I'm a bit confused about this problem, from chapter 4 of Spivak's Calculus. enter image description here

In particular, I'm not sure what a straight line is defined as in Spivak. Earlier in the text, Spivak defines a straight line as a certain collection of pairs, including, among others, the collections {$(x, cx)$: x a real number}. However, it doesn't seem right to use this definition of a straight line - that straight line is a set of all points such that {$(x, -(A/B)x + C)$: x a real number}, as it would make the problem trivial. So what should I treat the definition of a straight line as, for this problem?

Thanks in advance.

Also, could you avoid giving any hints to this problem, as I would still like to attempt solving it myself. Thank you.

  • 1
    $\begingroup$ That seems okay. Alternatively, a line is a curve whose second derivative is $0$. $\endgroup$
    – Tavish
    May 6 at 12:35
  • $\begingroup$ @Tavish I see, thanks! $\endgroup$
    – Ethan Chan
    May 6 at 12:37

A straight line, if it's not vertical can be written in the form of $(x, mx+c)$. If it's vertical, then it can be written as $(x_0, y)$ where $x_0$ is fixed.


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