Warm up dealt with piecewise functions and limits, recognizing that values at a specific value are not important when dealing with limits, but rather values less and greater than (albeit very less than and barely greater than). Looked at a non-polynomial problem dealing with horizontal asymptotes which showed that considering a function's dominance (which grows faster) is useful in considering which functions have H.A. (perhaps y=0) and which have none (when the top grows faster). Looked at the squeeze theorem and how it can prove 2 special cases of trig limits, and then worked through an analytical example using the sin(x)/x rule (see here for a cool proof of the sine limit rule: link). Finally started to look at absolute value limit problems which can be rewritten as piecewise--absolute values will not be on Wednesday's assessment.
Notes from board
Homework
handout #1-18 (can skip 2 and 4 if you're not ready) [blank copy] [numerical answers]
study for assessment (will be in DS on Wed ) practice assess: [blank copy] [SOLUTIONS]
Resources
piecewise limits: link
special trig limits, example: link
algebraic approach to horizontal asymptotes/limits at infinity: link
find vertical asymptotes of a function (video made by me!!) link
one sided limits link
more one sided limits link
finding asymptotes with limits: link
limits at infinity, involving radicals: link
limit by rationalization: link
Took assessment on advanced limits and asymptotes/infinity in DS. Looked at limits involving e during the warm up and went over homework. Continued with absolute value limits and then took stock of our limit toolkit. Started discussing continuity with the roads and bridges activity, which will be finished for homework.
Notes from board
Homework
Finish the roads and bridges activity (8 graphs, 3 tasks per graph)
Warm up involved an absolute value limit problem. Used groups to produce hw answers from roads/bridges activity leading to development of definition of continuity at a point. Explicitly defined continuity on an interval as well as the graphical and algebraic interpretations of jump, removable, and infinite discontinuities. Saw how to find and classify discontinuities algebraically, then started on 4 examples worked on in groups.
Notes from board
Homework
p. 80 #39-57 (multiples of 3) (for each, sketch a graph and find/classify all discontinuities using limits)
be sure to check calcchat.com for help also
Resources
limits involving absolute value: link
review of limits and continuity: link
checking to see if a function is continuous at a point: link
finding and classifying discontinuities algebraically and graphically: link