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|MAT245 - Calculus 3|
Catalog Description: Covers infinite series, curves in parametric and polar forms, vectors, partial differentiation, and multiple integrals. Applications of these topics focus on analysis of functions and surfaces in 3 dimensional space. Fall only. Prerequisite: MAT 142 with a grade of ‘C’ or higher or by placement. May be taken concurrently with MAT 142 with instructor permission.
Lecture: 4 hrs.
Course Student Learning Outcomes (CSLOs):
Upon completion of Calculus III, students will be able to do the following, either by hand on an exam (using a graphing calculator for computations) or, where possible, using a computer algebra system software for a collected assignment:
1. Given a function in parametric or polar form, sketch the graph of the resulting curve.
2. Given a set of vectors in space, algebraically define magnitudes, dot-products, cross-products, unit vectors, and triple scalar products of these vectors.*
3. Find the equation and give a (rough) sketch of any of the following surfaces in 3-space: sphere, plane, cylinder, quadric.
4. Evaluate a function of two variables at a specific point, and/or determine and sketch its domain.
5. Given a function of two or more variables, find and evaluate any partial derivatives, directional derivatives and/or gradient vectors.*
6. Find the minimum and maximum value of a function of two variables over a defined interval.
7. Use a double integral in either rectangular or polar form to find the volume of the solid defined by a function of two variables.*
8. Given an infinite series, determine the convergence or divergence of the series using any of the following tests (as appropriate): divergence, ratio, root, comparison, limit comparison, geometric, harmonic, integral, p-series.
9. Given a power series, find the interval of convergence.
10. Given a function of a single variable, find and evaluate a Taylor polynomial or Taylor series for the function at a given point.
* This course objective has been identified as a student learning outcome that must be formally assessed as part of the Comprehensive Assessment Plan of the college. All faculty teaching this course must collect the required data and submit the required analysis and documentation at the conclusion of the semester to the Office of Institutional Research and Assessment.
I. Infinite Series
1. Introduction: Taylor polynomials and approximations
3. Series and convergence
4. The Integral Test and p-Series
5. Comparisons of series
6. Alternating series
7. The Ratio and Root Tests
8. Power series
II. Plane curves, parametric equations, and polar coordinates
1. Plane curves and parametric equations
2. Parametric equations and calculus
3. Polar coordinates and polar graphs
III. Vectors and the geometry of space
1. Vectors in the plane
2. Space coordinates and vectors in space
3. The dot product of two vectors
4. The cross product of two vectors in space
5. Lines and planes in space
6. Surfaces in space
7. Cylindrical and spherical coordinates
IV. Vector-valued functions
1. Vector-valued functions
2. Differentiation and integration of vector-valued functions
3. Velocity and acceleration
4. Tangent vectors and normal vectors
V. Functions of several variables
1. Introduction to functions and several variables
2. Limits and continuity
3. Partial derivatives
5. The Chain Rule
6. Directional derivatives and gradients
7. Tangent planes and normal lines
8. Extrema of functions of two variables
9. Applications of extrema of functions of two variables
10. Lagrange multipliers
VI. Multiple integration
1. Iterated integrals and area in the plane
2. Double integrals and volume
3. Change of variables: Polar coordinates
4. Center of mass and moments of inertia
5. Surface area
6. Triple integrals and applications
VII. Vector analysis (Optional)
1. Vector fields
2. Line integrals
3. Surface integrals
Effective Term: Fall 2012