A point moves along a parabola y^2 = 3x. Find the Approximate change in its distance from the origin as its x coordinate changes from 1 to 1.1
The answer is 0.125
I assume origin is (0,0) ( I am not sure if this is correct; maybe it meant origin of the original?) Anyway so I try:
At point 1:
y^2 = 3(1); I get the coordinates (1, sqrt(10) ) with a distance of sqrt(17) from the origin)
At point 2:
y^2 = 3(1.1); I get the coordinates (1.1, sqrt(3.3) ) with a distance of sqrt(2.12376) from the origin)
The difference between both distances are 1.0386…
Why am I getting wrong answer? Why "Approximate"?
Best Answer
Distance from origin is $\sqrt{x^2+y^2} = \sqrt{x^2+3 x}$. Let $x_0=1$, $h=0.1$; then the change in distance from the origin is
$$\sqrt{(x_0+h)^2+3 (x_0+h)} - \sqrt{x_0^2+3 x_0} \approx h \left [\frac{d}{dx} \sqrt{x^2+3 x} \right ]_{x=x_0} = \frac{2 x_0+3}{2 \sqrt{x_0^2+3 x}}h = 0.125$$
Typically, problems asking for small changes in a function in response to small changes in their arguments are asking for the evaluation of a derivative of the function.