Question

Let $S$ be a square with sides of length $x$. If we approximate the change in size of the area of $S$ by ${h\cdot \frac{dA}{dx}|}_{x=x_0}$, 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. $2h{x}_0$
• C. $x_0^2$
• D. $h$

Answer 1

Answer: (A)

$A=x^2$
$\frac{dA}{dx}=2x\text{. So }{\left(\frac{dA}{dx}\right]}_{x=x_0})h=2x_{0}h\text{. }$
The exact change in the area of S when $x$ is changed from $x_0$ to $x_0+h$ is
${\left(x_0+h\right)}^2-x_0^2=x_0^2+2x_{0}h+h^2-x_0^2=2x_{0}h+h^2$
The difference between the exact change and the approximate change, is
$2x_{0}h+h^2-2x_{0}h=h^2$