- I’m out of town second part of next week (Wednesday afternoon on)
- Working on a guest lecture for class that day
- Colloquium
- “An Elementary Construction of Tessellated Surfaces”
- Jonathan Pakianathan
- University of Rochester
- Monday, March 2, 4:00 - 5:00, Newton 201
- Extra credit for summaries
- Exam 1
- Next Tuesday (March 3)
- Material since start of semester (e.g., ln and ex, L’Hôpital’s Rule, inverse trig functions, hyperbolic functions, integration by parts, integration via trig substitution, partial fractions, numeric integration, improper integrals)
- 4 - 6 short-answer questions, similar to problem set questions
- Whole class period
- Open book, notes, computer for reference; closed person, no CASes
- Polynomial division / synthetic division?
- Simpson’s rule vs trapezoid rule
- Area under curve = sub-interval size times weighted sum of y values
- Trapezoid rule weights = 1 2 2 2 2 ... 2 1 (over 2)
- Simpson’s rule weights = 1 4 2 4 2 4 2 ... 4 1 (over 3)
- Riemann sums weights = 1 1 1 1 ... 1
- Simpson’s rule example
- y = 3x2 between 1 to 5 in 4 sub-intervals
- y = 3x2 between 1 to 5 in 4 sub-intervals
- Area under curve = sub-interval size times weighted sum of y values
- Section 8.8
- Rules for integrals of the form integral from a to ∞ or
-∞ to b or -∞ to ∞
- Limits as bound approaches +/- ∞
- Or 2 limits and finite value for -∞ to ∞
- Rules for integrals of f(x) where f(c) = +/- ∞
- Limit of integral as x approaches c from appropriate side
- Rules for integrals of the form integral from a to ∞ or
-∞ to b or -∞ to ∞
- Examples
- Integrate 1/x2 from 1 to ∞
- Integrals of 1 / ( √x (√x - 1) )
- This function has 2 vertical asymptotes
- An integral from 0 that involves the first
- Substitution and other integration techniques work here
- But note that substitution has implications for value approached in limit
- This is a converging integral
- On the other hand, integrals that involve the asymptote at x = 1 are infinite, they diverge
- Integrate 1/x2 from 1 to ∞
- (After exam)
- Introduction to differential equations
- Section 7.4