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3.6:
a. Find the derivative
b. Is the graph periodic? If so, what is the period? (Hint: adjust the xmin and xmax in “window”)
c. What is the domain?
d. Find one value of ϴ (approximately) where graph has relative extremum (you may use your min/max finder in calculator). Over the interval , how many relative extrema does the graph appear to have?
a) Using the quotient rule [(secxtanx)(1-tan^3x)-secx(-3tan^2xsec^2x)]/(1-tan^3x)^2
b) Periodic with period 2 pi (see attached graph)
c) Domain: there are 3 things that can go wrong.
1. secx is not defined when cosx is 0, i.e., for odd multiples of pi/2.
2. tanx is not defined, but those are the same places as sec x
3. denominator is 0, i.e., when 1-tan^3x=0 or tanx=1 or x= pi/4+pi*k where k is integer, in other words every other odd multiple of pi/4.
Putting 1 and 3 together D= R {x | x=odd mult of pi/2 or x= pi/4+pi*k where k is integer}
From the graph, we see that odd mult of pi/2 are removable discontinuities whereas for x= pi/4+pi*k where k is integer, the discontinuities are infinite (vertical asymptotes).
d) From the 2nd attached file, there seem to be 3 relative extrema over the interval pi/4 to 5 pi/4
Max around x=1.5 and 3.2 and a min about x=2.8. Use your calculator (min or max in calc menu) to find an exact value.
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