Day 17, Mat129: Calculus I

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Today:

Announcements
- We meet in the lab today
- Quizzes are returned
- Homework is still out: grading p. 136, #56, 60, 92
  We'll need to take a look at 92...
Today I want you to practice your chain rule (using this file as a starting place).
1. Given the functions $f(x)=x^2$ and $g(x)=\sin(x)$, create and plot on reasonable windows the following:
  1. $f(g(x))$
  2. $g(f(x))$
  3. $f(f(x))$
  4. $g(g(x))$
  Write out the form of each, and simplify. You can compare your results to those produced by Mathematica by simply typing, e.g.,
  f[g[x]]
  on its own line. You may need or want to use the "Simplify" command.
  Notice that the order in which you do the composition is critical: $f(g(x)) \ne g(f(x))$ in general. In this particular case, one composition represents amplitude modulation, and the other represents frequency modulation. Which is which?
  Notice also that my sample code doesn't graph $f(f(x))$ or $g(g(x))$ (you'll need to add those).
2. Now, by hand, use the chain rule to compute the derivatives, using the chain rule: that is,
  for $F(x) = f(g(x))$,
  
  $F^\prime(x) = f^\prime(g(x))g^\prime(x)$
  Compare to the graphs of the derivatives you obtain using Mathematica.
  Do the graphs make sense?
3. Now try this process again with the functions $f(x)=\frac{1}{x}$ and $g(x)=\cos(x)$. You'll need to change some of the domains and ranges, perhaps.
One of your homework problems to hand in (next Friday) is #58, p. 154, which requires a computer to do some graphics. Work that problem now.
I have a file about Compositions, which looks at them in an unusual way. Check it out if you wish!

Website maintained by Andy Long. Comments appreciated.