Week 4 lesson plans: ------------------------------- Mon 5.9 Approximate Integration: in-class discussion on why we learn/need approximate integration techniques (when you don't know the antideriv and when you don't have the formula for the function) go over Riemann sums (left & right endpoints) introduce Midpoint Rule introduce Trapezoidal Rule introduce Simpson's Rule Activity 1: Comparison of Methods illustrate accuracy by approximating the same thing using different rules ------------------------- Tue go over activity error bounds for the above rules note: Simpson's rule is often the most accurate also note that we can sometimes tell when you have an over- or under-estimate Activity 2: Using Error Bounds go over activity ------------------------- Wen 5.10 Improper Integrals (only Type 1): define type 1 improper integrals Note: don't forget that divergence could mean limit just doesn't exist rather than having infinite area do some examples: comparison tests for conv/div do some examples, suggestions: Int_1^Inf 1/(x^2+x+1) dx converges (compare to 1/x^2) Int_4^Inf dx/(x^2-2x) converges by comparing to 2/x^2 for x \geq 4 Note: do example like above where f is not greater than g for all x, just the relevant x Int_1^Inf dx/(x^3+7x^2+2x+1) converges ------------------------- Thu/Fri explain induction & do example (formula for sum of i^2) Activity: Sigma detectives revisted (use induction in order to do problem 50*) (**then in chapter 8, early on, do induction again**) hand out extra reading material on induction