Finished the group work on poster paper on reverse chain rule, then did a gallery walk. Did another u-sub problem for practice.
Notes from the board
Homework
#1-8 on this handout [NUMERICAL ANSWERS: here]
assessment scheduled for Monday 1/29
Resources
Reverse chain rule example (made by me) link
U-Substitution example (made by me) link
U-sub vs Reverse chain rule: demonstration how u-sub can give same answer: (made by me) Link
more u sub examples: link1 link2
Reviewed summation notation in DS, building up to infinite sums. Had two 'tricky' indefinite integrals to do as a warm up in class, before launching into a different topic, approximating area of weird curvy spaces defined by curves. Saw that one way to do this was to split them up into smaller shapes (like rectangles) and find their total areas. So a sum of areas will approximate the total area. Defined this procedure as a "Riemann Sum" which deals with rectangles of equal width (for simplicity; the width need not be equal) and then formalized a procedure to find the total area as a sum of y-values defined by a function multiplied by a fixed width (for height times base). Which value defined the height separates a left vs right approximation.
Notes from board
Homework
p. 263 #25-30 (for each, sketch a graph and then find the LRAM and RRAM using the indicated # of rectangles)
Resourcse
LRAM and RRAM example by me :) link
another RRAM example by me :) link
more good examples, by another person: link
Warmup was to find a right rectangle approximation for the area under a curve but this time the delta-x's were not 1 so the calculations were a little more tedious. Looked at using the trace function and the table in the calculator to expedite calculations. Reviewed discussion on area from Wednesday, looking at the parts that make up the notation for Riemann sums and then watched a video (first five minutes of this) to help connect the idea of antiderivatives with area (velocity's area gives position). Thought about how to find more precise measures of area, and to get the exact area would involve infinitely many rectangles of negligible, but non-zero width. This led to the formulation of Riemann's definition of the definite integral, which we wrote out and examined bit for bit in the context of area under a curve. We will work more with this notation in the near future. Passed out practice assessments.
Notes from board
Homework
do the practice assessment (blank copy) (SOLUTIONS)
assessment on these topics Monday
Definite Integral Resources
basic premise of approximating a definite integral: link
finding a right Riemann sum: link
understanding the Riemann definition of the definite integral: link1 link2
Assessment Study Resources
I-A1
rationale behind reverse power rule: link
many great integral examples link
I-A2a
reverse chain rule concept video example video
some more examples link link 2, cool old australian guy
I-A2b
U-Substitution example link
Some more: link1 (starts at 6:30) link2 (starts at 7:20, alt method)