Recently, a car rental firm charged 36 per day or portion...

Recently, a car rental firm charged $\$ 36$ per day or portion of a day to rent a car for a period of 1 to 5 days. Days 6 and 7 were then free, while the charge for days 8 through 12 was again $\$ 36$ per day. Let $A(t)$ represent the average cost to rent the car for $t$ days, where $0< t \leq 12$ . Find the average cost of a rental for the following number of days. a. 4 b. 5 c. 6 d. 7 e. 8 f. Find $$\lim _{x \rightarrow 5^{-}} A(t)$$ g. Find $$\lim _{x \rightarrow 5^{+}} A(t)$$ h. Where is A discontinuous on the given interval?

Recently, a car rental firm charged $\$ 36$ per day or portion of a day to rent a car for a period of 1 to 5 days. Days 6 and 7 were then free, while the charge for days 8 through 12 was again $\$ 36$ per day. Let $A(t)$ represent the average cost to rent the car for $t$ days, where $0< t \leq 12$ . Find the average cost of a rental for the following number of days.
a. 4
b. 5
c. 6
d. 7
e. 8
f. Find $$\lim _{x \rightarrow 5^{-}} A(t)$$
g. Find $$\lim _{x \rightarrow 5^{+}} A(t)$$
h. Where is A discontinuous on the given interval?

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Key Concepts

Discontinuity

Discontinuity refers to a break or gap in the graph of a function, where the function may jump or not have a defined value. In the context of pricing models or piecewise functions, discontinuities occur when the function's value abruptly changes at certain points or when the left-hand and right-hand limits at a point do not agree.

Average Cost Calculation

Average cost calculation involves determining the total cost divided by the number of units (such as days) over which the cost is spread. This concept is used to understand how the cost per unit changes when different pricing bands or free periods are applied in the model.

Piecewise Function

A piecewise function is defined by different expressions or rules on different intervals of the domain. This concept is crucial when a system, such as a rental pricing model, charges differently over various ranges, necessitating a function that has distinct formulas for each segment of the input values.

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a specific point from one side—either from the left or the right. This concept is important when analyzing potential discontinuities, especially in piecewise functions, where the function's rules change at boundary points.

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