Answer:
Explanation:
To find the maximum rate of change in y concerning x, we need to find the derivative of y concerning x and then determine the maximum value of that derivative.
Given that y = sin 3x + 4/3 cos 3x, we can find the derivative by applying the chain rule.
First, let's find the derivative of sin 3x:
The derivative of sin 3x concerning x is cos 3x, and we multiply it by the derivative of the inner function (3x) concerning x, which is 3. So, the derivative of sin 3x is 3cos 3x.
Next, let's find the derivative of 4/3 cos 3x:
The derivative of 4/3 cos 3x concerning x is -4/3 sin 3x, and we multiply it by the derivative of the inner function (3x) concerning x, which is 3. So, the derivative of 4/3 cos 3x is -4sin 3x.
Now, we can find the derivative of y:
The derivative of y = sin 3x + 4/3 cos 3x with respect to x is:
dy/dx = 3cos 3x - 4sin 3x.
To find the maximum rate of change, we need to find the maximum value of dy/dx. This occurs when dy/dx is equal to zero or does not exist.
Setting dy/dx = 0, we have:
3cos 3x - 4sin 3x = 0.
Dividing both sides by 3, we get:
cos 3x - (4/3)sin 3x = 0.
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
cos 3x - (4/3)sin 3x = cos^2(3x) + sin^2(3x).
Rearranging the equation, we have:
cos^2(3x) + sin^2(3x) - cos 3x + (4/3)sin 3x = 0.
Simplifying further, we get:
(5/3)sin 3x - cos 3x = 0.
Now, we can solve for x.
One solution is x = 0, but there are other solutions as well.
To find the maximum rate of change, we need to find the maximum value of dy/dx. To do this, we can substitute the values of x where dy/dx = 0 into dy/dx and compare the values to find the maximum rate of change.
However, without additional information or constraints on the domain of x, we cannot determine the exact value of the maximum rate of change.
In summary, the maximum rate of change in y concerning x occurs when dy/dx is equal to zero or does not exist. However, without additional information or constraints on the domain of x, we cannot determine the exact value of the maximum rate of change.