Instantaneous Velocity on the Moon An astronaut throws a...

Instantaneous Velocity on the Moon An astronaut throws a ball down into a crater on the moon. The height $s$ (in feet) of the ball from the bottom of the crater after $t$ seconds is given in the following table: $$ \begin{array}{|cc|} \hline \begin{array}{l} \text { Time, } \boldsymbol{t} \\ \text { (in seconds) } \end{array} & \begin{array}{l} \text { Distance, } \boldsymbol{s} \\ \text { (in feet) } \end{array} \\ \hline 0 & 1000 \\ 1 & 987 \\ 2 & 969 \\ 3 & 945 \\ 4 & 917 \\ 5 & 883 \\ 6 & 843 \\ 7 & 800 \\ 8 & 749 \\ \hline \end{array} $$ (a) Find the average velocity from $t=1$ to $t=4$ seconds. (b) Find the average velocity from $t=1$ to $t=3$ seconds. (c) Find the average velocity from $t=1$ to $t=2$ seconds. (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous velocity at $t=1$ second.

Instantaneous Velocity on the Moon An astronaut throws a ball down into a crater on the moon. The height $s$ (in feet) of the ball from the bottom of the crater after $t$ seconds is given in the following table:
$$
\begin{array}{|cc|}
\hline \begin{array}{l}
\text { Time, } \boldsymbol{t} \\
\text { (in seconds) }
\end{array} & \begin{array}{l}
\text { Distance, } \boldsymbol{s} \\
\text { (in feet) }
\end{array} \\
\hline 0 & 1000 \\
1 & 987 \\
2 & 969 \\
3 & 945 \\
4 & 917 \\
5 & 883 \\
6 & 843 \\
7 & 800 \\
8 & 749 \\
\hline
\end{array}
$$
(a) Find the average velocity from $t=1$ to $t=4$ seconds.
(b) Find the average velocity from $t=1$ to $t=3$ seconds.
(c) Find the average velocity from $t=1$ to $t=2$ seconds.
(d) Using a graphing utility, find the quadratic function of best fit.
(e) Using the function found in part (d), determine the instantaneous velocity at $t=1$ second.

Key Concepts

Average Velocity

Average velocity is calculated as the change in position divided by the change in time over a given interval. This concept reflects how much displacement occurs per unit of time and is foundational in understanding motion. It is often used as an approximation before transitioning to the concept of instantaneous velocity.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. In calculus, it is defined as the derivative of the position function with respect to time. This concept refines the idea of average velocity by considering the limit of the average velocity over an increasingly small time interval, providing a precise measure of speed and direction at a single point in time.

Quadratic Regression (Curve Fitting)

Quadratic regression is a statistical method used to model the relationship between a dependent variable and an independent variable by fitting a quadratic function to data. It is particularly useful when the trend in the data is better represented by a parabolic curve rather than a straight line. This process minimizes the differences between the observed data points and the values predicted by the quadratic model, offering a best-fit representation of the underlying pattern.

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that represents the rate at which the function's value changes with respect to its input. It is essential for calculating instantaneous rates of change, such as velocity. The derivative provides a mathematical means to understand and predict the behavior of dynamic systems by analyzing how small changes in one quantity affect another.

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