Assessment on area under curves and area between curves. Took up most of class.
Homework:
- Keep working on practice assessment ahead of next Monday's big honkin' assessment on variety of skills.
- Watch and takes notes on this video: link
(Here is the original Khan Academy video I took the problem from: link)
Worked on practice assessment in DS and began an accumulation function activity using a geometric approach. Continued it in class. Reviewed some of the basic background knowledge for the Fundamental Theorem of Calculus (FTC). Talked about inverses and how they "cancel" each other, reviewed limits, reviewed the limit definition of derivative (using delta x) [See here for an example using h ; see here for an example using delta x].
Then developed a demonstration of the Fundamental Theorem of Calculus, bridging an accumulation function, the limit definition of derivative, and the concept of antiderivative and area. Did this using a guided lessons with area. [Link to blank worksheet here; completed worksheet here]
Will prove the second part of the FTC on Friday, which deals with the method of evaluating definite integrals.
Homework:
p. 290 #75-86 [I-U4]
For 75-80, just evaluate the antiderivative as usual and then plug in x and 0 and subtract the results. Then, take the derivative of the result--you should end up with the integrand as is written in the book, except with an x instead of a t. For #80-86, look back at the example from class--these really are super easy!!!
Resources:
algebraically applying the FTC to find the derivative of an accumulation function: link
first few examples here (the later examples involve the chain rule, which we will do next) link
Went over applying the inverse property of the FTC homework to start class. It is pretty easy to recognize the pattern. Went over some more examples, including what to do when the integration bounds are not correct (flip them and make the integral negative, using the properties of definite integrals) and more importantly how to use the Chain Rule for Derivatives (NOT the reverse chain rule) when the x term of the definite integral is composite (that is, not just plain old "x"). See resources for examples,
Also learned about average value. The formula is obvious: just the sum of the set (integral from a to b) divided by the size of the set (b minus a). See resources for examples dealing with average value.
Then began AP-prep problems in small groups on chart paper dealing with a variety of ideas: accumulation, net change, Riemann sums, FTC, and more. We will continue these on Wednesday.
Whiteboard notes
Homework:
Big Honkin' Assessment is on Monday! here is the blank study guide
Here are the study guide solutions
additional detail on #5 which deals with Riemann sums from tables: link
12 skills! has a lot of weight in powerschool! Study and do well!
FTC and Avg Value Resources
Examples like the FTC chain rule examples we did in class, Note the last example is what we will do next week. link
Average value example, including graphical interpretation: link
another average value example: link
Assessment Review videos:
connection between riemann sum definition and definite integral (I-U1): link
convert a riemann sum to a definite integral, then evaluate it (I-U2): link
approximate a definite integral using left and/or right endpoints analytically (I-U3a): link
midpoint and trapezoid approximations from functions (I-U3b) midpoint link trapezoid link link2
riemann sums from a table, not an algebraic function (note that delta x may not be consistent) (I-U3c) link
area bounded by a curve, vertical lines, and axis (I-A4a) link
evaluate definite integral (won't use as hard a rule as inverse trig tomorrow) (I-A5): link
curve sketching (D-AD13): link
reverse chain rule antiderivatives (I-A2a): link
advanced antiderivatives (I-A2b) link
u substitution, when reverse chain rule fails (I-A2b): link
finding C (I-A3) link