Opened with speeding car example dealing with the mean value theorem. After talking about what "calculus" is [brief mention of Newton and Leibniz and their quarrel], we talked about functions, what they are, and how they behave. This led to a discussion on well-behaved functions and predictability.
This led to an introduction of the concept of the limit, what it is, and how to read them (notation). Click here for a summary similar to our talk. We did our first limit, a simple case that shows the useful of direct substitution. But we closed with an example where direct substitution fails--and thought about how to fix that.
Notes from board
Homework
Finish reading "A Mathematician's Lament" excerpt from last week link
Have parent fill out survey if needed: here
Resources
Mean value theorem concept from warm up (a little technical at the end) link
What is a limit? link
Wednesday
Warm up on 4 limit problems, some a little challenging and unusual. Then looked at how to solve a simple rational function limit graphically by using the TI 84 and then how to get the same answer algebraically. Some problems are more easily solved using one method or the other. Then practiced graphically determining limits through a simple case and then a more complicated case involving letters rather than numbers, including infinity. (blank version and solutions below). Closed with what to do when direct substitution, cancellation/factoring both fail--multiplying by the conjugate. Will explore this in more detail.
Notes from board
Limits (Graphically) Handout (solutions here)
First Quarter Learning Targets
Homework
p. 55 #17-19, 23, 24 [F-L2a]
p. 67 #18-33 (multiples of 3), 41-50 [F-L1a]
First test on Monday will cover F-L1a (Simple Limits), F-L1b (One sided and infinite limits), and F-L2a (Limits from graphs).
Resources
remember that calcchat.com has all the odd-numbered problems worked out
how to "read" a graph to give you limits: link
limits from a graph, shorter video link
limits algebraically, by factoring and cancelling: link
bunch of worked out examples: link
Started class on whiteboards and sketched out graphs that met certain criteria regarding limits. See notes for the worked out answers if you still aren't sure about yours. Went over homework in detail, which dealt with limits from a graph and limits that involve direct substitution as well as factoring and cancelling.
Worked out another example of rationalization to solve a limit, involving multiplication by the conjugate. Then discussed asymptotes and how they relate to limits (basically, an asymptote occurs when infinity is either the result or the "x approaches..." part of the limit). We will define asymptotes according to limits next week. Worked out one-sided limits using a numerical approach.
Notes from board
Homework
p. 79: #7-12, 19, 20 [F-L1b]
Test on Monday! What to study? Notes, homework we went over today, warm up from Wednesday, handout from Wednesday (limits from graphs), and today's warm up.
Resources
sketching a graph from limits (like our warm up); link
another curve sketching video: link
limit by rationalization: link
one sided limits using numerical method (helps with homework) link