Please help..

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.

Please show steps-by-steps, thank you.

To determine the instantaneous rate of change of the function y = cos(x) on the interval x ∈ [0, 2π], we can use the derivative of the function. The derivative represents the rate of change of a function at any given point.

Step-by-step Solution:

1. Start by finding the derivative of the function y = cos(x). The derivative of cos(x) is -sin(x). So, we have r(x) = -sin(x).

2. Now, we need to find the value of the instantaneous rate of change at x = π/4. To do this, substitute π/4 into the derivative function r(x) = -sin(x).

r(π/4) = -sin(π/4) (Plug in x = π/4 into r(x))
= -1/√2 (Simplify the sine value for π/4, which is 1/√2)

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

Explanation:

To find the instantaneous rate of change of a function, we use the concept of derivatives. The derivative of a function gives us the rate of change of the function at any point. In this case, the derivative of cos(x) is -sin(x).

By finding the derivative, we obtain an equation, r(x), that represents the instantaneous rate of change of the function y = cos(x) on the interval x ∈ [0, 2π]. To find the instantaneous rate of change at a specific point, we substitute that point into the derivative equation.

In this problem, we substitute x = π/4 into r(x) = -sin(x) to find the exact instantaneous rate of change at x = π/4.