No warm up. Reviewed L'hopital's Rule and derivatives of inverses for those who were out or needed a refresher. Worked on practice assessment. See below for blank copy + solutions.
Calendar of upcoming content + assessments before end of Q2
Work on practice assessment (blank here) [SOLUTIONS]
Resources
L'hopitals rule: link link 2 link 3 link 4
Derivatives of Inverses: link
Horizontal and vertical tangents from implicit derivative (harder than our problem) link
Second derivative implicitly: link1 link 2
Went over practice assessment in DS, then assessed in class before discussing the Mean Value Theorem, a fundamental property of differentiable curves. It states that a function differentiable over an interval is guaranteed to have an instantaneous rate of change (derivative value) that is the same as its average rate of change over the interval, at some value in the interval.
Notes from board
Homework:
none -_-
Resources
intuition of the mvt (similar to what was discussed in class) link
additional MVT example: link
Warm up on the mean value theorem and finding the value of c. Reviewed pre-calc concepts of intervals of increase and decrease and related to how derivatives describe a function's slope so increasing f implies positive f-prime while decreasing f implies negative f-prime and vice versa.
Introduced concepts of relative (local) max and mins, where a function is a peak or a valley. These occur where the slope (f-prime or dy/dx) is equal to zero or undefined (these locations are called critical numbers). In brief, they are points where the function is highest or lowest in a narrow vicinity (in its 'neighborhood'). Absolute Extrema, by contrast, occur over a defined interval and represent the highest or lowest value the function achieves on the interval. These occur either at relative extrema (meaning, the local max or min happens to also be the highest/lowest over the whole interval) or at an endpoint of the interval itself.
Used clickers to identify behavior of F based on a graph of F-Prime, noting that relative maxima occur when F-prime equals zero (or is undefined) and changes from positive to negative, while relative mins occur when F-prime equals zero (or is undefined) and changes from negative to positive.
Notes from board
Absolute Extrema example worked out in detail
Homework
p 167 #17-28
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