Warm up and lesson dealt with more implicit differentiation, including how to find the second derivative implicitly. Basically use implicit differentiation to find the first derivative, then take the derivative of the result (often uses the quotient rule) and replace all the y-primes (or dy/dx, same thing) with the result you got for the first derivative so that your final answer is just x's and y's.
Notes from board
Homework
Mixed Review handout, due 11/29 (blank copy)
Resources
implicit differentiation to find second derivative: LINK
another implicit second derivative: link
basic premise of implicit differentiation: link
implicit differentiation involving product rule: link
implicit differentiation involving product rule to find slope at a point: link
deep dive into the mechanics of implicit differentiation for those curious: link
Tuesday
Finished implicit differentiation by doing another second derivative and also working through an AP free response problem. Announced new retake policy for the end of the quarter, and talked about the format and scoring of the AP test.
Started unit on analyzing function behavior by reviewing what y, y', and y'' each mean. Looked at the s-curve as the basis of a discussion on concavity. This led to the definition of an inflection point. Talked about the four kinds of curvature, and how whether F is increasing and decreasing comes from F-prime's values, and how whether F is concave up or down comes from F-doubleprime's values. Matched derivative graphs to a given function for practice again, noticing the importance of peaks and valleys. These are actually called "relative maxima and minima" or local max and min. Noted that for a max or min to occur, the derivative must be 0 or undefined, and there must be a sign change in derivative (pos to neg for max, neg to pos for min). The locations of maxima and minima are found by identifying critical numbers, which is where the derivative is either zero or undefined. Then you test to find a sign change by making a sign chart on a number line. More on this Thursday.
Notes from board
Homework
Mixed Review handout, due 11/29 (blank copy)
Here are the answers to the "mixed review" handout. Sorry if they're messy. LINK
Thursday
Worked on absolute and relative extrema and how to justify them. Used Kahoot to practice interpreting derivative graphs. An absolute max or min is the biggest/smallest output value a function takes on over a specified interval. They occur either at endpoints or at critical numbers.
Notes from board
Homework
do the practice assessment (blank copy) (SOLUTIONS)
Resources
using a graph of f-prime to determine intervals of inc/dec and relative ext LINK
finding absolute extrema on an interval: link
finding and classifying relative extrema: basic example detailed look
algebraically finding intervals of increase/decrease: link
tangent line approximations: link
implicit differentiation and finding slope at a point: link1 link2
interpreting the derivative link1 link2 link3 link4