| Originator: | Rucnov, Marco Status: Approved Department: MAT Mathematics |
| Date Created: | 08/17/2016 Submitted: 09/29/2016 Completed: 12/27/2016 |
| Effective Semester: | Fall |
| Catalog Year: | 2017-18 |
| Course Prefix: | MAT |
| Course Number: | 241 |
| Course Full Title: | Analytic Geometry and Calculus III |
| Old course information: | |
| Reason for Evaluation: | Prerequisite Change Description Change Goals, Competencies and/or Objectives Change |
| Current Credit: | 4 |
| Lecture Hours: | 4 |
| Lab Hours: | 0 |
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| If the credit hour change box has been marked, please provide the new credit hour: | |
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| SUN Course?: | Yes |
| AGEC Course?: | No |
| Articulated?: | Yes |
| Transfer: | ASU NAU UA |
| Prerequisite(s): | MAT 231 with a C or higher. |
| Corequisite(s): | None |
| Catalog Course Description: | Calculus With Analytic Geometry III is multivariable differential and integral calculus(mathematical analysis)with three-dimensional analytic geometry, and extends the study of differential,integral, and vector calculus to transcendental functions and functions defined using parametric equations and polar coordinates. The course explores various techniques of integration, including numerical integration and the evaluation of improper integrals, as well as elementary techniques for solving first order linear differential equations. Infinite sequences, series, and their convergence are also emphasized. |
| Course Learning Outcomes: | 1. Extend the concept of limits, continuity, differentiation, and integration to vector-valued functions of two and three variables, and apply these concepts to solve problems of velocity, acceleration, vector components, arc length, and curvature in plane and space (3).
2. Apply the concepts of functions, limits, continuity, partial derivatives, differentials, and chain rules to functions of several variables, find directional derivatives, gradients, tangent and normal lines, extrema, and use these techniques to solve applications problems in science, technology, engineering, economics, and higher mathematics (3,5). 3. Create and evaluate multiple (double and triple) integrals solve problems involving area, volume, center of mass, moments of inertia, surface areas, using change of variables or Jacobian where appropriate (3). 4. Analyze vector fields, finding curvature, curl, and flux, and create and evaluate line integrals and surface integrals for vector fields. In solving related problems, students will use Green's Theorem, Stokes' Theorem and the Gauss' Divergence Theorem where appropriate (3,5). |
| Course Competencies: | Competency 1 Analyze planes, lines, surfaces, and curves in 2 and 3 dimensional space using coordinate systems and vectors, by applying the analytic geometry of space to solve application problems in geometry and physics.
Objective 1.1 Describe the surfaces in 2 and 3 dimensional space represented by equations or inequalities using various axes orientations. Objective 1.2 Convert vectors in component form to model force, displacement, velocity and other directed quantities in 2 and 3 dimensional space. Objective 1.3 Perform operations on vectors in component form. Objective 1.4 Solve the dot product of two vectors to find angles, and the vector projection to calculate work. Objective 1.5 Solve the cross product and triple scalar product to solve problems in geometry and physics by using the algebraic and geometric definitions. Objective 1.6 Calculate equations for lines, line segments, and planes in space, and calculate angles, distances, and intersections in space by using the concepts of scalar and vector products. Objective 1.7 Sketch cylinders and quadratic surfaces in space using the three-dimensional coordinate system. Competency 2 Create vector-valued functions to model the position of a moving body, by using calculus to analyze the paths, velocities, and accelerations of moving bodies. Objective 2.1 Calculate the limit and the derivative of the vector-valued functions for particle motion in 2 and 3 dimensional space. Objective 2.2 Calculate the integral for these vector-valued functions. Objective 2.3 Solve the vectors projectile motion in physics and sports. Objective 2.4 Calculate the arc length of a curve in 2 and 3 dimensional space. Objective 2.5 Describe motion of an object in space in terms of its speed and unit tangent vector. Objective 2.6 Calculate the circle of curvature at a point on a curve by computing the curvature and unit normal vectors for plane and space. Objective 2.7 Compute the bi-normal vector and torsion of a space curve to describe