© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (16) |
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**Reading.** Sections 1.4 and 1.5 of Larson's textbook.

**Next Quiz.** On Tuesday February 3, mostly problems from this
handout and from Larson's Sections 1.4 and 1.5.

**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 1.5.1). One morning it started snowing at
a heavy and constant rate. A snowplow started out at 8:00AM. At 9:00AM,
it had gone 2km. By 10:00AM, it had gone 3km. Assuming that the snowplow
removes a constant volume of snow per hour, determine the time at which
it started snowing.

**Problem 4** (Larson's 1.5.2).

- If $n\in\bbN$ and $2n+1$ is a square, show that $n+1$ is the sum of two successive squares.
- If $n\in\bbN$ and $3n+1$ is a square, show that $n+1$ is the sum of three successive squares.

**Problem 5** (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 6** (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 7** (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?

**Problem 8.** What is your favourite "choose an effective notation"
problem?