MAT 231HC - Calculus II: Honors

4 Credits, 4 Contact Hours
4 lecture periods 0 lab periods
Continuation of MAT 220 . Includes techniques and applications of integration, numerical integration, improper integrals, sequences, infinite series, polar coordinates, parametric equations, and other related topics. Also includes additional Honors  content.

Prerequisite(s): MAT 220  
Information: Must qualify for Honors program. Instructor or advisor/counselor approval may be required before registering for this course. Honors Content may include: Intensive theoretical-based and/or application- based projects using highest standards and best practices for the discipline. Also may include team problem solving projects in formats appropriate for the discipline with results presented in class or to a wider audience.
Gen-Ed: Meets AGEC - MATH; Meets CTE - M&S.

1. Evaluate indefinite and definite integrals using exact and approximation techniques.
2. Use integrals in various applications.
3. Determine convergence of infinite series.

1. Use definite integrals to calculate areas between curves, volumes of solids, work, centroids of lamina or centers of mass and some of the following applications: arc length, hydrostatic force, average value of a function, economics, or probability.
2. Calculate definite and indefinite integrals of various combinations of algebraic and transcendental functions, including powers and products of trigonometric functions, using integration by parts, U-substitution, partial fractions, and some of the following: trigonometric substitutions, rationalizing substitutions, and integral tables.
3. Estimate definite integrals using techniques such as left hand, right hand, trapezoid, midpoint, and Simpson’s rules.
4. Determine convergence or divergence of various forms of improper integrals, and in the cause of convergence, calculate the exact value or estimate the value of the integral as appropriate.
5. Determine if an infinite sequence is convergent or divergent.
6. Determine convergence (or divergence) of infinite series using the divergence test, integral test, comparison test, limit comparison test, alternating series, test, and ratio test.
7. Estimate the error associate with a partial sum approximation of a convergent infinite series using tests such as integral test, alternating series test, and/or comparison test.
8. Determine if a series converges absolutely or conditionally.
9. Determine radii of convergence and intervals of convergence of power series.
10. Find the power series representation for a given function by integrating or differentiating existing power series.
11. Determine Taylor and Maclaurin series using the definition.
12. Apply integration techniques to solve separable differential equations.
13. Determine slopes and arc length of two dimensional curves modeled with parametric equations.

Optional Objectives:

14. Use rationalizing substitutions to calculate indefinite integrals and exact values of definite integrals.

15. Generate the equation of a conic section given a graph, or sketch a graph given an equation.

16. Calculate error bounds for the numerical integration techniques.

17. Generate the binomial series expansion for appropriate functions.

18. Apply integration techniques to solve separable differential equations.

19. Use slope fields and/or Euler’s Method to estimate solutions to differential equations.

1. Techniques of Integration
   1. Substitution
   2. Integration by parts
   3. Products of powers of trigonometric functions
   4. Partial fractions
   5. Trigonometric substitution
   6. Rationalizing substitutions (optional)
   7. Numerical integration
      1. Trapezoid rule
      2. Midpoint rule
      3. Simpson’s rule
      4. Error bounds (optional)
   8. Improper integrals
2. Applications of the Integral
   1. Area between curves
   2. Volumes of solids
   3. Work
   4. Centroids or centers of mass
   5. Arc length
   6. At least 1 of the following:
      1. Hydrostatic force
      2. Average value of a function
      3. Economics
      4. Probability  
3. Sequences and Series
   1. Convergence/divergence of infinite sequences
   2. Convergence/divergence of infinite series
      1. Divergence test
      2. Integral test
      3. Comparison test
      4. Limit comparison test
      5. Alternating series test
      6. Ratio test
      7. Root test
   3. Estimate infinite series
   4. Absolute and conditional convergence
   5. Power series
      1. Radius and interval of convergence
      2. Integration and differentiation
      3. Taylor and Maclaurin series
      4. Binomial series (optional)
4. Parametric Equations and Polar Coordinates
   1. Slopes of parametric curves
   2. Arc length of parametric curves
   3. Slopes of polar curves
   4. Areas of polar curves
   5. Conic sections (optional)
5. Differential Equations
   1. Separable (optional)
   2. Slope fields (optional)
   3. Euler’s method (optional)