Warm up involved recognizing a scary looking limit problem as actually a derivative question and using the power rule to find the derivative of a quadratic function. Did another lengthy example "from scratch" of finding the slope of a tangent line using the limit definition. Then looked at the derivative ~as a function~ which takes inputs and produces a slope as its output (as opposed to a y-value or height). We saw this graphically with the trace of a dot whose value was the slope of the tangent line. (applets here: link1 and link 2) Did some more power rule examples involving rewriting a function using exponent and radical rules so that it's "ready for prom."
Notes from board
p. 114 #5-18, 25-35 (use power rule! don't hurt yourself with limit definition!)
see calcchat.com for help, plus resources below
limit definition of derivative, the long and hard and ugly way: link0 (made by me!) link1 link 2
using the power rule, simple examples: link
using the power rule, having to rewrite functions first: link (made by me!)
Went over homework in DS, worked through a few problems, then learned the alternate form of derivative at a point and used it to redo the problem from Monday's class. In class, looked at writing the equation of a line tangent to a curve at a given point by using point-slope form which means all that is needed is the slope (or derivative) evaluated at the given point. Worked through a challenging looking example involving the power rule by rewriting using both exponent properties and simplifying with division before taking the derivative.
Looked at several pairs graphs and tried to interpret which one was the original function and which was the derivative. A good strategy is that the peaks/valleys of the original function should line up with the roots (zeroes) of the derivative function. The uphill/increasing portions of the original function should line up with the positive values of the derivative, and likewise for decreasing F and negative F-prime. Sketched a derivative graph from a given function's graph as well, noting the discontinuity in the derivative when the function's slope suddenly changes.
Looked at how to find numerical derivatives (slope at a point, not the slope function itself) using the TI-84. Added basic trig functions to our derivatives toolkit by examining the reasons for why the derivative of sine is cosine (see this video from 12:37 to 16:44). Worked through an example with a basic trig derivative. Passed out practice test.
Notes from board
finish practice assessment [blank copy here] [SOLUTIONS]
real assessment is Friday
Resources to help with today's class stuff
Alternate definition of derivative: link
Write the equation of the tangent line: link
Simplifying using exponent rules before power rule: link
figuring out f and f prime from graphs link
sketching derivative graph from graph of f: link
derivative at a point, ti84: link
sine and cosine derivatives: link
Resources to help with assessment study
showing that a function has a root in a given interval: link
finding the value of c guaranteed by the IVT: link
using limit definition to find derivative: link (feat. me!)
power rule examples link (feat. me!)
approximate derivative value from table link
approximating derivative value using graph link1 link2
horizontal asymptotes using limits: link
vertical asymptotes using limits: link (feat. me!)
finding a and b to make piecewise function continuous: link
find and classify discontinuities, justify with limits: link
determine if piecewise function is continuous or not using def of continuity: link