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.