. What are the most important contributions by Galileo in the history of physics? Why do you think so? How do these contributions affect your life? Your journal entry must….

## . Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 4. Use the following information for Questions 4 and 5. Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT dt equals k times the quantity T minus 38 where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.

SHOW ALL WORK AND CALCUATIONS

**No calculator is permitted.**

**For Questions 1–3, use the differential equation given by , y > 0.**

- Complete the table of values

x |
−1 |
−1 |
−1 |
0 |
0 |
0 |
1 |
1 |
1 |

y |
1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

- On the axes below, sketch a slope field for the given differential equation using the nine points on the table that you found.

- Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 4.

Use the following information for Questions 4 and 5.

Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.

- To the nearest degree, what is the temperature of the lemonade after an
**additional**30 minutes?

- To the nearest minute, how long does it take for the lemonade to cool to 55°F?

**Use the following information for Questions 1–3:**

The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R of time t. The table below gives the rate as measured at various times in an 8-hour time period.

t (hours) |
0 |
2 |
3 |
7 |
8 |

R(t) (gallons per hour) |
1.95 | 2.5 | 2.8 | 4 | 4.26 |

- Use a trapezoidal sum with the four sub-intervals indicated by the data in the table to estimate . Using correct units, explain the meaning of your answer in terms of water flow. Give 3 decimal places in your answer.

- Is there some time t, 0 < t < 8, such that R′(t) = 0? Justify your answer.

- The rate of water flow R(t) can be estimated by W(t) = ln(t2 + 7). Use W(t) to approximate the average rate of water flow during the 8-hour time period. Indicate units of measure.

Use the following information for Questions 4 and 5.

f is a continuous function with a domain [−3, 9] such that and let .

- On what interval is g increasing? Justify your answer.

- For 0 ≤ x ≤ 6, express g(x) in terms of x. Do not include +C in your final answer.