Met for the full 90 minutes today. Warm up involved 2 limit problems that would have required special trig cases or factor-and-cancel but now can be done with L'hopital's Rule. Went over homework (answers included in notes link below) and then used implicit differentiation to find a second derivative (the acceleration function, or as we will discover, the concavity function). Basically you do implicit differentiation as you normally would, then do it again to the result. Your answer will probably have y-prime in it, so replace it with what you first got for y-prime and simplify if possible/reasonable.
Then reviewed an algebra concept known as inverse functions which literally invert input and output. Saw the special case that if f(x) and g(x) are inverses, then f(g(x))=x . Applying implicit differentiation to this and using the chain rule and then solving for g'(x) we get that g'(x)=1/f'(g(x)) when f and g are inverses. This rule can allow you to find slopes of inverse functions when finding the inverse function itself is a pain.
Notes from board
Homework
Finish L'hopital's rule hw p. 564 #5-17, 23-30 [D-AD0] for 11/28
Assessment Weds 11/30 in class: covers D-AD0 (L'hopital's Rule) D-AD6 (derivatives of inverses) and D-CD5 (Horizontal and vertical tangents)
Resources
implicit differentiation to find second derivative: LINK
one of the implicit y'' from class worked out: link
inverse functions' derivative example: link
great example of inverse function derivative with a table: link
l'hopitals examples: link link 2 link 3 link 4