SUNY Geneseo Department of Mathematics
Thursday, September 26
Math 221 06
Fall 2019
Prof. Doug Baldwin
SI session this afternoon 3:00 - 4:30, Fraser 104.
Section 3.5
Key idea: proof via the limit definition of derivative, with some apt uses of the squeeze theorem to find certain crucial limits.
Use these ideas to show that the derivative of cos x is -sin x.
Start with the limit definition applied to cosine:
Then use a trigonometric identity to expand cos(x+h):
Then rearrange to separate terms involving x and terms involving h (rationale: any terms that involve x are just constants as far as limits as h approaches 0 are concerned, and so will be easy to evaluate; remaining terms involving h will hopefully be evaluable via limit laws or similar).
Finally, the limits with h in them can be evaluated via the squeeze theorem. One goes to 0 and the other to 1, leaving a simple expression for the derivative.
Show that the derivative of sec x is sec x tan x.
Key idea: proof via the quotient rule, which is the crux of all derivations for trig functions besides sine and cosine.
secx = 1 / cosx
Key trigonometric antiderivatives, maybe examples of applying trigonometric differentiation rules.
Then the chain rule.
Read section 3.6 of the textbook.