A Cyclist is riding on a path whose elevation is modeled by the function f(x) = 0.08(16x-x^2) where x and f(x) are measured in miles. Find the rate of change of elecation when x=4. Would this be 0.64?
dx/dt would be 1.28 - 0.16x and then plug in four?
df/dx = .08(16-2x)
at x=4, df/dx = .08*8 = .64 as you figured.
To find the rate of change of elevation when x = 4, we need to differentiate the given function f(x) = 0.08(16x - x^2) with respect to x.
Taking the derivative f'(x) will give us the rate of change or the slope of the function at any given point.
To differentiate the function f(x), we can use the power rule and the constant multiple rule of differentiation.
Applying the power rule, the derivative of x^n, where n is a constant, is nx^(n-1). And applying the constant multiple rule, the derivative of k * f(x), where k is a constant, is k * f'(x).
Using these rules, let's differentiate f(x):
f'(x) = 0.08 * (16 - 2x)
Now, we have the derivative in terms of x.
To find the rate of change of elevation when x = 4, we substitute x = 4 into the derivative f'(x):
f'(4) = 0.08 * (16 - 2 * 4) = 0.08 * (16 - 8) = 0.08 * 8 = 0.64
So when x = 4, the rate of change of elevation is indeed 0.64.
Now, let's address your question about dx/dt.
The expression dx/dt represents the rate at which the independent variable x is changing with respect to the dependent variable t. Since the function f(x) does not involve t, we can't directly use dx/dt to find the rate of change of elevation.
Instead, we differentiate f(x) with respect to x, which gives us f'(x), the rate of change of elevation with respect to x. Then we can substitute the specific value of x to find the rate of change at that point.