The problem asks to estimate the instantaneous rate of change of the function $f(x) = x^3 + 4x^2 - 11x - 30$ at $x = 7$. The instantaneous rate of change is given by the derivative of the function at that point.

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 $x = 7$. The instantaneous rate of change is given by the derivative of the function at that point.

2. Solution Steps

We need to find the derivative of the function $f(x)$ and then evaluate it at $x = 7$.

The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

Applying this rule, we find the derivative of $f(x) = x^3 + 4x^2 - 11x - 30$:

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

$f'(x) = 3x^2 + 4(2x) - 11 - 0$

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

Now we 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

1

9

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