How do you find the intervals of increasing and decreasing using the first derivative given #y=cos^2(2x)#?

Answer 1

#y " is" { ("increasing", =>, pi/4 < x < pi/2, (3pi)/4 < x < pi, ...), ("stationary", =>, x=0, x=pi/4, ...), ("decreasing", =>, 0 < x < pi/4, pi/2 < x < (3pi)/2, ..) :} #

#y=cos^2(2x)# is # { ("increasing", <=>, dy/dx> 0), ("stationary", <=>, dy/dx= 0), ("decreasing", <=>, dy/dx< 0) :} #

Differentiating wrt #x# we have:

# dy/dx = 2cos(2x)(-sin(2x))(2) #
# :. dy/dx = -4cos(2x)sin(2x) #
# :. dy/dx = -2sin(4x) #

This is the graph of the derivative :

#y " is" { ("increasing", =>, pi/4 < x < pi/2, (3pi)/4 < x < pi, ...), ("stationary", =>, x=0, x=pi/4, ...), ("decreasing", =>, 0 < x < pi/4, pi/2 < x < (3pi)/2, ..) :} #

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Answer 2

Cameron Alexander

To find the intervals of increasing and decreasing using the first derivative given $y = \cos^2(2x)$ , follow these steps:

Differentiate $y = \cos^2(2x)$ with respect to $x$ to find the first derivative.

$\frac{dy}{dx} = -4\cos(2x)\sin(2x)$
Set the first derivative equal to zero to find critical points.

$-4\cos(2x)\sin(2x) = 0$

Critical points occur where $\cos(2x) = 0$ or $\sin(2x) = 0$ .
Solve for $x$ to find the critical points.

For $\cos(2x) = 0$ , $2x = \frac{\pi}{2} + n\pi$ , where $n$ is an integer. For $\sin(2x) = 0$ , $2x = n\pi$ , where $n$ is an integer.
Find the intervals on the domain of $y$ where the first derivative is positive or negative.

Test intervals between critical points and at endpoints of the domain.

Use the first derivative test:
- If $\frac{dy}{dx} > 0$ , the function is increasing.
- If $\frac{dy}{dx} < 0$ , the function is decreasing.
Record the intervals where the function is increasing or decreasing.

That's how you find the intervals of increasing and decreasing using the first derivative for $y = \cos^2(2x)$ .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.