Explain why the instameous rate of change for g(x)=-x^2 cannot be the same for any different value of x
The instantaneous rate of change, also known as the derivative, represents the rate at which a function is changing at a specific point. For the function g(x) = -x^2, the derivative can be found using the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).
Applying the power rule to g(x) = -x^2, we get:
g'(x) = -2x
Since the derivative of g(x) is -2x, the instantaneous rate of change for g(x) is dependent on the value of x. In other words, the rate at which g(x) changes depends on the specific input value of x.
The rate of change of -2x means that as x increases or decreases, g(x) changes at a constant rate of -2 times the current value of x. This implies that the instantaneous rate of change is not the same for any different value of x. As x changes, the rate of change changes accordingly.
Therefore, for the function g(x) = -x^2, the instantaneous rate of change cannot be the same for any different value of x.