Warm up was limits dealing with piecewise functions. Went over the assessment, then properties of limits (basically common sense; see more at this link) then re-learned the special trig rules as a consequence of the "squeeze theorem" and applied them in several examples.
Notes from board
Homework
Watch and take notes on this video starring moi :) link
Warm up dealt with piecewise functions and limits. Then looked at families of functions in chronological order of when they are studied. A good way to self-assess your progress in calculus is to ask yourself if you can do a new calculus skill to each function type.
Since absolute value functions are inaccessible to us with limits, we saw that absolute values are actually piecewise functions in disguise. The hard part is to rewrite the given absolute value as a piecewise. Reviewed asymptotes (see video above for verticals) and especially horizontals, which are limits at infinity that yield a finite result. Looked back at the 'degree rules' of h.a. from pre-calc and algebra 2 and saw how they are really a function of plugging in infinity and noticing which term(s) dominate the numerator and denominator. Extended this idea beyond polynomials to a discussion of function dominance, which allows other functions to be analyzed.
Notes from board
Homework
#1-18 on assorted limits practice handout (blank) (ANSWERS)
Resources
forthcoming
DS notes
Thursday
Warm up dealt with finding the asymptotes of a given rational function and justifying with limits. Went over hw, then began discussion of continuity. The basic definition of a continuous function is one that can be drawn without picking up the pencil, but this leads to a discussion on what kinds of ways something can be discontinuous. Through the metaphor of roads and bridges, we developed the different kinds of discontinuities (jump, removable, and infinite) and discussed the figures in terms of limits. This led to the definition of continuity itself, which says a function is a continuous at a point if and only if the left limit, right limit, and value of the function at the exact value are all 3 equal to each other.
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
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