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 find an equation that predicts the instantaneous rate of change of the function y = cos(x) on the interval x E [0, 2π], we need to consider the derivative of the function. The derivative represents the rate of change of the function at any given point.

The derivative of y = cos(x) is given by dy/dx = -sin(x).

Now, let's calculate the exact instantaneous rate of change at x = π/4 using the equation for the derivative.

We substitute x = π/4 into the derivative equation:

dy/dx = -sin(π/4)

To find the value of sin(π/4), we note that it is a special angle where sin and cos have the same value. In this case, sin(π/4) = cos(π/4) = 1/√2.

Substituting this value into the derivative equation:

dy/dx = -1/√2

Therefore, the exact instantaneous rate of change at x = π/4 is -1/√2.