I have been given a question which asks to draw the graph of $x\cos\left(\frac 1x\right)$ without a calculator. However, I'm unsure as to how to do this. It may involve differentiation as that was the theme of the question sheet.
Any help would be greatly appreciated.
Think of this function as $$f(x)=x\cos\left(\left(\frac{1}{x^2}\right)\cdot x\right)$$ What this shows you is that both the amplitude ($x$) and frequency ($\frac{1}{2\pi x^2}$) of the wave tends vary with $x$. As $|x|$ gets large, the amplitude goes to infinity but is offset by a frequency that diminishes to zero. But as $|x|\to0$, the frequency blows up to infinity and the amplitude diminishes to zero.
So as $|x|\to 0$, the frequency keeps increasing, but with a smaller and smaller amplitude.
What about when $|x|\to\infty$? What exactly is happening then? We know the amplitude is getting larger and larger and the frequency is getting smaller and smaller. In particular, notice that as $|x|\to\infty$, $\frac{1}{x}\to0$, so $\cos\left(\frac{1}{x}\right)\to1$, and so in the long run the function just approaches $g(x)=x$.
For the sake of comparison, think about the graphs of $x\cos x$ and $\frac{1}{x}\cos x$, too.