HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: MATH 2122: Calculus 3 CREDITS: 4 (4 Lec / 0 Lab) PREREQUISITES: Math 2111: Calculus 2 with a grade of C or better, or equivalent CATALOG DESCRIPTION: Calculus 3 focuses on three-dimensional coordinate systems, vectors, dot and cross products, lines and planes in space, cylinders and quadric surfaces, vector functions, projectile motion, arc length and the unit tangent vector, curvature and the unit normal vector, torsion and the unit binormal vector, functions of several variables, limits and continuity in higher dimensions, partial derivatives, the chain rule, directional derivatives and gradient vectors, tangent planes and differentials, extreme values and saddle points, Lagrange multipliers, partial derivatives with constrained variable, Taylor s formula for two variables, double integrals, double integrals in polar form, triple integrals in rectangular, cylindrical, and spherical form; areas, moments, and centers of mass, substitutions in multiple integrals; line integrals; vector fields, work, circulation, and flux; path independence, potential functions, and conservative fields; Green s Theorem; surface area and surface integrals; parameterized surfaces; Stokes Theorem; and the Divergence Theorem. MNTC goal area: (4) Math & Logical Reasoning. OUTLINE OF MAJOR CONTENT AREAS: I. Vectors and analytic geometry in space A. Three-dimensional coordinate systems B. Vectors C. The dot product D. The cross product E. Lines and planes in space F. Cylinders and quadric surfaces II. Vector-valued functions and motion in space A. Vector-valued functions B. Modeling projectile motion C. Unit tangent vector, unit normal vector, and unit binormal vector D. Arc length, curvature, and torsion III. Partial derivatives A. Functions of two or more variables B. Limits and continuity in higher dimensions C. Partial derivatives D. The Chain Rule E. Directional derivatives and gradient vectors MATH 2122 Hibbing Community College, a technical and community college, is 1
IV. F. Tangent planes and differentials G. Extreme values and saddle points H. LaGrange multipliers I. Partial derivatives with constrained variables J. Taylor s formula for two variables Multiple integrals A. Double integrals B. Areas, moments, and centers of mass C. Double integrals in polar form D. Triple integrals in rectangular coordinate E. Masses and moments in space F. Triple integrals in cylindrical and spherical coordinates G. Substitution in multiple integrals V. Integration in vector fields A. Line integrals B. Vector fields, work, circulation, and flux C. Path independence, potential functions, and conservative fields D. Green s Theorem E. Surface area and surface integrals F. Parameterized surfaces G. Stokes Theorem H. The Divergence Theorem COURSE GOALS/OBJECTIVES/OUTCOMES: Students will 1. interpret equations and inequalities geometrically in three dimensions. 2. graph equations in three dimensions. 3. find the distance between points in three dimensions. 4. find the center and radius of a sphere given its equation. 5. find the equation of a sphere given its center and radius. 6. define and determine the initial points, terminal points, and magnitudes of vectors in two and three dimensions. 7. use vector algebra operations. 8. define unit and standard unit vectors. 9. find a vector s direction. 10. express velocity as speed times direction. 11. find the midpoint of a line segment in three dimensions. 12. find the angle between two vectors. 13. define and calculate dot products. 14. define and apply orthogonal vectors in terms of the dot product. 15. find vector projections. 16. find the scalar component of a vector in the direction of a second vector. 17. write a vector as a sum of orthogonal vectors. 18. define the cross product of two vectors. 19. define parallel vectors in terms of their cross-product. MATH 2122 Hibbing Community College, a technical and community college, is 2
20. calculate the area of a parallelogram using the cross product. 21. calculate cross products using determinants. 22. use cross products to find vectors perpendicular to a plane, to find the area of a triangle, and to find a unit normal to a plane. 23. define and calculate triple scalar products. 24. describe lines using vector equations and parametric equations. 25. parameterize a line through a point parallel to a vector, a line through two points, and a line segment. 26. find the distance from a point to a line in space. 27. find an equation for a plane in space. 28. find a vector parallel to the line of intersection of two planes. 29. parameterize the line of intersection of two planes. 30. find the intersection of a line and a plane. 31. find the distance from a point to a plane. 32. find the angle between two planes. 33. define and examine the graphs of cylinders in space. 34. define quadric surfaces and examine examples of ellipsoids, paraboloids, cones, and hyperboloids. 35. graph a helix. 36. find limits of vector functions. 37. examine continuity of space curves. 38. calculate velocity, direction, speed, and acceleration of a particle moving along a smooth curve in space. 