Calc 1, Lec 31A: Important Mean Value Theorem Examples, Squeeze Theorem, L'Hopital's Rule Examples

(0:00) Upcoming schedule. I will review old content as we do new content. Lecture plan.
(0:51) Write Mean Value Theorem (MVT) statement and graph on the board.
(5:50) Example 2: a discontinuous function that does satisfy the conclusion of the Mean Value Theorem (by "chance"). In fact, it satisfies it at infinitely many values of c. Also remember that differentiable functions are continuous and therefore discontinuous functions are not differentiable.
(7:53) Example 3: a continuous function that is not differentiable at one point (the absolute value function) and fails the conclusion of the Mean Value Theorem.
(9:25) Example 4: a function that "just barely" satisfies the hypotheses of the Mean Value Theorem (it is not differentiable at the endpoints of the interval) (it's the inverse sine function). It must satisfy the conclusion because of the truth of the theorem. Be able to solve for c if asked on an exam.
(13:47) Example 5: a continuous but nondifferentiable function (the cube root function) that happens to satisfy the conclusion of the Mean Value Theorem.
(16:08) An application of the Squeeze Theorem: to evaluating the derivative of a wild function at a point.
(22:24) L'Hopital's Rule basic examples: sin(x)/x, sin(2x)/(3x), and (e^(5x)-1)/(4x) as x goes to zero.
(24:00) L'Hopital's Rule Example 1 and an intuitive justification using local linearity.
(29:28) L'Hopital's Rule Example 2.
(31:06) L'Hopital's Rule Example 3.