Instantaneous rate of exchange for f(x) = x - x^3 where x is -1
To find the instantaneous rate of exchange for f(x) = x - x^3 at x = -1, we need to find the derivative of the function and then evaluate it at x = -1.
Taking the derivative of f(x) = x - x^3 using the power rule, we get:
f'(x) = 1 - 3x^2.
Now, evaluating f'(x) at x = -1:
f'(-1) = 1 - 3(-1)^2.
Simplifying,
f'(-1) = 1 - 3(1) = 1 - 3 = -2.
Therefore, the instantaneous rate of exchange for f(x) = x - x^3 at x = -1 is -2.
To find the instantaneous rate of change at a specific point, we need to find the derivative of the function and then substitute the given value into it.
Let's find the derivative of f(x) = x - x^3 with respect to x:
f'(x) = 1 - 3x^2
Now, let's substitute x = -1 into the derivative:
f'(-1) = 1 - 3(-1)^2
= 1 - 3(1)
= 1 - 3
= -2
Therefore, the instantaneous rate of change of f(x) at x = -1 is -2.