Final answer:
The 10th derivative of y=cos(2x) is -4cos(2x). Its amplitude is 4, periodicity is π, maximum value is 4, and the rate of change's maximum magnitude is also 4.
Step-by-step explanation:
To find the 10th derivative of the function y=cos(2x), we must understand the differentiation pattern of sine and cosine functions. The derivatives of cosine are cyclic with a period of 4: cos(x), -sin(x), -cos(x), sin(x), and then back to cos(x). Since we are finding the 10th derivative, which is 4 times 2 plus 2, we'll end up at the second position in the cycle, which is the same as the second derivative.
The second derivative of y=cos(2x) is -4cos(2x), hence the 10th derivative will be the same. Now, let's address the student's questions based on the 10th derivative:
- a) Amplitude: The amplitude is 4, since the coefficient of cos(2x) in the 10th derivative is 4.
- b) Periodicity: The function's periodicity is π, as the factor of x in the argument of the cosine function is 2, and the period of cosine is π units with such a factor.
- c) Maximum value: The function's maximum value is 4, which is the amplitude of the cosine function.
- d) Rate of change: The function's rate of change at any given point will depend upon the current value of the cosine function. However, the rate of change's maximum magnitude will be 4, at the points where the derivative is ±4.