SOLUTION: Consider the function {{{y = 4sin(expr(pi/4)(x) - pi/2) - 3}}}. Identify an interval starting at x = 2 with an average rate of change that is a) positive b) negative, and

Algebra ->  Trigonometry-basics -> SOLUTION: Consider the function {{{y = 4sin(expr(pi/4)(x) - pi/2) - 3}}}. Identify an interval starting at x = 2 with an average rate of change that is a) positive b) negative, and       Log On


   

Question 1140452: Consider the function y+=+4sin%28expr%28pi%2F4%29%28x%29+-+pi%2F2%29+-+3. Identify an interval starting at x = 2 with an average rate of change that is
a) positive
b) negative, and
c) zero
With explanations
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The first obvious thing to do is use the identity 

sin%28theta-pi%2F2%29=-cos%28theta%29 to simplify.

y+=+4sin%28expr%28pi%2F4%29%28x%29+-+pi%2F2%29+-+3 becomes

y+=+-4sin%28expr%28pi%2F4%29%28x%29%29+-+3

Its average (not instantaneous!) rate of change over P%3C=x%3C=Q

is  



which simplifies to



Then we use the identity: sin%28alpha%29-sin%28beta%29=2sin%28%28alpha-beta%29%2F2%29cos%28%28alpha%2Bbeta%29%2F2%29

I won't go through the details, but that simplifies to

}
Now you finish by solving this equality:



to find some zeros which will be answers to c) and which
will also be critical numbers for a) and b).  Then solve this



for a)

and this



for (b)

Edwin