Category: Class Recaps (Page 2 of 9)

Schedule

• WebWork: “Derivatives – Logarithmic” and “Derivatives – Implicit” extended to Fri Dec 13
• Final Exam Review: hand in written solutions to the following exercises on Wed Dec 11:
  • #5 (logarithmic differentiation)
  • #7 (implicit differentiation)
  • #8 (tangent lines, linear approximations and differentials)
  • #16 & #17 (optimization)
  • #18 (using critical pts, intervals of increasing/decreasing, and intervals of concavity for sketching a graph)
  • note that we will cover linear approximation/differentials and optimization on Monday, including setting up the exercises above
  • we have covered logarithmic differentiation, implicit differentiation, and the concepts of #18 — please complete those now
  • the Final Exam Review is available as a pdf on OpenLab Files
• Exam #3:
  • take-home exercises for this exam will be handed out in class on Wed Dec 11, to be handed in on Monday Dec 16
  • there will also be a short in-class exam on Monday Dec 16, similar to the take-home exercises
  • we will use half the class period on Mon Dec 16 to do some additional review for the final
• Final exam: Wednesday Dec 18

Boardshots

We continued the example from the previous class on graph-sketching using the first and second derivatives, looking at a cubic polynomial f(x):

• using the first derivative f'(x) to
  • find the critical points (where f'(x) = 0)
  • to figure out the intervals on which f(x) is increasing (which corresponds to where f'(x) > 0) vs the intervals on which f(x) is decreasing (where f'(x) < 0)
  • to classify each critical point as a local max or a local min
• using the second derivative f”(x) to
  • figure out the intervals on which f(x) is “concave up” (where f”(x) > 0) vs the intervals on which f(x) is “concave down” (where f”(x) < 0)
  • to find the inflection points (where the graph changes concavity)
• using this information to sketch the graph

Please look at the various figures in Sec 4.5 which illustrate these concepts:

Current Topics

Please do the WebWork and work on the Final Exam Review exercises for the following topics!

Boardshots

We an example of logarithmic differentiation (i.e., using the properties of logarithms (specifically “ln”) to simplify and differentiate functions of the form y = f(x)^g(x) )

We then introduced the concepts of maxima and minima, and showed how we can find them by solving for the critical points, i.e., the x-values where f'(x) = 0. We started an example that we will finish on Wednesday:

Schedule

Boardshots

We reviewed the properties of logarithms, and showed how we can use them, together with the chain rule, to carry out logarithmic differentiation. We also introduced implicit differentiation, for finding dy/dt when y is given “implicitly” as a function of x via an equation: