Chapter 2 Conceptual Project:

Some years ago, it was common for long-distance phone companies to charge their customers in one-minute increments. In other words, the company charges a flat fee for the first minute of a call and another fee for each additional minute or any fraction thereof (see Exercise 82 in Section 2.5). In this project, we will explore in detail a function that gives the cost of a telephone call under the above conditions.

1. Suppose a long-distance call costs $75$ cents for the first minute plus $50$ cents for each additional minute or any fraction thereof. In a coordinate system where the horizontal axis represents time t and the vertical axis price p, draw the graph of the function $p=C\left(t\right)$ that gives the cost (in dollars) of a telephone call lasting t minutes, $0<t\le 5$.

2. Does $\underset{t\to \mathrm{1.5}}{\lim }C\left(t\right)$ exist? If so, find its value.

3. Does $\underset{t\to 3}{\lim }C\left(t\right)$ exist? Explain.

4. Write a short paragraph on the continuity of this function. Classify all discontinuities; mention one-sided limits and left or right continuity where applicable.

5. In layman's terms, interpret $\underset{t\to \mathrm{2.5}}{\lim }C\left(t\right)$.

6. In layman's terms, interpret $\underset{t\to 3^{-}}{\lim }C\left(t\right)$.

7. In layman's terms, interpret $\underset{t\to 3^{+}}{\lim }C\left(t\right)$.

8. If possible, find $C^{\prime }\left(3.5\right)$.

9. If possible, find $C^{\prime }\left(4\right)$.

10. Find and graph another real‑life function whose behavior is similar to that of $C\left(t\right)$. Label the axes appropriately and provide a brief description of your function.