Problem: The region in the first quadrant that is bounded above by the curve , on the left by the line , and below by the line is revolved about the y-axis to generate a solid. Find the volume of the solid by

1. the washer method.
2. the shell method.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

II) Integrals and Transcendental Functions

Problem: Suppose that a cup of soup cooled from to after 10 minutes in a room with a temperature of Use Newton’s Law of Cooling to answer the following questions.

1. How much longer would it take the cup of soup to cool to ?
2. Instead of being left to stand in the room, the cup of soup is put in a freezer with a temperature of How long will it take the soup to cool from to ?

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem : Sociologists sometimes use the phrase social diffusion to describe the way information spreads through a population. In a sufficiently large population, the number of people x who have the information is treated as a differentiable function of time t , and the rate of diffusion, , is assumed to be proportional to the number of people who have the information times the number of people who do not. This leads to the equation

,

where N is the number of people in the population. Suppose t is in days, , and two people start a rumor at time in a population of N =1000 people.

1. Find a as a function of t .
2. When will half the population have heard the rumor?

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem : Heat capacity is the amount of heat required to raise the temperature of a given mass of gas with constant volume by , measured in units of cal. / deg-mol. (calories per degree gram molecular weight). The heat capacity of oxygen depends on its temperature T and satisfies the formula

a) Find the average value of for and the temperature at which it is attained.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem : Does any non-degenerate conic section have all of the following properties?

1. It is symmetric with respect to the origin.
2. It passes through the point (1,0).
3. It is tangent to the line y=1 at the point (-2,1).

Give reasons for your answers.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem: A satellite is in an orbit that passes over the North and South poles of the earth. When it is over the South Pole, it is at the highest point of its orbit at 1,000 miles above the earth’s surface. Above the North Pole, it is at the lowest point of its orbit at 300 miles above the earth’s surface.

1. Assuming that the orbit is an ellipse with one focus at the center of the earth, find its eccentricity. (Take the diameter of the earth to be 8,000 miles.)
2. Using the north-south axis of the earth as the x-axis and the center of the earth as origin, find a polar equation for the orbit.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem: A ball is dropped from a height of 4 meters. Each time it strikes the pavement after falling from a height of h meters it rebounds to a height 0.75 h meters. Find the total distance the ball travels up and down.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

Problem: The series Converges to for all x .

1. Find a series for . How is it related to the series for ? Explain your answer.
2. Find a series for How is it related to the series for ? Explain your answer.

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

1. Explain the problem in your own words.
2. What mathematical concepts learned in this module apply to this problem?
3. Explain the steps you must take to solve this problem.
4. What is the most difficult aspect of solving this problem?
5. Explain exactly what the answer means from a mathematical perspective.

finite sum.

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