I need some help with these problems I have to find the critical numbers fo the problems.
First I need to differentiate the problem and then I need to set it equal to 0 and solve for the variable.
The first one I just need help differentiating:
This second one I differentiated but I don't know how to solve for theta:
I appreciate any help.
Well, for the first one, you take the derivative of each term
Keep in mind:
The derivative of cosx = -sinx
The derivative of sinx = cosx
The first term's derivative would be 2 * -sin(theta) = -2sin(theta)
The second term's derivative involves the chain rule
It is gotten by first using the power rule, and then differentiating the trig function inside the parenthesis:
So, it is 2sin(theta)*cos(theta)
So, f'(theta) = -2sin(theta) +2sin(theta)cos(theta)
I'll leave the factoring and solving for theta to you.
So, you set the right side equal to zero:
4 - sec^2(theta) = 0
Using a little algebra:
sec^2(theta) = 4
Take the square root of both sides to get:
sec(theta) = 2
Now remember, sec(theta) = 1/cos(theta)
So, 1/cos(theta) = 2
Take the inverse of both sides to get:
cos(theta) = 1/2
I'm sure you know how to do the rest. I hope this helps.
Also, for the first problem, do some review of the chain rule. It will help you understand how to get the derivatives a lot better.
Acutally I don't know what to do next. That's where I'm stuck.
Originally Posted by ajj86
Sorry for not replying, but I haven't been on the forum since thursday or so.
The last part of the second problem involves taking the arccos of both sides to yield:
theta = arccos(1/2)
Now, we think what values of theta will cause cos(theta) to equal 1/2.
well, we know cos(Pi/3) will yield a value of 1/2. Also, since the 4th quadrant is also positive for cosine, the theta value of 5Pi/3 will yield 1/2. Now, the trick is writing these so that it covers all values that will make cos(theta) equal to 1/2.
theta = n*Pi/3 for n = 1,5,7,11,13,17,...