Let's solve the problem step by step. We are given the cost function $C(x) = 7000 + 5x + 0.5x^2$, where $C(x)$ represents the cost of producing $x$ units of a commodity.

Part (a) - Average Rate of Change

The average rate of change of $C$ with respect to $x$ between two points $x_1$ and $x_2$ is given by the formula:

$\text{Average rate of change} = \frac{C(x_2) - C(x_1)}{x_2 - x_1}$

(i) From $x = 100$ to $x = 105$

Let's calculate the values of $C(100)$ and $C(105)$.

1. $C(100) = 7000 + 5(100) + 0.5(100^2) = 7000 + 500 + 0.5(10000) = 7000 + 500 + 5000 = 12500$
2. $C(105) = 7000 + 5(105) + 0.5(105^2) = 7000 + 525 + 0.5(11025) = 7000 + 525 + 5512.5 = 13037.5$

Now, apply the average rate of change formula:

$\text{Average rate of change} = \frac{13037.5 - 12500}{105 - 100} = \frac{537.5}{5} = 107.5$

So, the average rate of change from $x = 100$ to $x = 105$ is 107.5.

(ii) From $x = 100$ to $x = 101$

Similarly, let's calculate the values of $C(101)$.

1. $C(101) = 7000 + 5(101) + 0.5(101^2) = 7000 + 505 + 0.5(10201) = 7000 + 505 + 5100.5 = 12605.5$

Now, apply the average rate of change formula:

$\text{Average rate of change} = \frac{12605.5 - 12500}{101 - 100} = \frac{105.5}{1} = 105.5$

So, the average rate of change from $x = 100$ to $x = 101$ is 105.5.

Part (b) - Instantaneous Rate of Change (Marginal Cost)

The instantaneous rate of change of $C$ at $x = 100$ is found by taking the derivative of $C(x)$ and evaluating it at $x = 100$. The derivative of $C(x)$ is:

$C'(x) = \frac{d}{dx} \left(7000 + 5x + 0.5x^2 \right) = 0 + 5 + x = 5 + x$

Now, evaluate $C'(100)$:

$C'(100) = 5 + 100 = 105$

So, the instantaneous rate of change (marginal cost) at $x = 100$ is 105.

Would you like any additional details or clarification on any of the steps? Here are five related questions to expand on this problem:

1. How does the marginal cost function change for different values of $x$?
2. What is the economic interpretation of the average rate of change in this context?
3. How would the cost function change if the coefficient of $x^2$ were different?
4. What happens to the marginal cost as $x$ increases?
5. How would you interpret the marginal cost if it were negative?

Tip: The marginal cost function $C'(x)$ gives a direct estimate of the cost of producing one more unit of the commodity at a given production level $x$.