Wednesday
Finished group work on AP FRQ problems dealing with integration in small groups. Here are the worked out solutions if interested. Continued looking at accumulation functions, examining them through the lens of differential calculus, since the FTC says that differentiation and integration are inverse operations. Wrote the equation of a line tangent to an unseen accumulation function. Finished our proof of the FTC, including the 2nd part which shows how to evaluate definite integrals. Straightened out the meaning and use of both FTCs, emphasizing that FTC1 is used to differentiate accumulation functions while FTC2 is used to evaluate definite integrals by taking antiderivatives.
Looked at applications of FTC1 when the bound is not merely x but rather f(x), leading to a chain rule version of the fundamental theorem. We then returned to accumulation functions and found intervals of increase and inflection points algebraically as well as graphically.
Notes from board
Homework
Do this handout (blank copy)
Resources
video that makes the hw make more sense (link)
Warm up involved finding derivative and second derivative values of an accumulation function, this time involving the chain rule version of the FTC pt 1. Went over some of the homework questions on accumulation functions. Looked at FTC again graphically, noticing a pattern that helps use a graph to identify the values of the accumulation function via area, y-values, or slope for the function, its derivative, or its second derivative respectively. Passed out practice assessment.
Notes from board
Homework
do the practice assessment (blank copy) (SOLUTIONS)
assessment delayed to Weds 2/28
Resources
I-A4a: area under the curve using definite integration: link
I-U4: FTC, algebraically: link1 link2
I-U7: properties of definite integrals: link1 link2 (starts at 4:04)
I-U5: evaluating definite integrals: link1 link2 link3
I-U9: FTC, graphically: link1 link2 link3