Final answer:
To determine the instantaneous rate of change of h(x) at x=2, one must calculate the derivative h'(x) using the chain rule, and then evaluate it at x=2. Without the explicit function f(x), the exact rate cannot be determined.
Step-by-step explanation:
The question is asking for the instantaneous rate of change of the function h(x) = f(x)^3 at x=2. To find this, we need to calculate the derivative of h(x), which is h'(x), and then evaluate it at x=2. This is essentially calculating the slope of the tangent to the graph of h(x) at that particular point.
Assuming that we know the function f(x) and its derivative f'(x), we use the chain rule to find h'(x). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Therefore, h'(x) will be 3f(x)^2 × f'(x). Once we have the expression for h'(x), we substitute x=2 into this expression to find the specific instantaneous rate of change of h(x) at that point.
An example of this calculation, using hypothetical values for f(2) and f'(2), would look like this:
- Let's say f(2) = 4 and f'(2) = 6.
- Using the chain rule, h'(x) = 3f(x)^2 × f'(x).
- Then h'(2) = 3 × 4^2 × 6, which simplifies to 3 × 16 × 6 = 288.
Accordingly, the instantaneous rate of change of h(x) at x=2 would be 288. However, without the specific function f(x), we cannot find the exact rate for h(x).