Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Im seeking for help on the bullet point that starts with 'What do you notice about the t values for...', I would also appreciate help on the questions below this bullet point. Thank you!
Transcribed Image Text:You are studying the impacts of rising sea levels on an estuary and are modeling how the salinity of a particular area changes with the tidal cycle. The salinity is also impacted seasonally by snowmelt increasing river flows, so measurements are often taken in early autumn for this particular area. The mixed-tide cycle on this part of the coast has a period of approximately 25 hours, giving the salinity fluctuation of the estuary a similar cycle. Twenty years ago, the early autumn salinity was modeled by the function \( s(t) = 12 \sin \left( \frac{3\pi}{25} t \right) \cos \left( \frac{\pi}{25} t \right) + 15 \), where \( t \) is in hours and \( s(t) \) is the salinity in parts per million (ppm). But you have determined that the model \( S(t) = 14 \sin \left( \frac{3\pi}{25} t \right) \cos \left( \frac{\pi}{25} t \right) + 17 \) more closely fits the current data.
1. **Graph both \( s(t) \) and \( S(t) \) using technology. What do you observe about the two functions? How are they the same? How are they different?**
2. **Find both \( s'(t) \) and \( S'(t) \), describing what differentiation rules you use for each, and showing your process.**
3. **Use technology to determine the values of \( t \) for which \( s(t) \) and \( S(t) \) have horizontal tangents. (Focus on the first period of the graphs, so \( t < 25 \).**
- What do you notice about the \( t \) values for which the two functions have horizontal tangents?
- Use these \( t \) values, and the graphs of the two salinity functions, to determine the highest and lowest salinity for the estuary using the historical model and the current model.
- What do you notice? How does this relate to what you are studying?
4. **Graph both derivatives using technology and use these to determine the values of \( t \) for which \( s'(t) \) and \( S'(t) \) have horizontal tangents. (Focus on the first period of the
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