Passed back assessments in DS, went over them in small groups. Learned how to do midpoint rectangle sums (MRAM) using a given function. Then did a trapezoid rule example problem based on the formula extracted from the video above. Also looked at how to do left and right sums from a table when not given an explicit function, noting that this is sometimes more like real-world behavior with only selected values rather than a perfect function modeling something. Learned how to move fluently from an infinite Riemann sum to the definite integral notation and vice versa.
Notes from board
Homework
#1-12 on this handout (link) (numerical answers)
Resources
write a definite integral from an infinite Riemann sum: link1 link2
write an infinite Riemann sum from a definite integral: link
LRAM and RRAM from table: link (ignore the 1/10th in the 2nd example)
Midpoint sum given a function: link (first half)
trapezoid rule examples: link1 link2 (starts at 5:27)
Warm up was to rewrite integrals as infinite sums and vice versa. Did a midpoint rectangle from a table, which is pretty easy as far as calculus goes. Looked at several different 'trick' problems to calculate definite integrals (which, remember, have numerical answers) by noticing certain properties regarding the area, whether utilizing symmetry, an interval width of zero, or using geometry to find area.
Started to prepare for the Fundamental Theorem of Calculus by looking at a simple example of linear motion involving a snail on a stick, and remembering that distance equals rate * time, and that this can also be thought of as the area under the velocity graph, since the horizontal axis unit (time) would cancel part of the vertical axis unit (distance/time) to leave an "area" answer in "distance" units. Noted that this would just give displacement (how far something has moved) and NOT position, since there is no frame of reference or starting position.
Gave a starting position for the snail example, involving a ladybug at one end of the stick and saw that we can calculate position by adding the distance traveled to the initial position. Also noted that we can calculate the distance traveled by subtracting the initial position from the final position (s(8)-s(0) in the example).
Generalized this idea to note that, by analyzing units, the area underneath any time vs rate function would give the accumulated quantity. (Example in notes uses hours vs. megabytes per hour of phone usage; area of all the rectangles in that space would have unit "megabytes").
Looked at a motion problem with variable velocity, asking for distance traveled in first 4 seconds. Recognized from the position-velocity-acceleration hierarchy that the antiderivative of velocity would give us position, with a pesky +C. But, thinking back to the snail example, distance traveled can be found by subtracting final position minus initial position, so we found s(4) and s(0) and subtracted them, and the C disappeared, leaving a numerical answer. This is the way to find the value of a definite integral (since the area under the velocity curve is ALSO distance traveled). This is the Fundamental Theorem of Calculus Part 2.
Notes from board
Homework
none
Resources
forthcoming