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Calculus

A. Create a real-world science problem that requires the use of differentiation to solve by doing the following:

1. Provide a description of each of the following within the context of the selected real world application using the appropriate

units:

a. Dependent variable

b. Independent variable

c. Domain

d. Range

2. Explain the specific issue that the created problem is addressing.

3. The created problem should involve taking the first and second derivative of a selected functional relationship f(x), which

includes the following components:

a. Describe how f (x) describes the behavior of f(x) within the context of the real world application.

b. Describe how f(x) describes the behavior of f(x) and f(x) within the context of the real-world application.

4. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.

B. Create a real-world science problem that requires the use of integration to solve by doing the following:

1. Provide a description of each of the following within the context of the selected real world application using the appropriate units:

a. Dependent variable

b. Independent variable

c. Domain

d. Range

2. Explain the specific issue that the created problem is addressing.

3. The created problem should involve calculating the volume of a solid of revolution obtained by revolving a planar curve about a straight line in the plane (e.g. x-axis or y-axis), which includes the following components:

a. Describe how to calculate the volume of the solid of revolution using either the disk method or the washer method.

b. Describe how to calculate the volume of the solid of revolution using the shell method.

4. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.

Note: Demonstrate that you can generate the same answer for volume by using two different integration methods.

C. Compare the methods that were used within both of the problems you have created for parts A and B to explain the

relationship between differentiation and integration.

D. Create a real-world science problem that requires the use of both parts of the Fundamental Theorem of Calculus to solve by doing the following:

1. Provide a description of each of the following within the context of the selected real world application using the appropriate

units:

a. Dependent variable

b. Independent variable

c. Domain

d. Range

2. Explain the specific issue that the created problem is addressing.

3. The created problem should require use of the first part of the Fundamental Theorem of Calculus to solve that includes

the following components:

State the first part of the Fundamental Theorem of Calculus. a. Explain how the first part of the Fundamental Theorem of Calculus was used to solve part of the created problem.

4. The created problem should require use of the second part of the Fundamental Theorem of Calculus to solve that includes the following components:

State the second part of the Fundamental Theorem of Calculus. a. Explain how the second part of the Fundamental Theorem of Calculus was used to solve part of the created problem.

5. Provide a solution to the created problem using both parts of the Fundamental Theorem of Calculus.

6. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.