© | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: | (18) |
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**Reading.** Sections 1.4 and 1.5 of Larson's textbook.

**Next Quiz.** On Thursday February 11, mostly problems from this
handout and from Larson's Sections 1.4 and 1.5.

**Problem 0.** Put $2^n$ $n$-dimensional oranges of radius $1$ at the
corners of the $n$-dimensional cube $[-1,1]^n$, let $B_n$ be the
largest blue ball centered at $0$ and bound by these oranges and let
$C_n$ be the smallest black cube containing these oranges. Compute
\[ \lim_{n\to\infty}\frac{\operatorname{Vol}(B_n)}{\operatorname{Vol}(C_n)} \]
(working mathematicians are often shocked by the answer).

**Old Problems.**

**Problem 1** (Larson's 1.4.4). Compute $\displaystyle \int_0^\infty
e^{-x^2}dx$.

**Problem 2.** Compute the volume of the $n$-dimensional sphere
$S^n:=\{x\in\bbR^{n+1}\colon|x|=1\}$ in $\bbR^{n+1}$ and the volume of
the $n$-dimensional ball $D^n:=\{x\in\bbR^n\colon|x|\leq 1\}$ in $\bbR^n$.

**Problem 3** (Larson's 1.5.3). In a triangle $ABC$, $AB=AC$, $D$
is the mid point of $BC$, $E$ is the foot of the perpendicular drawn
from $D$ to $AC$, and $F$ is the midpoint of $DE$. Prove that $AF$ is
perpendiculr to $BE$. (Hint: use analytic geometry and be clever about
the choice of coordinate system).

**Problem 4** (Larson's 1.5.4). Let $-1 < a_0 < 1$ and define
recursively for $n>0$,
\[ a_n=\left(\frac{1+a_{n-1}}{2}\right)^{1/2}. \]
What happens to $4^n(1-a_n)$ as $n\rightarrow\infty$?

**Problem 5** (Larson's 1.5.6). Guy wires are strung from the top
of each of two poles to the base of the other. What is the height from
the ground where the two wires cross?

**New Problems.**

**Problem 6** (Larson's 1.5.7). The bottom of this handout (width
8.5in) is folded up and to the left, so that the bottom right corner would
align with the left edge of this page. Given the angle of the fold
$\theta$, what is the length of the crease $L$? See the figure on the
right.

**Problem 7** (Larson's 1.5.8). Let $P_1,P_2,\ldots,P_{12}$ be the
vertices of a regular 12-gon. Are the diagonals $P_1P_9$, $P_2P_{11}$ and
$P_4P_{12}$ concurrent?