Submitted by Surfer Rosa on Tue, 03/28/2017 - 14:55
Forums:
Paper I 2016 Q4 defines the signed curvature K (kappa) in terms of f'(x) and f''(x).
It then says "use this definition to determine all curves for which the signed curvature is a non-zero
constant".
I got this completely wrong. This was my reasoning:
I thought of K as a function and called it g(x)
I figured if g(x) is constant then g'(x) = 0
(Incidentally if it's non-zero then by the definition give f''(x) is not equal to zero, but I thought I'd look out for that later.)
So I set off and differentiated. I did get an expression involving f', f'', and f''', but I couldn't interpret it at all.
Can you explain why this is not the correct approach?
Thanks
It's not an incorrect
It's not an incorrect approach, it's just not a useful one. Essentially, you have an expression involving $f''(x)$ and $f'(x)$ and from this you want to determine $f(x)$. So differentiating that expression so that it involves $f''', f''$ and $f$ is only going to complicate things. For example, if you had a DE (say $f''(x) + xf'(x) = 2x^2$ for example) would you start solving this DE by differentiating it to get $f'''(x) + xf''(x) + f'(x) = 4x$? That would only serve to complicate matters.
Instead, you want to minimize the order of derivatives, so simply setting $\kappa = c$ for some $c$ and getting a final answer involving $c$. To further the point of why this is the method you should pick, you know that your answer is going to be a family of functions, it would make sense for this family to depend on $\kappa$, so differentiating to get rid of the constant $c$ is going to lead to some arbitrary constant at the end anyway that you'll need to try and relate to $\kappa$ anyway; so it makes more sense to simply stick with $\kappa$ from the start.
Thanks
I haven't looked at DEs for years so they were totally off my radar.
I think this part is key: "Essentially, you have an expression involving f″(x) and f′(x) and from this you want to determine f(x)."
I suppose (given my ignorance of DEs) I was hoping all the algebra would magically drop out and I'd have a recognisable f'(x), say, from which I could deduce the required properties. This didn't happen :-)
In the back of my mind, given the term signed curvature, I was thinking - very vaguely - about increasing/decreasing functions. Don't often get these questions completely wrong, but I did here.
Like I said I think if I'd been working with DEs the correct approach would have sprung to mind, hopefully.
Thank you for the prompt answer.
No worries, glad you've got
No worries, glad you've got the key bits!