School

Science, Technology, Engineering and Math

Division

Mathematics

Department

Mathematics

Academic Level

Undergraduate

Course Subject

Mathematics

Course Number

280

Course Title

Calculus III

Credit Hours

5.00

Instructor Contact Hours Per Semester

77.00 (for 15-week classes)

Student Contact Hours Per Semester

77.00 (for 15-week classes)

Grading Method

A-E

Pre-requisites

MATH-183 with a C or better

Catalog Course Description

Covers topics such as the calculus of vector-valued functions, the differential calculus of functions of more than one variable, directional derivatives, gradients, partial derivatives, multiple integration, vector fields, and line integrals. Various applications are covered throughout the course. Utilizes numerical, graphical, and algebraic approaches whenever possible. Requires a non-graphing scientific calculator for some formal assessments, access to a free graphing utility application to support learning and during informal assessments, and access to an online homework assessment.

## Goals, Topics, and Objectives

Goal Statement

- To study the nature and significance of calculus for students of science, technology, engineering, and mathematics disciplines.
- To demonstrate various applications of calculus to problems from the social sciences, physical sciences, and engineering.
- To present an exposition of calculus that incorporates graphical, numerical, and algebraic analysis, without undue emphasis on theoretical abstraction or routine mechanical manipulation.
- To use technology to illustrate calculus concepts and verify calculus solutions to application problems.
- To provide students with an exposure to the logical reasoning of mathematics.

Core Course Topics

- Vectors and Vector -Valued Functions
- Perform vector addition and scalar multiplication of vectors algebraically and geometrically.
- Identify and solve problems involving vectors in the plane and in 3-space both geometrically and algebraically.
- Compute the dot product for vectors in 2 and 3 dimensions and the cross product for vectors in 3 dimensions.
- Use the dot and cross products in applications such as computing the angle between 2 vectors, vector projections, scalar components, work and torque.
- Construct vector-valued functions in order to describe lines and curves in 3-space.
- Determine limits of vector-valued functions using numerical, algebraic and graphical methods.
- Determine continuity or discontinuity of vector-valued functions.
- Determine derivatives and integrals of vector-valued functions.
- Find the vector components of velocity and acceleration for particle motion in 3-space.
- Find the length of a curve in 3-space and be able to parameterize a given curve with respect to arc length.
- For a given curve, determine the unit tangent, unit normal, unit binormal and curvature.

- Functions of Several Variables
- Determine the equation of a plane in 3-space using both the vector form and the scalar form.
- Write the equation for a given quadratic surface.
- Draw accurate and useful renditions of a surface given an equation by hand and using appropriate graphing application software.
- Use graphs of level curves (contour maps) to obtain information about an arbitrary surface.
- Determine limits of functions of several variables using numerical, algebraic and graphical methods.
- Determine continuity or discontinuity of functions of several variables.
- Find the first and second partial derivatives of functions using differentiation rules.
- Determine the gradient and directional derivatives for a function of several variables.
- Use the appropriate chain rule to differentiate composite functions of several variables.
- Use the gradient and directional derivatives to find instantaneous rates of change and related rates of change in application problems.
- Use the gradient and directional derivatives and the Second Partials Test to find properties of functions of several variables: maxima and minima and saddle points, and use this information to sketch the graph of functions.
- Solve optimization problems using the method of Lagrange Multipliers.
- Use the gradient and tangent plane to determine the linear approximation of a function of several variables.

- Integration of Functions of Several Variables
- Integrate functions of 2 variables over regions in the xy-plane using Riemann sums and iterated integrals with respect to rectangular and/or polar coordinates.
- Integrate functions of 3 variables over 3-dimensional regions using Riemann sums and iterated integrals with respect to rectangular, cylindrical, or spherical coordinates.
- Use multiple integrals to solve application problems involving areas, volumes, masses, first and second moments, centers of mass, and moments of inertia.

- Vector Fields
- Sketch vector fields by hand.
- Interpret vector fields as models of fluid flow.
- Compute line integrals over curves in 2- or 3-space.
- Determine whether a vector field is conservative or not.
- Use the Fundamental Theorem for Line Integrals to compute line integrals under appropriate conditions.
- Use Green’s Theorem in the plane in flux form or in circulation form to compute line integrals.
- Compute the 2-dimensional divergence and curl for a vector field in 2-dimensions.
- Use line integrals to solve application problems involving work, mass, circulation and heat flux.
- Compute the 3-dimensional divergence and curl for a vector field in 3-dimensions.
- Compute surface integrals of scalar-valued functions.
- Compute surface integrals of vector fields.
- Use Stokes’ Theorem to evaluate line and surface integrals.
- Use the Divergence Theorem to calculate flux.

## Assessment and Requirements

Assessment of Academic Achievement

- All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives. This final exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this final exam may be a common final exam administered to all sections of Math-280.
- All students will be required to complete at least two proctored on-campus exams. The cumulative value of those exams must be at least 40% of a student’s final grade.
- All students will be required to complete online homework. This online homework must be weighted in such a manner so that it is worth between six percent (6%) and twelve percent (12%) of the final course grade.
- Additional assessment of student achievement may include assignments, quizzes, and exams.
- For proctored in-person formal assessments (quizzes, tests, and exams) the only technology students can use is a non-graphing scientific calculator. Quizzes, tests, and exams may have non-calculator parts. Class projects and informal assessments will require students to use a free graphing application to support learning.
- Application problems must not only be included on chapter exams but also on the final exam.

General Course Requirements and Recommendations

- A non-graphing scientific calculator is required for formal assessments.
- Free graphing applications (such as DP-Graph Computer program, 3D-GeoGebra, or other free 3D graphing software) will be used to support learning, informal assessments, and class work.
- Access to an online homework management system is also required.
- Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.

## Outcomes

General Education Categories

- Mathematics

Institutional Outcomes

- Quantitative Literacy

MTA Categories

- Category 3: Mathematics (College Algebra Track)

Satisfies Wellness Requirement

No

Effective Term

Fall 2024