Question

$$ \begin{array}{l}{\text { Analyzing Motion Data Priya's distance } D \text { in meters from a }} \\ {\text { motion detector is given by the data in Table 4.1. }}\end{array} $$ $$ \begin{array}{llll}{t(\text { sec) }} & {D(\mathrm{m})} & {t(\mathrm{sec})} & {D(\mathrm{m})} \\ \hline 0.0 & {3.36} & {4.5} & {3.59} \\ {0.5} & {2.61} & {5.0} & {4.15} \\ {1.0} & {1.86} & {5.5} & {3.99} \\ {1.5} & {1.27} & {6.0} & {3.37}\end{array} $$ $$ \begin{array}{llll}{2.0} & {0.91} & {6.5} & {2.58} \\ {2.5} & {1.14} & {7.0} & {1.93} \\ {3.0} & {1.69} & {7.5} & {1.25} \\ {3.5} & {2.37} & {8.0} & {0.67} \\ {4.0} & {3.01}\end{array} $$ $$ \begin{array}{l}{\text { (a) Estimate when Priya is moving toward the motion detector; }} \\ {\text { away from the motion detector. }} \\ {\text { (b) Writing to Learn Give an interpretation of any local }} \\ {\text { extreme values in terms of this problem situation. }}\end{array} $$ $$ \begin{array}{l}{\text { (c) Find a cubic regression equation } D=f(t) \text { for the data in }} \\ {\text { Table } 4.1 \text { and superimpose its graph on a scatter plot of the data. }} \\ {\text { (d) Use the model in (c) for } f \text { to find a formula for } f^{\prime} . \text { Use this }} \\ {\text { formula to estimate the answers to (a). }}\end{array} $$

Short answer

(a) Priya moves towards the motion detector from 0 to 2 seconds and 4 to 8 seconds. She moves away from it between 2 to 4 seconds. (b) Local extreme values represent when Priya changes her direction. (c & d) The precise cubic regression equation and its derivative will depend on the specific statistical software used, but they allow more precise confirmation of the answers to (a) and (b).

Step-by-step solution

Identify when Priya is moving toward or away from the motion detector

By inspecting the data, when D decreases, Priya is moving towards the motion detector. When D increases, she is moving away. For t=0 to 2 seconds, D is decreasing, so she is moving towards it. Then, for t=2 to 4 seconds, D is increasing, so she is moving away, and lastly, for t=4 to 8 seconds, D is decreasing so she is moving back towards it.

Interpret any local extreme values

The local extreme values are the minimum and maximum points. In our data, the minimum is at t=2 seconds, and the maximum is at t=4 seconds. These represent when Priya changes direction.

Find a cubic regression equation

Utilize statistical software or a graphing calculator to generate a cubic regression model from the given data set. This function, f(t), describes Priya's distance from the motion detector over time.

Calculate the derivative and estimate answers to (a)

Find the derivative of the model found in step 3, f'(t), to determine the rate of change at any given time. Use this to confirm the answers in step 1; when f'(t) is negative, D is decreasing, and when f'(t) is positive, D is increasing.