Given Part: Clearly list all known variables, constants, or conditions provided in the problem.
To Find Part: Clearly state what needs to be determined or calculated.
To estimate the speed when
t
=
1
,
we need to estimate
vavg
=
7
.
2
8
−
1
.
8
6
h
when h approaches
?
.
This yields the following.
vavg
=
?
m
/
s
Formulas Used: Write out all relevant formulas that will be used to solve the problem. Ensure formulas are correct and applicable.
Concept Used: Explain the main concept or principle that applies to the problem, including any theories or laws involved. The description should be clear and concise, providing a good understanding of the concept.
Complete Calculation: Show every step of the calculation without skipping any part. Each step should include the following:
Substitution of values into the formulas
Arithmetic or algebraic operations performed
Simplification or rearrangement of equations where necessary
Explanation: After every step, include a brief explanation of what was done and why it was done. This ensures that each step is clear and logically connected to the next.
Final Answer: Provide a clear and crisp final answer, emphasizing the key result or outcome of the calculation.
Additional Requirements:
Avoid the use of pronouns and ing forms in the steps.
Structure the answer in a way that is easy to understand and follow.
Ensure accuracy in every step, calculation, and explanation.

### Problem Breakdown

#### Given Part:
- **Time (t):** \( t = 1 \) second
- **Average velocity (\(v_{\text{avg}}\)):** \( v_{\text{avg}} = \frac{7.28 - 1.86}{h} \) as \( h \) approaches a small value.

#### To Find Part:
- Determine the limit of the average velocity \( v_{\text{avg}} \) as \( h \) approaches 0.
- Final average velocity \( v_{\text{avg}} = ? \) m/s

### Formulas Used:
- **Average velocity formula:** 
\[
v_{\text{avg}} = \frac{\Delta s}{\Delta t}
\]
where \( \Delta s \) is the change in position and \( \Delta t \) is the change in time.

- **Limit definition of derivative:**
\[
v = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}
\]
This formula calculates the instantaneous velocity.

### Concept Used:
The problem involves estimating the instantaneous velocity of an object at \( t = 1 \) second. The average velocity \( v_{\text{avg}} \) as \( h \) approaches 0 will approximate the instantaneous velocity at that specific moment. This concept is based on the derivative from calculus, which provides the rate of change of position with respect to time, or velocity.

### Complete Calculation:

1. **Substitute the given values:**
   \[
   v_{\text{avg}} = \frac{7.28 - 1.86}{h}
   \]
   This expression calculates the average velocity over the interval \( [t, t+h] \) where \( t = 1 \) and \( t+h \) is close to \( t \).

   **Explanation:** Substituting the provided values into the formula for average velocity.

2. **Calculate the difference in the position:**
   \[
   \Delta s = 7.28 - 1.86 = 5.42 \text{ meters}
   \]
   **Explanation:** The difference between the positions at \( t = 1+h \) and \( t = 1 \) is calculated as 5.42 meters.

3. **Express the average velocity:**
   \[
   v_{\text{avg}} = \frac{5.42}{h} \text{ m/s}
   \]
   **Explanation:** The average velocity is calculated as the ratio of the position difference over time difference.

4. **Take the limit as \( h \) approaches 0:**
   \[
   v = \lim_{h \to 0} \frac{5.42}{h}
   \]
   Since \( h \) is approaching 0, the formula simplifies to:
   \[
   v = \frac{5.42}{0} = \text{undefined}
   \]
   **Explanation:** Direct substitution into the limit gives an undefined result, which suggests the need for further information on how the position function behaves as \( h \) approaches 0.

### Final Answer:
The average velocity formula as \( h \) approaches 0 becomes undefined with the given information. The problem likely requires additional context, such as the position function or a different approach to evaluate the limit.

### Additional Requirements:
Ensure further details about the position function or context are provided to resolve the issue with the limit calculation.

---

### Follow-up Questions:
1. What is the exact position function \( s(t) \) used in this problem?
2. What is the correct interpretation of the limit in this context?
3. How can the derivative be directly calculated from the given data?
4. Is there any specific method for evaluating the undefined limit?
5. How can the instantaneous velocity be interpreted physically in this situation?

### Tip:
When dealing with limits, always check if the context requires a different mathematical approach, such as using L'Hopital's rule for indeterminate forms.

Explore how to estimate instantaneous velocity using limits in calculus. This problem involves determining the average velocity as h approaches 0, covering key concepts in calculus such as derivatives and limits. Suitable for advanced high school students and calculus enthusiasts.

Math Problem Statement

Given Part: Clearly list all known variables, constants, or conditions provided in the problem. To Find Part: Clearly state what needs to be determined or calculated. To estimate the speed when t

1 , we need to estimate vavg

7 . 2 8 − 1 . 8 6 h when h approaches ? . This yields the following. vavg

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