Day 52 in Mat229

• the exam
• the key
• Some comments:
  • It's necessary to give reasons for one's conclusions. Too much information is much better than too little....
  • In a few cases I accepted multiple answers. In particular, in the first six problems you could either present exact answers or errors of using \(S_{10}\).
  • In the case of the comparison for #7, different series used for comparison would result in different answers.
  • In #10, one could provide a bound for the whole interval, or for a specific value of \(x\) on that interval.

That being said, only one of you ever asked me about it...:(

• We're just "changing coordinates" -- when one system makes more sense than another. If things are orbiting some center, say, then it makes more sense to use a polar coordinate system, like the one below, which shows how the angle \(\theta\) is measured:
  

  If \(r\) is the distance of a point \((x,y)\) from the origin, then we can write that

  \[ x(r,\theta) = r\cos(\theta) \hspace{1in} y(r,\theta) = r\sin(\theta) \]

  This section features some really beautiful curves that one can easily create using polar coordinates, such as the spiral inside a chambered nautilus:

  

• Materials:
  • Polar Coordinates (.nb)
  • Polar Coordinates (.pdf)
  • Polar plotting example from our handout