the error in our approximation, is

• A. $ h^2$
• B. $2 hx _0$
• C. $x_0^2$
• D. $h$

Answer 1

Answer: (A)

$A=x^2$
$\left.\frac{ dA }{ dx }=2 x \text {. So }\left(\frac{ dA }{ dx }\right]_{ x = x _0}\right) h =2 x _0 h \text {. }$
The exact change in the area of S when $x$ is changed from $x_0$ to $x_0+h$ is
$\left( x _0+ h \right)^2- x _0^2= x _0^2+2 x _0 h + h ^2- x _0^2=2 x _0 h + h ^2$
The difference between the exact change and the approximate change, is
$2 x _0 h + h ^2-2 x _0 h = h ^2$