MATH 1770 - Analytic Geometry & Calculus 2

Credit Hours: 4.00

Objectives: Students will:

1. Calculate the volumes of solids of revolution using circular discs and washers.
2. Calculate the volumes of solids of revolution using the cylindrical shells method.

Outcome 2: Upon completion of this course, the student will be able to determine the arc length and the surface area of solids of revolution.

Objectives: Students will:

1. Determine the arc length by using the arc length formula.
2. Use formulas to find the surface area of solids at revolution.

Outcome 3: Upon completion of this course, the student will be able to evaluate integrals using trigonometric substitution, integration by parts, partial fractions, and by the integral tables.

Objectives: Students will:

1. Evaluate integrals using trigonometric substitution.
2. Use the trigonometric formulas from memory.
3. Evaluate integrals using integration by parts and the tabular method.
4. Evaluate integrals by utilizing partial fractions.
5. Evaluate the integrals for powers of trigonometric functions.

Outcome 4: Upon completion of this course, the student will be able to evaluate improper integrals.

Objectives: Students will:

1. Recognize improper integrals.
2. Recognize if an improper integral is convergent or divergent.
3. Evaluate an improper integrals with a discontinuous integrand, as well as one with an unbounded domain.
4. Use the direct comparison test.

Outcome 5: Upon completion of this course, the student will be able to determine the convergence or divergence of sequences.

Objectives: Students will:

1. Recognize infinite sequences.
2. Use the squeeze theorem for convergent or divergent sequences.
3. Recognize bounded and unbounded sequences.

Outcome 6: Upon completion of this course, the student will be able to determine convergence or divergence of infinite series.

Objectives: Students will:

1. Recognize infinite series.
2. Recognize geometric series and determine convergence or divergence.
3. Utilize the test for divergence.
4. Recognize the harmonic series.
5. Use the integral test and comparison tests for convergence or divergence of infinite series.
6. Use the p-series to determine convergence or divergence.
7. Use the alternating series test (Leibniz test) to determine convergence or divergence.
8. Use the ratio and root tests to determine the absolute convergence of infinite series.
9. Use the properties of series.

Outcome 7: Upon completion of this course, the student will have a working knowledge of power series.

Objectives: Students will:

1. Recognize if the power series centered at the origin or at another number.
2. Find the radius and the interval of convergence of a power series.
3. Differentiate and integrate power series.

Outcome 8: Upon completion of this course, the student will be able to represent functions by using Taylor and Maclaurin series.

Objectives: Students will:

1. Represent functions like sin x, cos x, e^x, ln x, and arctan x by using Maclaurin and Taylor series.
2. Find the Taylor polynomial of certain functions of degree < 5.

Outcome 9: Upon completion of this course, the student will be able to solve problems, use algorithms, and comprehension of concepts related to parametric equations and calculus with parametric and polar curves. 

Objectives: Student will:

1. Express the equation of a conic in parametric form.
2. Eliminate the parameter from parametric equations and graph the equation.
3. Find slope and concavity of a curve described parametrically without eliminating the parameter.
4. Find the slope of a polar fuction without converting to rectangular coordinates.
5. Find the area enclosed by and length of a polar curve.

• Communication: The graduate can communicate effectively for the intended purpose and audience.
• Critical Thinking: The graduate can make informed decisions after analyzing information or evidence related to the issue.
• Global Literacy: The graduate can analyze human behavior or experiences through cultural, social, political, or economic perspectives.
• Information Literacy: The graduate can responsibly use information gathered from a variety of formats in order to complete a task.
• Quantitative Reasoning: The graduate can apply quantitative methods or evidence to solve problems or make judgments.
• Scientific Literacy: The graduate can produce or interpret scientific information presented in a variety of formats.

1. Review
   1. Rules for derivatives, constant and the power rule, product and quotient rule, trigonometric functions, logarithmic and exponential base e, composite functions and the chain rule, implicit differentiation
   2. Indefinite integrals, required department formulas, the fundamental theorem of calculus, initial value problems, area between curves
2. Solids of revolution
   1. Disks, washers, and shell methods
   2. Arc length and surface area
3. Methods of integration
   1. Basic integration formulas, integration by parts and partial fractions
   2. Algebraic and trigonometric substitutions
   3. Powers of trigonometric functions
4. Improper integrals
5. Limits and properties of sequences
6. Series
   1. Convergence of series
   2. The nth term test for divergence
   3. The integral test
   4. The direct and comparison tests
   5. The Leibniz alternating series test
   6. The ratio and root tests
   7. Absolute and conditional convergence
7. Power series
   1. Center, radius, and interval of convergence
   2. Taylor and Maclaurin series
   3. Differentiation and integration of power series
   4. The binomial series (optional)
8. Calculus with Parametric Equations and Polar Coordinates

1. Express the equation of a conic in parametric form
2. Eliminate the parameter from parametric equation and graph the equation
3. Find slope and concavity of a curve described parametrically without eliminating the parameter
4. Areas and lengths in polar coordinates