Joe is investigating the rate of change of the function y=cos x on the interval xE[0,2π]. He notices that the graph of y=cos x passes through the x-axis at 45°. He also determines the instantaneous rate of change at x = 0, π, and 2π by inspection. Based on this information, determine an equation r(x) to predict the instantaneous rate of change of the function y=cos x on the interval xE[0,2π]. Then use the equation to calculate the exact instantaneous rate of change at x = π/4.
Thanks!
To determine an equation r(x) to predict the instantaneous rate of change of the function y=cos(x) on the interval x ∈ [0, 2π], we can use the fact that the instantaneous rate of change is equal to the derivative of the function.
First, let's find the derivative of y=cos(x):
dy/dx = -sin(x)
Now, we can use the derivative to predict the instantaneous rate of change at different points.
For x = 0, we have dy/dx = -sin(0) = 0.
For x = π, we have dy/dx = -sin(π) = 0.
For x = 2π, we have dy/dx = -sin(2π) = 0.
Since the derivative is constant and equal to 0 at all points in the interval [0, 2π], the equation r(x) to predict the instantaneous rate of change is:
r(x) = 0
Now, let's calculate the exact instantaneous rate of change at x = π/4.
Using the derivative dy/dx = -sin(x), we have:
dy/dx = -sin(π/4) = -sqrt(2)/2
Therefore, the exact instantaneous rate of change at x = π/4 is -sqrt(2)/2.
Hope this helps! Let me know if you have any further questions.
To determine an equation r(x) to predict the instantaneous rate of change of the function y = cos x on the interval x ∈ [0, 2π], we need to find the derivative of the function.
The derivative of y = cos x can be found using calculus. The derivative of cos x is given by:
dy/dx = -sin x
Now, to determine the instantaneous rate of change at x = 0, π, and 2π, we can plug these values into the derivative equation:
At x = 0:
dy/dx = -sin(0) = 0
At x = π:
dy/dx = -sin(π) = 0
At x = 2π:
dy/dx = -sin(2π) = 0
We notice that the instantaneous rate of change, as determined by inspection, is 0 for all these values of x.
Therefore, the equation r(x) that predicts the instantaneous rate of change of y = cos x on the interval x ∈ [0, 2π] is:
r(x) = 0
To calculate the exact instantaneous rate of change at x = π/4 using r(x) = 0, we can plug in the value of x:
r(π/4) = 0
So, the exact instantaneous rate of change at x = π/4 is 0.
Please note that the graph of y = cos x passing through the x-axis at 45° (π/4) would provide more information about the behavior of the function at that point. However, based on the given information, we can only determine that the instantaneous rate of change at x = π/4 is 0.