Passed back assessments from Nov 20, went over a few in depth. Looked at the Mean Value Theorem, which states that for a function differentiable on an open interval, the instant rate of change (derivative value) must be the same as the average rate of change on the interval (slope of secant line) for at least 1 value in the open interval. Saw this geometrically on a graph as parallel lines and why smoothness (differentiability) was a requirement. Worked one example of finding the value of c guaranteed to exist by the MVT.
Notes from board
Homework
p. 174 #31-40 (Due Friday)
Resources
mean value theorem algebraically: link
intuition of the mvt (similar to what was discussed in class) link
additional MVT example: link
Warm up involved finding values of c that satisfied the Mean Value Theorem, noting that the answer(s) need to fall in the interval (a,b) so the endpoints are excluded. Looked at several generic graphs of F to then choose from options which given graph accurately graphed F-prime, the derivative of F. Talked about how peaks and valleys of F signify the x-intercepts/roots of F-prime, and that positive F-prime values correlate with increasing/uphill F, and similarly for negative dy/dx and decreasing y.
Reviewed concept of intervals of increase/decrease with interval notation and precise vocabulary, being sure to justify answers by using the derivative values (e.g. increasing F because F-prime >0). Gave the idea of a peak and valley the formal names of relative (or local) maximum and minimum, respectively. Defined critical numbers as those where the derivative is either equal to zero or undefined, noting that extrema must occur at these critical numbers but not every critical number necessarily is a max or min (a sign change must ALSO occur). Practiced this skill graphically using the voting eggs and looking at graphs of F-prime to determine certain behaviors of F.
Closed by introducing the algebraic approach to finding max and mins, which we will do more of on Friday.
Notes from board
Homework
due Friday: p. 174 #31-40 (deals with Mean Value Theorem)
due Monday: p. 183 #18-39 (mult of 3) [ignore book instructions, just (a) find and classify extrema and (b) find and justify intervals of increase/decrease.
Resources
how to find the value guaranteed by the MVT: link
Warmup dealt with MVT, involving the product rule. Went over hw, then worked out in detail how to find, classify, and justify a function's relative extrema. Did one together and then did one independently. Noted that finding intervals of increase and decrease is basically similar to finding relative extrema.
Introduced the idea of absolute extrema, which must occur on an interval. The EVT (Extreme Value Theorem) states that a function must have a max and a min value on an interval. Absolute extremes must occur either at endpoints of the interval or critical numbers. Used whiteboards to draw functions that met certain criteria involving relative and absolute extrema.
Notes from board
Supplemental to help with p. 167 hw
Homework
** next assessment is Wednesday #18-39 (mult of 3)
** due Monday: p. 183 #18-39 (mult of 3) [ignore book instructions, just (a) find and classify extrema and (b) find and justify intervals of increase/decrease.
**due Wednesday: p. 167 #17-27 (odd), 52 (see supplemental notes above for help)
Resources
some background on relative vs absolute extrema: link
finding absolute extrema on an interval: link
what are critical numbers: link
extrema and critical numbers, graphical concept: link
identifying relative extrema: link