Day 26 in Mat228

Last time: Exam 2

Next time: The dot (inner) product

Today:

Announcements
- Homework 13.1 hasn't been graded yet.
- Homework from 13.2 will be due next time (Tuesday, 12/1).
- Read Section 13.3 for next Tuesday responding to the worksheet for section 13.3. There will be a quiz next Tuesday over the reading.
- The second exam is graded. Craig was right: disaster loomed.
  1. Two series, type of convergence/divergence:
    1. ${\cos(\pi{k})=(-1)^k}$ -- only one student realized this (or stated it in such a way that it was clear).
      Let's be honest, folks: you don't know your trig functions.
    2. This is not geometric. It is teed up for a root test. Everyone who performed it forgot the absolute values, however.
  2. Three series, converge or diverge:
    1. Comparison, or limit comparison shows that it diverges.
    2. Integral test or ratio test shows that it converges.
    3. Diverges by the divergence test (no one got this).
      Let's be honest, folks: you don't know your logarithmic functions.
  3. Choice of "story problem". Ironically, this was the most successful problem (for those who chose part b):
    1. ${\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}+\ldots}$
    2. ${2+2\cdot{1}+2\cdot{\frac{1}{2}}+2\cdot{\frac{1}{4}}+2\cdot{\frac{1}{8}}+\ldots}$
    Both are solved through the use of geometric series.
  4. Taylor series. Steps:
    1. Compute derivatives ${f^{(k)}(x)}$
    2. Evaluate them at ${c=1}$ : ${f^{(k)}(c)}$
    3. Put them into the formula for a Taylor series:
      ${f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(c)}{k!}(x-c)^k}$
  5. Series choice:
    1. Find the Taylor series using the geometric series: ${f(x)=\frac{-2x}{3-x^2}=\frac{-2x}{3}\cdot\frac{1}{1-(x^2/3)}=\frac{-2x}{3}\sum_{n=0}^{\infty}\left(\frac{x^2}{3}\right)^n}$
      Valid for ${|\frac{x^2}{3}|<1}$ or ${|x|<\sqrt{3}}$ (and not on the boundaries, since the geometric series doesn't work there).
    2. The second one again illustrated that we're missing "log sense": many failed to show that the indeterminate form
      ${\lim_{n\rightarrow\infty}\frac{\ln(n)}{\ln(n+1)}=1}$
      I also got some who wrote the following with an equal (=) sign: ${\frac{\ln(n)}{\ln(n+1)}\ne\frac{\ln(n)}{\ln(n)+\ln(1)}}$
- The exams were replete with algebra errors, often dealing with rational expressions. For example, not everyone realized that
  ${\frac{-2x}{3-x^2}\ne\frac{-2}{3}x+\frac{2}{x}}$
  This kind of naive simplification would be wonderful, but it's not the algebra we use in calculus.
  If you know your rational functions, you can easily see that the two expressions are not equal:
  1. The one on the left blows up at ${\pm\sqrt{3}}$
  2. The one on the right blows up at 0.
- Problems #1a and #2 (all of the series) were provided by another professor. We are using those for common assessment of your ability to handle series.
- Stats:
  - Max: 53.5/80
  - Min: 24/80
  - Mean (of 100%): 45.2%
  - Standard Deviation: 13.2%
- Do the exam again, correcting your errors, and see me for 40% of your missed points back.
We were finishing some examples from section 13.1:
- More physics:
  - #58, p. 665
    - First and foremost: Newton's laws of motion (in particular ${\underline{\bf{F}}=m\underline{\bf{a}}}$ ), and statics.
    - The parallelogram rule
    - We can think of the force vectors on the cables as being composed of vector components (as well as coordinate components).
  - #60, p. 665
  - Let's use a little calculus and the parameterization of the position vector in uniform circular motion given in one of our videos
    
    ${x(t)=R\cos(\frac{2\pi}{T}t)}$ ${y(t)=R\sin(\frac{2\pi}{T}t)}$
    
    to derive the velocity and acceleration vectors, and check the video's claims.
Additional material from section 13.2: three-dimensional vectors
- The right-hand rule
- Vocabulary:
  - coordinate planes (p. 666)
  - octants
  - translates/translation (p. 669)
  - parallel vectors
  - equivalent vectors
  - standard basis vectors
- Distance formula in R³:
  
  examples:
  - #1, p. 673
  - #2, p. 673
  - #14, p. 673
- Equations of Spheres and Cylinders
  What are a sphere and a right cylinder?
  - A sphere: all points equidistant from a given center.
  - A right cylinder: all points equidistant from a given line.
  examples: Draw the following:
  - The set of points satisfying
    ${(x-1)^2+(y+1)^2=4,\qquad{-1\le{z}\le{1}}}$
  - The set of points satisfying
    ${(x-1)^2+(y+1)^2+(z-1)^2=4,\qquad{0\le{y}}$
- Parametric Equations of Lines:
- Additional examples:
  - #27, p. 674
  - #39, p. 674
  - #51, p. 674

Website maintained by Andy Long. Comments appreciated.