2 Marks Questions

Question 12 Marks

For the function $y = x^2$, if $x = 10$ and $\triangle\text{x}=0.1$. Find $\triangle\text{y}.$

Answer

$\text{y}=\text{x}^2$
$\triangle\text{x}=0.1$
$\text{x}=10$
$\frac{\text{dy}}{\text{dx}}=2\text{x}$
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}=10}=20$
$\Rightarrow\triangle\text{y}=\text{dy}=\frac{\text{dy}}{\text{dx}}\text{dx}=20\times0.1=2$

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Question 22 Marks

If the relative error in measuring the radius of a circular plane is $\alpha,$ find the relative error in measuring its area.

Answer

Let x be the radius and y be the area of the circular plane.
We have $\frac{\triangle\text{x}}{\text{x}}=\alpha\ \text{and}\ \text{y}=\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=2\text{x}$
$\Rightarrow\frac{\triangle\text{y}}{\text{y}}=\frac{2\text{x}}{\text{y}}\text{dx}=\frac{2\text{x}}{\text{x}^2}\times\text{ax}=2\alpha$
Hence, the relative error in the area of the circular plane is $2\alpha$

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