the motion of an object in space in terms of the Tangent-Normal-Binormal (TNB) frame. Competency 3 Expand the concepts of limits and derivatives from single variable calculus to functions of several variables, and apply these concepts to solve related problems in probability, statistics, fluid dynamics, electricity, economics, and other natural and industrial applications. Objective 3.1 Graph the functions of two and three variables by using level curves, graphs, and level surfaces. Objective 3.2 Evaluate the limit of a multivariable function to and determine whether a function is continuous. Objective 3.3 Compute partial derivatives of all orders, by applying the Mixed Derivative Theorem to articulate the relationship between the continuity and differentiability of functions of several variables. Objective 3.4 Solve the problem by extending the concepts of the chain rule and implicit differentiation from single variable calculus to multivariable calculus and functions with two or more independent and/or intermediate variables. Objective 3.5 Calculate the directional derivative, gradient for a multivariable function, and the direction in which a function has the greatest increase or decrease. Objective 3.6 Calculate the equation of the normal line and tangent plane to a point on a smooth surface. Objective 3.7 Interpret sensitivity of a variable to change by applying the concept of linearization to functions of two or more variables using differentials. Objective 3.8 Solve problems by determining critical points, local and absolute extrema, and saddle points for functions of two variables using the first and second derivative tests. Objective 3.9 Use the method of Lagrange Multipliers to Calculate extreme values of constrained functions by using the method of Lagrange Multipliers. Competency 4 Create integrals in rectangular, cylindrical and spherical coordinates by expanding the concepts of the integral from single variable calculus to functions of several variables. Objective 4.1 Calculate the volume in rectangular regions with double integrals, by applying Fubini's theorem. Objective 4.2 Calculate double integrals over general regions in the plane by using Fubini's theorem. Objective 4.3 Calculate the area of bounded regions in the plane and the average value of a function of two variables by evaluating double integrals. Objective 4.4 Evaluate double integrals and their applications using the polar coordinate system. Objective 4.5 Evaluate triple integrals using rectangular coordinates to find volume and the average value of a function over a three-dimensional region. Objective 4.6 Calculate mass, moments, and centers of mass of two dimensional plates and three-dimensional solids by using double and triple integrals. Objective 4.7 Evaluate triple integrals and their applications using cylindrical and spherical coordinate systems. Objective 4.8 Evaluate the multiple multiple integral by substitution and the Jacobian. Competency 5 Solve problems by applying vector calculus and multiple integration to vector fields and related physics applications through line and surface integrals, Green's, Stokes', and the Gauss' Divergence Theorem to recognize the interrelationships among these three theorems. Objective 5.1 Evaluate line integrals to integrate over a smooth curve in space, by applying these line integrals to calculate mass and moments of inertia. Objective 5.2 Calculate the integrals over paths through vector fields to find work, circulation, and flux by extending concepts of vectors. Objective 5.3 Create potential functions for vector fields by determinig whether a vector field is conservative by expressing work and circulation integrals in differential form. Objective 5.4 Evaluate line integrals demonstrating independence of path. Objective 5.5 Evaluate line integrals that represent the circulation or flux across a closed curve in the xy-plane by determinig the divergence (flux density) of a vector field in plane and by applying Green's Theorem. Objective 5.6 Compute surface area using the three forms of the surface area differential and parametrization of two variables as well as level surfaces of functions of three variables. Objective 5.7 Compute flux, mass,center of mass, and moments of inertia by applying surface integrals. Objective 5.8 Determine the circulation of a vector field over a surface by finding the curl for vector field as it relates to Stokes' Theorem to Green's Theorem. Objective 5.9 Calculate the divergence of a vector field, by relating the Gauss' Divergence Theorem to Green's Theorem. Objective 5.10 Determine the outward flux of a vector field across a surface by applying the Gauss' Divergence Theorem. |