my Calculus class followup

DPatrick

Here are some comments on some of the outstanding items from my calculus class last week. If you have any questions/comments/issues with problems (whether discussed below or not) please feel free to post them.

Thurs #8. Hint: try it at the endpoints, and then use the fact that $\sqrt{u^2 + v^2} \le u + v$ (assuming $u$ and $v$ are positive).

Fri #9. Note that $f'' < 0$ implies that $f'$ is strictly decreasing. If we ever have $f'(a) = m < 0$ then this means that $f(a+x) < f(a) + mx$ for $x>0$ ; show this leads to a contradiction. On the other hand, if $f' > 0$ everywhere then replace with $g(x) = f(-x)$ .

Sat #1. We conjectured that we need m = 0 or 3 (mod 4). Replace $\cos(nx)$ with $\frac{e^{inx} + e^{-inx}}{2}$ and multiply out. This will be nonzero if and only if there is a $e^0$ term.

Sat #2. I can't remember if we got all the way through this, but the list of functions is 0, $2x$ , $\frac{2}{c}\sin(cx)$ for any nonzero $c$ , and $\frac{2}{c}\sinh(cx)$ for any nonzero $c$ .

Sat #3. I think the general case is that if we replace 2009 with n, then the answer is 2n - 2: computing out the derivative gives a degree 2n polynomial in general in the numerator, but with careful choices we can make the two leading terms vanish but not the next term.

Sat #4. We defined a linear operator $(Lh)(x) = \int_0^1 h(y)K(x,y)\,dy$ ; note that if $h \ge 0$ , then the resulting function is strictly positive unless $h = 0$ everywhere on $[0,1]$ . We also noted that $L^2(af + bg) = af + bg$ . So let $r$ be the minimum of the ratio $f/g$ on $[0,1]$ (this minimum must exist and is positive since $f$ and $g$ are continuous and positive on a closed interval). Then $f-rg$ is nonnegative but not strictly positive; since $L^2(f-rg) = f-rg$ , we must have $f-rg = 0$ , hence $f$ and $g$ differ by a constant. Finishing from here is fairly easy.

Sat #5. Hint: look at [0,1/2] and [1/2,1] separately, depending on whether f(1/2) >= 1/2 or f(1/2) <= 1/2. Note that $f''(x) < 4$ implies $f(x) < 2x^2$ for $f$ positive on [0,1].

Sat #6. Hint: multiply both sides by $e^{-x}y'$ , then integrate.

Sat #7. Hint: repeatedly integrate by parts both integrate, and compare what you get. You should (eventually) get something of the form
"polynomial" = "trig expression"
What does that imply?

Sat #8. Hint: try to graph it as a parametric curve on the yz-plane. If you can prove that the curve is closed and doesn't have y' = z' = 0 anywhere, then it will loop forever, and thus the parametric functions will be periodic.

Sat #9. This is problem B4 from the 1999 Putnam. The other "hardest Putnam problem ever" is the very next problem, B5 from 1999. The problems and solutions are at http://www.unl.edu/amc/a-activities/a7-problems/putnamindex.shtml. (Problem B2 is also an interesting calculus problem, albeit much easier!)

**Posted:** Wed Jul 29, 2009 10:40 am

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