AP: Began discussion of u-substitution for integrals. Contrasted it to reverse chain rule for integrals--main place to use it is for when the chain rule fails (that is, the derivative of the inner function of the integrand is not the outer function, nor easily changeable to it). When this is the case, set variable u equal to that inner function and take the derivative with respect to x, separate variables, and replace all x-terms with U, and dx with the appropriate du. Lecture notes here. Supplemental videos here: link 1 (useful example at 6:34), link 2.
Homework (due thurs) - p. 297: 15-33 (multiples of 3), 40-50 (evens), 57, 58
Honors: Turned in related rates homework. Examples here. Began discussion of applications of derivatives to build toward curve sketching. Lecture notes here.
Tuesday
AP: Looked at differential equations in a bit more detail by solving dy/dx = 3xy for its general solution, which revealed exponential growth. Then returned to anti-derivatives and another u-substitution example for when the reverse chain rule fails to work. Note: u-substitution will also work for reverse chain rule problems, so it is a good fall back option if you get stuck. The AB curriculum does not include "integration by parts" which is what you would do when even u-substitution fails.
Then worked on more examples in class of anti-derivatives of all varieties, including simple polynomials, rationals, trigonometric functions, and log/exponential functions. Lecture notes.
Announced next test scheduled for Feb 5th.
No new homework.
Wednesday
AP: Worked on homework problems from the textbook in DS.
Honors: Began analysis of function behavior. Looked at generic curve, gave it a story, examined local max and min and contrasted with absolute max and min. Constructed definitions of relative max and min, then developed procedure of how to find it. Practiced these skills on a worksheet "Critical Values and Extrema". Lecture notes
Thursday
AP: Turned in book work. Took a detour and examined the derivatives of the inverse trigonometric functions, rounding out differentiation of every family of function studied. We showed how to find the derivative of arcsine, and the process is similar for the other two.
Then practiced more on indefinite integrals, using the reverse chain rule and u-substitution, or perhaps just simplifying the integrand and then doing the power rule for integrals or something else.
Lecture notes
Homework: Finish "Integrals of Varying Difficulty" worksheet for Monday; Finish AP Multiple Choice problems ahead of next Thursday.
Honors
AP: Looked at some of the more difficult indefinite integrals on our worksheet. Allowed time to work on them in class.
Honors; Continued examining critical numbers and showing maximum and minimum. Looked at absolute value graph, where the relative minimum occurs where the derivative is not equal to zero, but does not exist at all. This is a reminder that critical numbers occur both where the derivative equals zero OR does not exist. You then test each for a sign change using the number line to see if it is a local max, local min, or neither. Lecture notes
Homework: Finish worksheet "Critical Values and Extrema"