The problem asks to estimate the instantaneous rate of change of the function $f(x) = x^3 + 4x^2 - 11x - 30$ at $x = 7$. This is equivalent to finding the derivative of the function at $x = 7$, which is $f'(7)$.

1. Problem Description

The problem asks to estimate the instantaneous rate of change of the function $f(x) = x^3 + 4x^2 - 11x - 30$ at $x = 7$. This is equivalent to finding the derivative of the function at $x = 7$, which is $f'(7)$.

2. Solution Steps

First, find the derivative of the function $f(x)$ with respect to $x$:

$f'(x) = \frac{d}{dx} (x^3 + 4x^2 - 11x - 30)$

Apply the power rule to each term:

$\frac{d}{dx}(x^n) = nx^{n-1}$

Therefore,

$f'(x) = 3x^2 + 8x - 11$

Now, evaluate the derivative at $x = 7$:

$f'(7) = 3(7)^2 + 8(7) - 11$

$f'(7) = 3(49) + 56 - 11$

$f'(7) = 147 + 56 - 11$

$f'(7) = 203 - 11$

$f'(7) = 192$

3. Final Answer

The instantaneous rate of change of the function $f(x)$ at $x=7$ is 192.