Math 199 (Assignments)

[Not to be turned in, but we will discuss in class on Friday]

1. Suppose $f$ is a differentiable function, and consider the following two statements: $$\begin{array}{l} \hbox{If $x_0$ is a critical point for $f$, then $f'(x_0) = 0$, $\quad$ and}\\ \hbox{If $f'(x_0)=0$ then $x_0$ is a critical point for $f$.} \end{array}$$

   Recall that relative maxima are critical points of $f$. Which of these statements allows you to find the maximum of a function by locating the points where $f'(x)=0$ and selecting the one where $f(x)$ is largest? Why? And why not the other one?

2. For a homework problem, a student would like to show that $\sin x - \cos x = \sqrt{1-\sin 2x}$ for all $x$, and so writes the following sequence of equations $$\displaylines{ \sin x - \cos x = \sqrt{1-\sin 2x}\cr (\sin x - \cos x)^2 = (\sqrt{1-\sin 2x})^2\cr \sin^2 x - 2\sin x \cos x + \cos^2 x = 1 - \sin 2x\cr (\sin^2 x + \cos^2 x) - 2\sin x \cos x = 1 - \sin 2x\cr 1 - \sin 2x = 1 - \sin 2x\cr }$$ and claims that this proves it.
   1. What argument is the student making in this sequence of equations? (That is, what words need to be inserted into this computation in order to make an argument. Certainly there are some before and after the computation, but be particularly careful about the words that go in between the equations. It might help you to try to frame this in terms of formal logic.)
   2. The wiley professor points out that the initial equation does not hold for all $x$, in particular for $x=0$, since $\sqrt{1-\sin 0}=\sqrt{1-0}=\sqrt 1 = 1$ while $\sin 0 - \cos 0= 0 - 1 = -1$, so the student's argument is flawed. Explain what the student did wrong and why it was wrong.

      Hint: The problem is not that the professor is mistaken and that $\sqrt 1=\pm1$. The square root operation is a function, so it must have only one answer for any given input; recall that $\sqrt x\ge 0$ for all $x$. Also, the identity $\sin 2x = 2\sin x\cos x$ is true, so that's not it either. Finally, it's not that the domains have to be restricted to $x\ge 0$; the final identity is true for all $x$, so why should there be a restriction for the initial equation?

      The fact of the matter is that the computations are all correct. The problem is one of logic, not computation, so your answer should address the logical construction of the argument, not details of computation. There is a logical flaw with the very form of the argument, regardless of the computations involved, that you need to identify.

      Note that you are not trying to fix the student's proof. That can't be done, since the claim is actually false. There is nothing you can do to make what they did be correct (other than change what they were trying to do, but that doesn't answer the question, which is "what was wrong with what they did do?").

      Because this type of argument is flawed in its very nature, you should not use this type of so-called "proof" yourself.

Math 199 (Fall 2019) web pages Created: 18 Sep 2019 Last modified: 18 Sep 2019 at 3:39 PM Comments to: dpvc@union.edu

Assignments