1. Let's model the usage cost of a mobile network like MTN or Airtel using a piecewise function, which is common in billing systems where different rates apply based on usage. 2. Suppose the cost function $C(x)$ depends on the number of minutes $x$ used in a month. 3. A typical piecewise function for billing might be: $$ C(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq 50 \\ 0.10x & \text{if } 50 < x \leq 200 \\ 15 + 0.05(x - 200) & \text{if } x > 200 \end{cases} $$ 4. Explanation: - For the first 50 minutes, the service is free. - For minutes between 51 and 200, the cost is 0.10 per minute. - For minutes above 200, a base cost of 15 applies plus 0.05 per additional minute. 5. This piecewise function is continuous and models real-world billing where rates change after certain thresholds. 6. To analyze this function using calculus, we can find the derivative $C'(x)$ to understand the rate of change of cost with respect to usage: $$ C'(x) = \begin{cases} 0 & 0 < x < 50 \\ 0.10 & 50 < x < 200 \\ 0.05 & x > 200 \end{cases} $$ 7. The derivative shows the marginal cost per minute in each interval. 8. This model helps users understand how their cost increases as they use more minutes and can be used by the company to optimize pricing. Final answer: The piecewise function $C(x)$ above models the billing system with its corresponding derivative $C'(x)$ representing the marginal cost.