39. differentiate vector functions. 40. integrate vector functions. 41. investigate ideal projectile motion. 42. calculate arc length along a space curve. 43. define and calculate unit tangent vectors. 44. define and calculate curvature. 45. define and calculate unit normal vectors. 46. find circles of curvature for plane curves. 47. define and calculate torsion. 48. define and calculate unit binormal vectors. 49. calculate tangential and normal components of acceleration. 50. evaluate functions of several variables. 51. find domains and ranges of functions of several variables. 52. define interior and boundary points, open regions, closed regions, bounded regions, and unbounded regions. 53. define and describe level curves. 54. graph functions of two variables. 55. calculate limits of functions of two or more variables. 56. determine continuity and/or discontinuity for functions of several variables. 57. apply the two-path test to show a limit does not exist. 58. calculate partial derivatives. 59. interpret partial derivatives of functions of two variables geometrically. 60. find partial derivatives implicitly. MATH 2122 Hibbing Community College, a technical and community college, is 3
61. find the slope of a surface in the x- or y- direction. 62. calculate higher order partial derivatives. 63. apply the chain rule for functions of several variables. 64. calculate directional derivatives. 65. interpret the directional derivative geometrically. 66. calculate gradient vectors. 67. find the directional derivative using the gradient. 68. find directions of maximal, minimal, and zero change. 69. find the tangent line to a level curve. 70. find tangent planes and normal lines. 71. estimate change in a specific direction using differentials. 72. find linearizations. 73. define and calculate total differentials. 74. apply the derivative tests for local extreme values. 75. identify saddle points. 76. apply the second derivative test for local extreme values. 77. find absolute extrema. 78. find extreme values with constraints. 79. use the method of Lagrange Multipliers to find local extrema. 80. find partial derivatives with constrained independent variables. 81. use Taylor s formula to find quadratic approximations. 82. calculate multiple integrals. 83. calculate volume using multiple integrals. 84. use Fubini s Theorem for multiple integrals. 85. use double integrals to calculate area, moments, and centers of mass. 86. find average value using double integrals and triple integrals. 87. evaluate double integrals in polar form. 88. convert double integrals from Cartesian form to Polar form. 89. calculate masses and moments in three dimensions using triple integrals. 90. evaluate triple integrals in cylindrical and spherical coordinates. 91. convert between Cartesian, spherical, and cylindrical coordinates. 92. evaluate multiple integrals using substitutions. 93. evaluate line integrals. 94. find gradient fields. 95. find work done by a variable force over a space curve. 96. find flow, circulation, and flux. 97. define path independence and conservative fields. 98. define and find potential functions. 99. state and utilize the Fundamental Theorem of Line Integrals. 100. state and utilize the closed-loop property of conservative fields. 101. use the component test for conservative fields. 102. define and calculate the divergence of a vector field. 103. define and calculate the k -component of curl of a vector field. 104. utilize Green s Theorem to evaluate line integrals, find outward flux, and counterclockwise circulation. 105. evaluate surface integrals. MATH 2122 Hibbing Community College, a technical and community college, is 4
106. define an orientable surface. 107. parameterize surfaces. 108. calculate surface area for a parameterized surface. 109. calculate parametric surface integrals. 110. use Stokes Theorem to determine circulation. 111. use the Divergence Theorem to determine outward flux. MNTC GOALS AND COMPETENCIES MET: Mathematical/Logical Reasoning a, b, c, and d HCC COMPETENCIES MET: Communicating Clearly & Effectively Thinking Creatively & Critically STUDENT CONTRIBUTIONS: The student will attend class regularly, participate in class discussion, complete daily assignments, in-class exercises, exams, and a comprehensive final examination. The student will spend a minimum of two hours completing assignments for every hour in class. These must be accomplished in such a way that they meet minimum standards set by the instructor STUDENT ASSESSMENT SHALL TAKE PLACE USING INSTRUMENTS SELECTED/DEVELOPED BY THE COURSE INSTRUCTOR. SPECIAL INFORMATION: (SPECIAL FEES, DIRECTIVES AND HAZARDOUS MATERIALS): The student may be required to provide a calculator for this course. If a specific calculator model is required, this model will be specified by the instructor on the course syllabus. Examples of calculators which may be required include but are not limited to the following: the TI89 and the TI Voyage 200. AASC APPROVAL: April 29, 2015 REVIEW DATE: April 2020 MATH2121: so 042915 MATH 2122 Hibbing Community College, a technical and community college, is 5