Question

For y = 4x6 + 5x, if the error in x is 0.04, the percentage error in y at x = 1 is?

Answer 1

If a quantity x (eg, side of a square) is obtained by measurement and a quantity y (eg, area of the square) is calculated

as a function of x, say y = f(x), then any error involved in the measurement of x produces an error in the calculated

value of y as well

Answer 2

Given a function y and error in x, find the percentage error in y.

Explanation:

1. given a function in 'x' $y=f(x)$ and absolute error in x as $\Delta x$ .
2. the absolute error in the function is given as ,                                                    $\Delta y= \frac{dy}{dx}\Delta x$   ----(a)
3. percentage error is given by,  $\%_{error}=\frac{\Delta y}{y}(100)$    ------(b)
4. hence given is a function , $y=4x^6+5x$
5. derivative of the function is , $\frac{dy}{dx}= 24x^5+5x$    ----(c)
6. now the absolute error in 'x' is given , $\Delta x=0.04$  
7. hence from (a) and (c) the absolute error in 'y' for $x=1$ is ,                                  [tex]\Delta y=\frac{dy}{dx} |_{x=1}(\Delta x)\\ \Delta y= (24(1^5)+5)(0.04)\\ \Delta y=1.16[/tex]      
8. hence putting this value of absolute error in (c) we get,                                        [tex]\%_{error}=\frac{1.16}{4(1^6)+5(1)}(100)\\ \%_{error}=12.889\%[/tex]  
9. hence, the error percentage in y is $13%$$\%$ (approximately).