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Nov 21, 2024
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MATH& 153 Calculus III (5 credits)
Distribution Area Fulfilled Natural Sciences; Quantitative and Symbolic Reasoning; General Transfer Elective
Formerly MATH 126 - CCN
Prerequisite Completion of MATH& 152 with a grade of 2.0 or higher or instructor permission.
Course Description
Sequences and series. Vectors and geometry of space. The calculus of vector functions and parametric surfaces. Polar, cylindrical and spherical coordinates.
Student Outcomes
The student should be able to:
Sequences and series component
1. Determine the terms of a recursively defined sequence.
2. Estimate the limit of a convergent sequence using graphical and numerical techniques.
3. Determine the convergence of a sequence and find the exact limit, if it exists.
4. Determine the convergence of a geometric series and find its sum, if it exists.
5. Apply the integral, comparison, limit comparison, ratio, and alternating series tests to determine the convergence of a given series.
6. Use the integral test and alternating series test to bound the error in estimating the sum of a convergent series via a partial sum.
7. Determine the radius and interval of convergence of a given power series.
8. Represent appropriate functions as a power series via geometric series and Taylor’s theorem.
9. Differentiate and integrate power series.
10. Solve problems in mathematics and/or the sciences via Taylor polynomial representations of a function.
11. Explore the convergence of series and sequences utilizing an appropriate computer algebra system.
Vectors component
12. Determine the distance between points in three dimensional space.
13. Determine the equation of a sphere centered at a given point and of a given radius.
14. Determine the magnitude of a vector.
Dot and cross product component
15. Interpret the dot product as the work done by a constant force.
16. Compute the dot product of two vectors.
17. Determine the angle between two vectors.
18. Compute the projection of one vector onto another.
19. Determine the area of the parallelogram spanned by two vectors via the magnitude of the cross product.
20. Determine the volume of a parallelepiped spanned by three vectors via the magnitude of the scalar triple product.
21. Interpret the cross product as the torque produced by the moment of a force along an axis.
22. Compute the cross product of two vectors.
23. Determine the volume of the parallelepiped spanned by three vectors via the scalar triple product.
Calculus of vector functions, space curves, and parametric surfaces component
24. Determine the parametric equation of a line given sufficient information (e.g, a point and a parallel vector, two points).
25. 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).
26. Determine the angle of intersection between two planes.
27. Use the technique of level curves to sketch the graph of a function of two variables.
28. Use appropriate computer technology to graph a function of two variables.
29. Determine the cylindrical and spherical coordinates of a point in three dimensional space.
30. Determine the Cartesian coordinates of cylindrical and spherical points.
31. Identify and/or sketch a solid defined by inequalities or equations in cylindrical or spherical coordinates.
32. Identify the graphs of standard equations (e.g, sphere, paraboloid, cylindrical surfaces, helix).
33. Sketch the graph of a vector valued function.
34. Integrate and differentiate vector valued functions.
35. Determine tangent lines to space curves.
36. Determine the arc length of a space curve or polar curve.
37. Determine the area enclosed by a polar curve.
Applications to physics component
38. Determine the position, velocity, and acceleration of an object moving along a given trajectory.
39. Use Newton’s Second Law of Motion to determine the force acting on an object moving radially.
40. Use Newton’s Second Law of Motion to determine the trajectory of a projectile fired at a given angle and with a given initial velocity.
41. Solve basic problems in astronomy via the calculus of vector curves (e.g, Kepler’s Laws of Planetary Motion).
General Content
42. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.
43. Write clear, coherent, and correct mathematical proofs at a basic level, including construction of counter examples and proof by contradiction.
44. Link graphical, numeric, and symbolic approaches when interpreting situations and analyzing problems.
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