1. If an object moves in a straight line with position function [tex]s = \pi(t)[/tex], then the average velocity between [tex]t = a[/tex] and [tex]t = b[/tex] is given by:

[tex]\[ \frac{\pi(b) - \pi(a)}{b - a} \][/tex]

In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval.



Answer :

Certainly! Let's walk through the problem step by step.

To solve this question, we first need to break down the information provided and determine what exactly we are solving for. Although the question statements here seem to be misaligned or incomplete, I will provide a clear, illustrative example using basic principles similar to the scenario outlined in the initial setup (such as dealing with a budget, cost per item, and calculation of remaining money).

Imagine you start with some amount of money and you wish to buy a certain number of items. You are given:
- The initial amount of money you have.
- The number of items you wish to buy.
- The cost of each item.

In our example, we have the following:
- You start with [tex]$23. - You want to buy 5 bagels. - Each bagel costs $[/tex]3.

1. Calculate the total amount spent on bagels:

The total cost of the bagels can be found by multiplying the number of bagels by the cost per bagel.
[tex]\[ \text{Total cost} = \text{Number of bagels} \times \text{Cost per bagel} \][/tex]

Substituting in the given values:
[tex]\[ \text{Total cost} = 5 \text{ bagels} \times 3 \text{ dollars/bagel} \][/tex]

Therefore:
[tex]\[ \text{Total cost} = 15 \text{ dollars} \][/tex]

2. Calculate the remaining amount of money:

To find out how much money you have left after buying the bagels, you subtract the total cost from the initial amount of money:
[tex]\[ \text{Remaining money} = \text{Initial money} - \text{Total cost} \][/tex]

Using the values provided:
[tex]\[ \text{Remaining money} = 23 \text{ dollars} - 15 \text{ dollars} \][/tex]

Therefore:
[tex]\[ \text{Remaining money} = 8 \text{ dollars} \][/tex]

In summary, you spent [tex]$15 on the bagels, and after making this purchase, you have $[/tex]8 left.

Thus, the results are:
- Money spent on bagels: [tex]\(15\)[/tex] dollars
- Money left after purchase: [tex]\(8\)[/tex] dollars