\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \)
© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: (16) Next: Blackboards for Thursday January 29
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Handout for January 29, mostly "Choose an Effective Notation"

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).

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