APPLY IT Car Rental A car rental firm normally charges 42 per day or portion of a day to rent a

step 1 So we're looking at section 3 .2 problem number 38. They deal here with a rental car company. Th

APPLY IT Car Rental A car rental firm normally charges 42...

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions depending on the interval of the input value. In problems like this one, the cost function changes based on the number of days rented because promotional offers introduce different rates or free days past certain thresholds. Understanding how to break the overall function into distinct segments is critical for analyzing its behavior and for performing calculations such as limits and discontinuities.

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specific value either from the left (approaching from smaller values) or from the right (approaching from larger values). In contexts where a function is piecewise-defined, these limits are often different at the points where the function’s expression changes, providing useful information about the function's continuity and its behavior at those transition points.

Function Discontinuities

Discontinuities occur when a function is not continuous at a certain point, meaning there is a 'jump' or 'break' in the graph. In piecewise functions with distinct formulations over different intervals, discontinuities often arise at the boundaries where the definition changes. Analyzing these discontinuities helps in understanding the overall behavior of the cost function and determining if there are errors or abrupt changes in the modeled process.

Average Cost Analysis

The concept of an average cost function is used to determine the cost per unit over a given period, taking into account any adjustments like free days or discounts. In practical applications, it involves dividing the total cost by the number of days, which can lead to computations that highlight how promotional offers effectively reduce the cost on average. This analysis is crucial for both consumers and businesses to understand the financial impact of such pricing strategies.

Transcript

00:01 So we're looking at section 3 .2 problem number 38.

00:05 They deal here with a rental car company.

00:08 This rental car company charges $42 a day or every fraction of a day to rent a car.

00:15 So $42 per day.

00:18 And they have a promotion that every fourth day, so on every fourth day, you get a free day.

00:28 So every fourth day of the rental, you get one free day.

00:39 So let's look at what the cost function looks like.

00:43 So it looks like if i were to look at the cost of renting this car, so what is this going to cost me? so it looks like it's going to cost me 42x on days one to three.

00:58 Okay, so i pay $42 for a day.

01:02 And then what happens? when you get to day four, instead of getting four times that charge, you only get three times that charge.

01:08 Okay, so you're getting one.

01:09 Less.

01:11 So when you rent for four days, you pay for three.

01:13 Rent for five days, you pay for four.

01:16 So what happens there is 42 times x minus one, and that's in effect on days four through seven.

01:26 And then when you get up to day eight on this particular car rental, so eight to 11, you're going to get two free days.

01:34 So you're going to pay $42 times the number of days minus two.

01:39 Okay.

01:40 So what would the average cost be okay so the average cost so let's look at that they want us to find the average cost this is the actual total cost find the average cost for three days okay so we'll let let let a of um we'll let a of t that is the average cost over t days okay so what is a of three so a of three is going to be what is the total cost so it's it's $42 a day for three days divided by three.

02:16 So that average cost is $42.

02:19 So that is your average cost at three days.

02:22 Okay, so at four days, what are you paying for four days? you're going to be paying at four days, you're paying $42 times three.

02:32 And so that is going to be $126.

02:35 So this is $126 divided by four.

02:40 So 31 .5.

02:43 So over four days.

02:45 Days, your average cost per day is 31 .5.

02:50 Then they ask us for the average cost for five days.

02:54 So the average cost for five days is going to be what, 42 times four.

03:01 So that's 42 times four, 168 divided by five, is 3360.

03:19 And then the average costs for six days.