Worked out a volumes by cross section example involving an isosceles triangle with a hypotenuse in the plane of the circular base. See notes below for this worked out. Also introduced big picture look at differential equations and their applications, including the mathematical basis of exponential growth. Looked at slope field generation and matching (see videos linked below)
Notes from board
Homework:
Work on practice assessment: blank here
SOLUTIONS HERE
Resources:
Cross section volumes
squares equilateral tri semicircles isosceles triangles dy isosceles right tri with hyp. in plane rectangles
slope field creation: link
slope field usage: link
Took assessment on cross sectional volume and slope fields in DS. Reviewed basic properties of logs (see here for review link 1, link 2). Then worked through several examples of translating a written explanation of physical phenomena into a differential equation, usually involving direct variation (proportional changes).
Then solved our first separation of variables differential equation, the classic dy/dx = ky equation which yields exponential growth (p 8 in the notes). It involves "surgically splitting" the dy from the dx, using algebra to re-arrange terms so y's are on the same side as dy and x's are on the same side as dx. Then we integrate both sides and consolidate our C's (just a constant of integration, could be any number...so C-C is still C, not necessarily 0). Finally we solve for y and get Y = Ce^(kt). It is useful to remember the shortcut on page 9. Moreover, that big capital C can be easily shown to be the "initial value" or "initial population" (as in, the output for t=0). [see here for a video working out the same problem LINK ]
Worked out an AP problem dealing with exponential population growth in order to solve for K. See p 10 in the notes for this. Then did a harder problem involving separation of variables on p 11. We did not get to the cooling problems.
NOTES FROM BOARD
BLANK HANDOUT THAT GOES WITH NOTES
Homework:
p 413 #17, 18, 56
p 421 #3, 7, 8 [D-DE3]
Resources:
for examples seen in class, definitely see the notes above including the videos embedded within the text
finding a particular solution to separable diff eq: link
population growth, exp growth application: link
separation of variables example: link
another example, finding particular solution: link
[Assessment that was scheduled for today is delayed to next Wednesday DS.] Briefly reviewed connection between general solutions to differential equations and slope fields. This motivated a discussion of how a specific point on the plane allows us to single out 1 unique curve that is "channeled" by the slope field, yielding a particular solution. Algebraically, this basically boils down to plugging in the x-y coordinates of a point to find C and plug it back into the general solution, producing a particular solution. Passed out practice assessment ahead of Wednesday's real assessment.
Notes from the board
Homework:
- Remember, Monday, AP-test takers will take a timed, no-calculator section of an AP test. Non-test takers will work on their roller coaster project.
- Work on the practice assessment ahead of Wed's real one (blank copy) (SOLUTIONS)
Watch and take notes on this video: LINK (BTW...Monday night's HW video is this one, if you want to work ahead: LINK)