MAT 241 CALCULUS WITH ANALYTIC GEOMETRY III

Transcription

1 MAT 241 CALCULUS WITH ANALYTIC GEOMETRY III PRESENTED AND APPROVED 12/02/11 EFFECTIVE FALL

2 Prefix & Number MAT 241 Course Title: Calculus With Analytic Geometry III Purpose of this submission: New Change/Updated Retire If this is a change, what is being changed? Update Prefix Course Description (Check all that apply) Title Course Number Format Change Credits Prerequisite Competencies Textbook/Reviewed Competencies-no changes needed Does this course require additional fees? No Yes If so, please explain. Is there a similar course in the course bank? No Yes (Please identify) Articulation: Is this course or an equivalent offered at other two and four-year universities in Arizona? No Yes (Identify the college, subject, prefix, number and title: ASU (MAT 272), NAU (MAT 238), UA (MAT 223), AWC (MAT 241), Chandler Gilbert (MAT 241), Coconino (MAT 241), Eastern (MAT 240), Estrella Mountain (MAT 241), Gateway (MAT 241), Glendale (MAT 241), Mesa (MAT 241), Paradise Valley (MAT 241), Phoenix (MAT 241), Rio Salado (MAT 241), Scottsdale (MAT 241), South Mountain (MAT 241), Yavapai (MAT 241) Is this course identified as a Writing Across the Curriculum course? No Yes Course Textbook, Materials and Equipment Textbook(s) Title Thomas Calculus, Twelfth Edition Author(s) Weir/Hass Publisher Pearson ISBN Barnes & Noble Price $ Software/ Equipment Modality Check all that apply Title Author(s) Publisher ISBN Barnes & Noble Price Textbook Bundled with MyMathLab access and Calculus Review Card. On-ground On-line Hybrid ITV Web-enhanced Course Assessments Description of Possible Course Assessments (Essays, multiple choice, etc.) Exams standardized for this course? Midterm Final Other (Please specify): Weekly homework and quizzes, midterm and final exams. Assessments will typically be short answer. Are exams required by the department? No Yes If Yes, please specify:

3 Where can faculty members locate or access the required standardized exams for this course? (Contact Person and Location) Example: NCK Academic Chair Office Any Resident Faculty Member Student Outcomes: Identify the general education goals for student learning that is a component of this course. Check all that apply: 1. Communicate effectively. a. Read and comprehend at a college level. b. Write effectively in a college setting. 2. Demonstrate effective quantitative reasoning and problem solving skills. Method of Assessment Weekly homework, quizzes, midterm and final examinations. 3. Demonstrate effective qualitative reasoning skills. 4. Apply effective methods of inquiry. a. Generate research paper by gathering information from varied sources, analyzing data and organizing information into a coherent structure. b. Employ the scientific method. 5. Demonstrate sensitivity to diversity a. Experience the creative products of humanity. b. Describe alternate historical, cultural, global perspectives. Office of Instruction Use only: CIP Code: ONET Code: Minimum Qualifications:

4 COURSE INFORMATION Initiator: Clark Brown Date of proposal to Curriculum Sub-Committee: October 26, 2011 Effective Semester/Year Fall 2012 Spring Summer Prefix & Number: MAT 241 Full Title: (100 character limit) Calculus With Analytic Geometry III Short Title: (30 character limit) Calculus III Catalog Course Description: Calculus II extends the study of differential and integral 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. SUN Course Number: MAT 2230 Credit Hours: 4 Lecture Hours: 4 Lab Hours: 0 Prerequisite(s) Grade of C or better in MAT 230 or 231 Co-requisite(s) Intended Course Goals By the end of the semester, students will be able to: 1. Extend the concept of limits, continuity, differentiation, and integration to vector-valued functions, and apply these concepts to solve problems of velocity, acceleration, vector components, arc length, and curvature. 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 application problems in science, engineering, economics, and higher mathematics. 3. Create and evaluate multiple (double and triple) integrals to solve problems involving area, volume, center of mass, moments of inertia, surface areas, using change of variables or Jacobians where appropriate. 4. Analyze vector fields, finding curvature, curl, and flux, and create and evaluate line integrals for vector fields. In solving related problems, students will use Green s Theorem, Stokes Theorem and the Divergence Theorem where appropriate.

