I recently saw this word problem:
Given $$y = \frac12x^2$$ if $x$ doubles, $y$ changes by what?
Someone solved it using basic numbers and substitution. The answer is $4$. However, my gut feeling when I see the question is to differentiate it, obtaining:
$$\frac{dy}{dx} = x$$
i.e. the slope of the parabola at point $x$ is $x$ itself, which leads me nowhere near the answer. Is there a way to go from $\frac{dy}{dx} = x$ to the answer? Is there a way to solve this using Calculus? What am I missing here?
Calculus wont work because $dy$ and $dx$ are infinitesimally small. The change in x or ${\Delta x}$ in the question is very large and is comparable to the values to x.
The way to solve this would be- $$y={1\over 2}x^2$$ $$y'={1\over 2}(2x)^2 = {4\times {1\over 2}x^2} = 4y$$
Hence when x is doubled, y will be quadrupled.