MATH& 163 Calculus 3 (5 credits)

1. Generate the terms of a sequence from an explicit or recursive equation.

2. Determine the formulas for simple sequences (geometric, arithmetic, power).

3. Determine the limit of a sequence, if it exists.

4. Determine the convergence of simple series (e.g., harmonic series) by comparing to improper integrals.

5. Explore the convergence of series and sequences using technology.

6. Determine the convergence of a geometric series and find its sum if it exists.

7. Determine the radius of convergence of a power series using the ratio test.

8. Represent appropriate functions via geometric series and Taylor series.

9. Differentiate and integrate power series.

Vectors

10. Determine the distance between points in three dimensional space.

11. Determine the equation of a sphere centered at a given point and of a given radius.

12. Determine the magnitude of a vector.

13. Perform basic computations with vectors numerically and graphically.

14. Compute the dot product of two vectors.

15. Determine the angle between two vectors.

16. Compute the projection of one vector onto another.

17. Compute the cross product of two vectors.

18. Determine the area of the parallelogram spanned by two vectors via the magnitude of the cross product.

Vector Functions and Space Curves

19. Determine the parametric equation of a line given sufficient information (e.g., a point and a parallel vector, two points).

20. Determine the scalar equation of a plane given sufficient information (e.g., three points, a point and a normal vector, a point and two non-parallel vectors).

21. Sketch the graph of a vector valued function (parameterized curve).

22. Identify the graphs of standard space curves (e.g., line, parabola, circle, helix).

23. Integrate and differentiate vector valued functions.

24. Determine tangent lines to space curves.

25. Determine the arc length of a space curve.

Functions of Several Variables and 3D Surfaces

26. Evaluate functions of several variables numerically, graphically, and symbolically.

27. Use the technique of level curves to sketch the graph of a function of two variables.

28. Graph functions of several variables utilizing technology as appropriate.

29. Identify the graphs of standard surfaces (e.g., sphere, paraboloid, ellipsoid, cylindrical surfaces).

30. Compute limits of functions of several variables.

31. Determine the domain and continuity of a function of several variables.

Partial Derivatives

32. Utilize the definition of the partial derivative of a function of several variables to solve rate of change problems.

33. Compute partial derivatives symbolically utilizing the basic techniques from single variable calculus.

34. Determine implicit and parametric equations for the tangent plane to a surface that is defined by the graph of a function.

35. Compute partial derivatives via the chain rule and through implicit differentiation.

36. Apply directional derivatives to solve rate of change problems in arbitrary directions. Determine the direction of maximal and minimal change of a function of several variables.

37. Apply techniques of partial derivatives to solve problems in the sciences and engineering.

General Content

38. Solve application problems in the sciences, including determining position, velocity, and acceleration of an object moving along a given trajectory.

39. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.

40. Link graphical, numeric, and symbolic approaches when interpreting situations and analyzing problems.

Degree Outcomes
Quantitative & Symbolic Reasoning: Graduates utilize mathematical, symbolic, logical, graphical, geometric, or statistical analysis for the interpretation and solution of problems in the natural world and human society.

Critical, Creative and Reflective Thinking: Graduates will evaluate, analyze, synthesize, and generate ideas; construct informed, meaningful, and justifiable conclusions; and process feelings, beliefs, biases, strengths, and weaknesses as they relate to their thinking, decisions, and creations.

Effective Communication: Graduates will be able to exchange messages in a variety of contexts using multiple methods.

Lecture Contact Hours 50
Lab Contact Hours 0
Clinical Contact Hours 0
Total Contact Hours 0

Potential Methods
A. Traditional Quizzes and Examinations. Computational, short answer, written proofs, and construction of counter examples.
B. Group work. In-class worksheets to explore more advanced topics from the sciences and the impact of multivariate calculus upon them. Long term projects.
C. Traditional and/or online homework.
D. Presentations – Individual and/or group presentations of problems, projects, and/or technology. Oral and/or written.
E. Informal Assessment – Self evaluation, peer evaluation, or teacher observation.