Calculus with compositions–an interesting juxtaposition– the hardest differentiation technique is the chain rule, while the easiest integration technique (other than ooh! I see a derivative I know!) is simple substitution.
Theorems of Calculus – EVT, IVT, MVT: mostly a repeat of what we did in October, though we got into a cool discussion. f(2) = 1; f(5) = -2; f(13) = 6, for f twice differentiable. Thought 1: Does f have an absolute maximum on [2,13]? Does f have an absolute maximum on (2,13)? Does f have an absolute minimum on [2,13]? How about (2,13)? I like the thought path that leads you to conclude that based on the closed interval having endpoints higher than a known point on the open interval, there has to be a non-endpoint minimum on the closed, therefore on the open. Nice.
Second thought here: 1. y = cuberoot(x). Can’t apply MVT, but the conclusion of the MVT holds. Piecewise f(x) = (1) sqrt(1-(x-1)^2), 0≤x≤2; (2) -sqrt(1-(x+1)^2), -2≤x≤0. Conclusion of MVT holds here too. Maybe a measure of a graph where you can draw all tangent lines?
Cognitive leaps in mathematics: Quantities, Place Value, Operations, Symbols, Expressions involving variables, Find the numbers that make equations true; Functions – all expressions that have a relationship; calculus – the rate of change of output with respect to input. differential equations – equations whose solutions are families of functions.
Jeopardy – working with cool problems in a fast paced format. Easy to edit file for creating calculus jeopardy games.
Exploring some weird functions: f(x)=.2x+x^2sin(1/x); f'(0)=.2, however, no interval around 0 is increasing.