Intro Real Analysis, Lec 17: Mean Value Theorem Corollaries, Definition of Riemann Integral

Introduction to Real Analysis, Lecture 17.

(0:00) Announcements.
(1:23) Lecture plan.
(1:41) Warnings related to the Increasing Function Theorem: 1) the converse of the Increasing Function Theorem is false (f(x) = x^3 is strictly increasing over an interval but f'(0) = 0), 2) just because f'(c) is positive at some number c, does NOT mean that f is increasing on some open interval containing c (example: f(x) = x/2 + x^2*sin(1/x) when x is nonzero and f(0) = 0, f'(0) = 1/2 but f is not increasing on any open neighborhood of the origin).
(7:50) Constant Function Theorem (CFT). This is not contradicted by the floor function. You should be able to prove the CFT with the Mean Value Theorem (MVT).
(11:43) Any two antiderivatives of the same function over some interval differ by a constant.
(13:10) Other applications: 1) first derivative test, 2) and 3) general solutions of differential equations (do a proof related to this), 4) Fundamental Theorem of Calculus.
(26:57) Prove (1 + x)^(1/3) is less than or equal to 1 + x/3 for all positive x with the Mean Value Theorem (and give statement of a more general case).
(31:30) Be able to use L'Hopital's Rule (do a couple examples).
(38:35) Derivatives satisfy the intermediate value property and various other definitions and facts (related to, for example, concave up and concave down).
(42:52) Definition of Riemann integrability (what it means for a function to be Riemann integrable over a closed and bounded interval, using the idea of a tagged partition and corresponding Riemann sum).

Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN.

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[KK-fvbs]

I just chanced upon this playlist while I was searching for this topic so much and OMG this is pure gold. Deserves so much more views.

[drewjackson]

I like this definition of Riemann-integrability much better than the one in my text (which uses the sup/inf of upper and lower sums) . Fun to work through both and see different ways of conceptualizing the integral

[GH-zqkq]

Thank you so much for these videos! Which book are you referring to in them?