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 that predicts the instantaneous rate of change, we can start by finding the slope of the function y = cos x at the given points. The slope is the rate of change of y with respect to x.
At x = 0, the graph of y = cos x intersects the x-axis, which means the slope at this point is zero. This is because the tangent line to the graph is horizontal, and if the tangent line is horizontal, the slope is zero.
At x = π, the graph of y = cos x is decreasing, so the slope at this point is negative.
At x = 2π, the graph of y = cos x intersects the x-axis again, which means the slope at this point is zero.
We can express these observations mathematically by using the derivative of the function y = cos x. The derivative represents the rate of change of y with respect to x at any given point.
The derivative of y = cos x is given by: dy/dx = -sin x
Since dy/dx represents the instantaneous rate of change, we can define r(x) as the equation for the instantaneous rate of change:
r(x) = -sin x
Now, let's calculate the exact instantaneous rate of change at x = π/4 by plugging this value into the equation for r(x):
r(π/4) = -sin(π/4)
Using the knowledge that sin(π/4) = 1/√2, we can calculate:
r(π/4) = -1/√2
Therefore, the exact instantaneous rate of change at x = π/4 is -1/√2.