5 Course Competencies and Objectives By the end of the semester, students will be able to: Competency 1 Describe planes, lines, surfaces, and curves in 2 and 3 dimensional space using coordinate systems and vectors, and apply the analytic geometry of space to solve application problems in geometry and physics. Objective 1.1 Describe and sketch the surfaces in 2 and 3 dimensional space represented by equations or inequalities using various axes orientations. Objective 1.2 Create vectors in component form to model force, displacement, velocity and other directed quantities in 2 and 3 dimensional space, and perform operations on these vectors. Objective 1.3 Evaluate and interpret the algebraic and geometric definitions of the dot product of two vectors and its relationship to vector projection, and use the dot product to find angles, and the vector projection to calculate work. Objective 1.4 Evaluate and interpret the algebraic and geometric definitions of the cross product of two vectors, and use the cross product and triple scalar product to solve problems in geometry and physics. Objective 1.5 Extend the concepts of scalar and vector products to create equations for lines, line segments, and planes in space, and calculate angles, distances, and intersections in space. Objective 1.6 Describe and sketch cylinders and quadric surfaces in space using the threedimensional coordinate system. Competency 2 Apply the concepts of vectors to create vector-valued functions to model the position of a moving body, and use calculus to analyze the paths, velocities, and accelerations of moving bodies. Objective 2.1 Create vector-valued functions for particle motion in 2 and 3 dimensional space, calculate and interpret the limit and the derivative, and evaluate and interpret the integral for these functions. Objective 2.2 Extend the concept of vectors to model projectile motion, and use calculus and algebra to solve related problems in physics and sports. Objective 2.3 Calculate the arc length of a curve in 2 and 3 dimensional space and describe motion of an object in space in terms of its speed and unit tangent vector. Objective 2.4 Compute the curvature and unit normal vectors for plane and space and apply these concepts to find and interpret the circle of curvature at a point on a curve. Objective 2.5 Compute the bi-normal vector and torsion of a space curve and describe the motion of an object in space in terms of the 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 Use level curves, graphs, and level surfaces to visualize functions of two and three variables. Objective 3.2 Evaluate the limit of a multivariable function, and determine whether a function is continuous. Objective 3.3 Compute partial derivatives of all orders, apply the Mixed Derivative Theorem, and articulate the relationship between the continuity and differentiability of functions of several variables. Objective 3.4 Extend the concepts of the chain rule and implicit differentiation from single variable

6 calculus to multivariable calculus and functions with two or more independent and/or intermediate variables. Objective 3.5 Calculate and interpret the directional derivative and gradient for a multivariable function, and determine the direction in which a function has the greatest increase or decrease. Objective 3.6 Find the equation of the normal line and tangent plane to a point on a smooth surface and apply the concept of linearization to functions of two or more variables, using differentials to calculate error and interpret sensitivity of a variable to change. Objective 3.7 Determine critical points, local and absolute extrema, and saddle points for functions of two variables using the first and second derivative tests, and use these concepts to solve applied optimization problems. Objective 3.8 Use the method of Lagrange Multipliers to find extreme values of constrained functions and solve related real-world problems. Competency 4 Expand the concepts of the integral from single variable calculus to functions of several variables to create multiple integrals in rectangular, cylindrical and spherical coordinates, and apply these concepts to solve related problems in geometry and physics. Objective 4.1 Create double integrals to find volume, and apply Fubini s theorem to evaluate these integrals over rectangular regions in the plane. Objective 4.2 Create double integrals to find volume, and apply Fubini s theorem to evaluate these integrals over general regions in the plane. Objective 4.3 Create and evaluate double integrals to find the area of bounded regions in the plane and the average value of a function of two variables. Objective 4.4 Evaluate double integrals and their applications using the polar coordinate system. Objective 4.5 Create and evaluate triple integrals using rectangular coordinates, and use triple integrals to find volume and the average value of a function over a three-dimensional region. Objective 4.6 Use double and triple integrals to find mass, moments, and centers of mass of two dimensional plates and three-dimensional solids. Objective 4.7 Evaluate triple integrals and their applications using cylindrical and spherical coordinate systems. Objective 4.8 Replace complicated multiple integrals with familiar forms using substitution and the Jabobian, and evaluate the resulting multiple integral. Competency 5 Apply vector calculus and multiple integration to vector fields and related physics applications through line and surface integrals, Green s, Stokes, and the Divergence Theorem, and recognize the interrelationships among these three theorems. Objective 5.1 Create and evaluate line integrals to integrate over a smooth curve in space, and apply these line integrals to calculate mass and moments of inertia. Objective 5.2 Extend concepts of vectors to visualize, sketch, and interpret vector and gradient fields, and evaluate and interpret integrals over paths through vector fields, and apply these integrals to find work, circulation, and flux. Objective 5.3 Determine whether a vector field is conservative, create potential functions for vector fields, express work and circulation integrals in differential form, and evaluate line integrals demonstrating independence of path. Objective 5.4 Determine the divergence (flux density) of a vector field in the plane and apply

7 Green s Theorem to evaluate line integrals that represent the circulation or flux across a closed curve in the xy-plane. Objective 5.5 Describe surfaces in space using a parametrization of two variables as well as level surfaces of functions of three variables, and compute surface area using the three forms of the surface area differential. Objective 5.6 Create and evaluate surface integrals and apply these integrals to find flux, mass, center of mass, and moments of inertia. Objective 5.7 Find the curl for a vector field, relate Stokes Theorem to Green s Theorem, and use Stokes Theorem to determine the circulation of a vector field over a surface. Objective 5.8 Calculate the divergence of a vector field, relate the Divergence Theorem to Green s Theorem, and apply the Divergence Theorem to determine the outward flux of a vector field across a surface.