What is the instantaneous rate of change at cos pi/4?
-1/root2
for what function ??
if it is y = cosx
then dy/dx = -sinx
and -sin(π/4) = -1/√2
so you are correct if y =cosx
To find the instantaneous rate of change at a point, we need to find the derivative of the function at that point. In this case, we want to find the instantaneous rate of change of the function f(x) = cos(x) at x = π/4.
The derivative of the function f(x) with respect to x is denoted as f'(x) or dy/dx, and represents the rate of change of the function with respect to x at any given point.
To find f'(x), we can use the derivative formula for the cosine function, which states that the derivative of cos(x) is equal to -sin(x).
So, f'(x) = -sin(x).
Now, let's find the derivative f'(x) at x = π/4:
f'(π/4) = -sin(π/4).
Since sin(π/4) equals 1/√2 or √2/2, depending on whether you are using radians or degrees, we get:
f'(π/4) = -√2/2 or -1/√2.
Therefore, the instantaneous rate of change of the function cos(x) at x = π/4 is either -√2/2 or -1/√2.