Today was a full B-day schedule. Warm up was 4 AP test non-calculator multiple choice problems. Went over homework, including an FTC graph problem in depth. Then reviewed the Mean Value Theorem for derivatives which said that the average rate of change of a differentiable function over an interval is equal to the instantaneous rate of change (derivative) for at least one value inside the interval. This is also true for integrals, in the sense that there exists some c in [a,b] where (b-a)*f(c) [or the base times the height of one big rectangle) is equal to the exact area under the curve from a to b. This value of f(c) is considered the "average value" of the function as it is the average of all infinitely many outputs f(x) between f(a) and f(b). Passed out 2003 AP Multiple Choice no-calculator section for hw.
Notes from board
Calendar of upcoming things (Last Q3 assessment is Monday)
Homework
Complete the 2003 AP multiple choice non-calculator section [blank here]
Resources
connection between MVT and the average value: link
concept + procedure of average value (and MVTi) link
Worked on an area between curves problem related to trigonometry without a calculator in DS. Then worked on implicit differentiation review from the AP practice test. Then gave some time to work on the problems.
In class, worked on u-substitution as it relates to a definite integral. Remember to convert the integration boundaries a and b into u-values as well. Then learned about finding volume of solids of revolution which is what happens when you take a region of the plane (either area under a curve or between two curves) and revolve it around a linear axis. The Riemann rectangles for area become thin cylinders whose radius is defined by the functions bounding the region. Worked several examples, including of solids with cavities whose cross sections are "washers" (annuli).
Notes from board
Homework
this handout, #1,2,5,6,7 [blank copy] [numerical answers]
Resources
disc method: link
disc example from class: link
washer method: link
x-axis rotation: link
washer method with axis above region: link
Warm up dealt with recognizing and describing average value and using the trapezoid rule with a table. Worked through a rotational volume problem around a vertical axis of rotation (disk method (dy)). Looked at two more washer method problems dealing with axes of rotation both above and below the region being revolved. Passed out practice assessment.
Notes from board
Homework
Complete the practice assessment [blank here] [SOLUTIONS HERE]
Assessment is on Monday
Resources
I-A4b: Area between two curves, split region: link
I-U7 properties of definite integrals example (starts at 4:04) link
I-U4: FTC, Algebraically: link [note that there's no chain rule in this one]
I-U9: FTC, Graphically: link [plus see storm day video from last week]
I-A7b: Accumulation/Net Change: link (part a only)
I-A7a: Average Value: link
I-A7a: Average Value, explaining what it is: link [starts at 2:16]
I-A5a: Volume by revolution, disk: link
I-A5a: Volume by revolution, washer: link