Prove each identity.

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Prove each identity.

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1. Find the exact global minimum value of on the domain

2. Find the exact global minimum and maximum values of on the domain

3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant , its pressure, , and volume, , satisfy the equation . (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?

4. (a) A hemispherical bowl of radius 10cm contains water to a depth of cm. Find the radius of the surface of the water as a function of . (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?

For this project, your task is to come up with an equation that models the height of the water in Dutch Kills as a function of time.

To do this, you will need to collect lots of data! Because of this, groups may share data. Think about what data you’ll need to collect and what a good plan for collecting it is.

The project is due by the end of the day on Friday, April 25th.

Here are the groups:

**MRS22-1**

- Oliwia K., Jade, Gabriel, Jonathan
- Kamil, Chris, Ignacio, Suraya
- Margaret, Myles, Prova, Rebecca
- Olivia L., Ashley, Lukasz, Andrew
- Auri, Cameron, Judah, Theo
- Saarah, Clofie, Kate, Julia

**MRS22-3**

- Hanaa, Raissa, Dominique
- Larry, Akash, Jim
- Valen, Tajreen, Christian
- Jetsun, Revana, Emily
- Adrian, Salina, Jessie
- Mia, Ava, Samena

You will be evaluated primarily on the *accuracy of your equation* and on your *description of your process*. You also must include the data collected either as a table or a scatter plot or both. An excellent project will include an equation for a function that comes within an hour and within a foot of each data point and an explanation of how you applied knowledge of trigonometric graphs and transformations to fit the function to the data.