Calculus 2, Lec 38A, Part 1, Pendulum Model, Start Review for Final Exam (Examples from Old Exams)

Part 1 of Lecture 38A for Calculus 2. This includes a review of a bunch of examples and solutions (problems and solutions) for the Calculus 2 Final Exam.

(0:00) Final exam next week. Course evaluations due soon. Financial Math for Actuarial Science.
(2:47) Differential equation and corresponding system for a pendulum ("swinging bob").
(5:04) Mathematica code involving NDSolveValue to approximate solutions and graph them, including in a phase plane on top of the level curves of a Hamiltonian function (representing total mechanical energy).
(8:51) Animation of the motion of the actual pendulum.
(14:06) Add a term to model model friction (so it's a damped pendulum).
(17:26) Increase the initial velocity to make the pendulum go over the top.
(20:12) Problems on the test will mostly be a sampling of problems similar to ones on old exams.
(21:35) Example 1: Use abstract properties of integrals (like "linearity") to solve for an unknown integral.
(23:20) Example 2: area under a parabola (be able to factor or use the quadratic formula to find roots).
(24:48) Example 3: average value of a function and geometric interpretation.
(27:16) Example 4: use the 2nd Fundamental Theorem of Calculus (Antiderivative Construction Theorem) and the Chain Rule (and be able to find the first positive critical point of the function defined by the integral).
(31:24) Example 5: a simple differential equation (a "pure antiderivative problem").
(32:26) Example 6: Approximate the final volume in a water tower when you are given the initial volume and the net flow rate (either use numerical integration or "count boxes" and use the area of each box).
(34:38) Example 7: total radiation emitted can be found by integrating the rate at which the radiation is being emitted (confirm algebra on Mathematica).
(38:57) Example 8: trigonometric substitution integral.
(44:46) Example 9: reminder about formula for Simpson's Rule.
(46:36) Example 10: improper integral calculated as a limit and using integration-by-parts.
(48:37) Example 11: Errors in numerical integration.
(55:21) Example 12: Start of a problem about graphing a parametric curve, find the speed and integrate to find the distance traveled (arc length), and draw position, velocity, and acceleration vectors.