Week 2 sketch of lesson plans

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(M/T) 5.4: Fund. Theorem, 1.5-2? days

idea: proof not as important as getting them to understand the statement.

1) lecture: state FTC1 and FTC2 (same as "Evaluation thm", 5.3).  

a) emphasize that FTC shows that d/dx and integrals are inverses.

b) why do we need to think of g(x) = \int_a^x f(t) dt as a function?  b/c
can't always directly integrate to get nice formula for g(x) (like Fresnel
function, see book), ALSO properties of f(t) give clues to properties of
g(x) (we'll explore this in activity).

2) (15 min) Activity #4 -FTC1 (stewart instructors guide - Activity #1
(integration potpurry), problem #1).

----- probably end of day 1 of 5.4

3) See workshop/discussion section of instructors guide for
more examples, answer questions.

4) ?If still time - proof/idea? FTC1 and/or FTC2 (in sec. 5.3)

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(T/W) 5.5: Substitution (+ even/odd functions), 1-1.5 day

1) lecture: introduce and state substitution rule, explain with Chain Rule
(may need example of computing d/dx using chain rule)

2) do some examples

 a) indefinite integrals first

 b) definite integrals two ways -- change limits, or 
	re-substitute (i.e. finding indefinite integral first)

3) (10 min) Activity #5 (stewart) "Unexpected Equality"

4) Trickier examples

 a) make the biggest substitution possible

 b) ex. like \int (p(x))^4 p'(x) dx, which could be done by
  multiplying out, but substitution easier.

5) (15 min) Activity #6 (stewart) "Loki's Dilemma"
(Loki = a norse god who's a trickster)

6) "Integrals of Symmetric Functions" rules and explanation (uses x -> -x
substitution)


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(R or F) 6.1: Areas, most of 1 day, (skip parametric)
	AND 15min-ish for starting Project #1

Make sure by this day, they have class list of emails and phone #'s.

1) Draw picture, area between graphs of two functions, 

(good source of examples - instructors manual, and it's group work #1)

2) maybe do two simple examples, one w/ dx, one w/ dy

3) trickier example, like Ex. 5 in sec. 6.1. (break up into two integrals)

4) if time, set up more examples, don't integrate

(no activity today, we already did enough this week)

5) (15 min)  project stuff
Project #1:  hand it out, hand out and go
through guidelines, and let them to pick groups and set up meeting
times. (we won't have time to do this next monday)

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