Samuel is investigating the rate of change of the function f(x) = cos x on the interval xE[0, 2pi]. He notices that the graph of f(x) passes through the x-axis at pi/2. He also determines the instantaneous rate of change at x = 0, pi and 2pi by inspection. Based on this information, determine an equation r(x) to predict the instantaneous rate of change of the function f(x) on the interval xE[0, 2pi] Then use your equation to calculate the exact instantaneous rate of change at x = pi/4

I bet f'=-sinx

To determine an equation r(x) to predict the instantaneous rate of change of the function f(x) on the interval xE[0, 2pi], you can use the concept of the derivative. The derivative of a function represents its instantaneous rate of change at a given point.

In this case, f(x) = cos x, and we want to predict its instantaneous rate of change on the interval xE[0, 2pi]. The derivative of cos x is -sin x.

To calculate the exact instantaneous rate of change at x = pi/4, we need to evaluate the derivative of f(x) at that point. So, substitute x = pi/4 into the derivative:

r(x) = -sin(x)
r(pi/4) = -sin(pi/4)

Calculating this, we get:
r(pi/4) ≈ -0.707

Therefore, the exact instantaneous rate of change at x = pi/4 is approximately -0.707.