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MOHAWK VALLEY COMMUNITY COLLEGE

UTICA, NEW YORK

COURSE OUTLINE

AND

TEACHING GUIDE

CALCULUS 3

MA253

REVISED BY GARY KULIS – 3/21

COURSE OUTLINE

TITLE: Calculus 3

CATALOG NO: MA253

CREDIT HOURS: 4

LAB HOURS: 0

PREREQUISITES: MA152 Calculus 2

CATALOG

DESCRIPTION: This is the third in a sequence of three courses in calculus. Topics include polar and space coordinates, parametric equations, the algebra and calculus of vectors, partial differentiation, and multiple integration, Applications are included. Prerequisite: MA152 Calculus 2.

Course Objectives:

1. To help the student learn the basics of multivariable calculus.

2. To raise the student’s level of logical thinking by requiring the student to write some basic calculus proofs.

3. To increase the student’s ability to use technology as a problem solving tool by requiring the student to use technology as an aid in solving applied problems.

4. To challenge the student, through both applied and theoretical problems, to appreciate

calculus as a problem solving art.

General Student Outcomes:

1. The student will demonstrate an ability to solve word problems using rigorous mathematical reasoning.

2. The student will demonstrate an ability to write proofs using rigorous mathematical reasoning.

3. The student will be able to state a problem correctly, reason analytically to a solution and interpret the results.

4. The student will be able to use technology to solve problems.

5. The student will demonstrate the ability to interpret and communicate mathematics in writing.

6. The student will be able to work effectively within a group by demonstrating openness toward diverse points of view, drawing upon knowledge and experience of others to function as a group member, demonstrating skill in negotiating differences and working toward solutions.

SUNY Learning Outcomes:

1. The student will develop well reasoned arguments by demonstrating an ability to write proofs.

2. The student will identify, analyze, and evaluate arguments as they occur in their own and other’s work.

3. The student will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics.

4. The student will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally.

5. The student will demonstrate the ability to employ quantitative methods such as arithmetic, algebra, geometry, or statistics to solve problems.

6. The student will demonstrate the ability to estimate and check mathematical results for reasonableness.

TOPIC 1: PARAMETRIC EQUATIONS AND POLAR COORDINATES

Students will be introduced to parametric forms including the calculus of equations in parametric form. Students will also be introduced to polar coordinates including how to convert between rectangular and polar forms, and how to apply polar forms to graphing, area and arc length.

Student Learning Outcomes:

The student will be able to:

1.1 Sketch a curve represented by a system of parametric equations and write the corresponding

rectangular equation by eliminating the parameter.

1.2 Find dy/dx for a system of parametric equations and use this concept to determine slopes of

tangents to a curve.

1.3 Find the arc length of a curve defined in parametric form.

1.4 Express a point both in rectangular and polar form.

1.5 Graph a polar equation including finding tangents to the pole and points of horizontal and

vertical tangency (if any).

1.6 Find area and arc length in polar coordinates.

TOPIC 2: ALGEBRA AND CALCULUS OF VECTORS

The algebra of vectors in two and three dimensions, and their applications to analytic geometry are introduced. Equations of lines and planes are studied. The calculus of vector-valued functions is then developed to include differentiation and integration.

Student Learning Outcomes:

The student will be able to:

2.1 Express vectors in two and three space in several formats including component form.

2.2 Perform vector operations including addition, scalar multiplication, dot product, and cross

product.

2.3 Use concepts such as dot product and cross product to determine the relationship of vectors

(orthogonal, parallel), find angles between vectors, projections, areas and volumes.

2.4 Write the equation of a line in space in both parametric and symmetric form.

2.5 Write an equation of a plane in space.

2.6 Solve application problems which relate lines and/or planes in space.

2.7 Find domains and ranges of vector-valued functions and sketch their graphs.

2.8 Differentiate and integrate vector-valued functions.

2.9 Find tangent and normal vectors.

TOPIC 3: SURFACES IN SPACE AND SPACE COORDINATES

Cylindrical surfaces and quadric surfaces will be discussed and graphed. This will include a brief study of some of the following surfaces: cylinders, ellipsoids, paraboloids, cones, and hyperboloids. Converting among the three coordinate systems (rectangular, cylindrical, and spherical) will be introduced.

Student Learning Outcomes:

The student will be able to:

3.1 Describe and sketch various surfaces, including finding traces.

3.2 Convert from one coordinate system to another.

TOPIC 4: FUNCTIONS OF SEVERAL VARIABLES

The calculus of functions of several variables will be discussed. Topics will include: limit, continuity, partial differentiation, the directional derivative, the gradient, the total differential, and maxima and minima of functions of several variables with side conditions including the use of Lagrange multipliers.

Student Learning Outcomes:

The student will be able to:

4.1 Find domains, ranges and sketch surfaces.

4.2 Find level curves and surfaces.

4.3 Find limits by using theorems and verifying that limits do not exist by examining paths.

4.4 Describe the continuity of a function.

4.5 Find partial derivatives by definition and by theorem.

4.6 Find higher order partial derivatives.

4.7 Find differentials.

4.8 Determine the differentiability of a function.

4.9 Apply the appropriate chain rule for functions of several variables.

4.10 Find directional derivatives and gradients and use these concepts in application problems.

4.11 Find tangent planes and normal lines to a surface in space.

4.12 Find extrema of functions in two variables using the second partials test

4.13 Solve applications using the method of the second partials test

4.14 Use Lagrange multipliers to solve optimization problems that have constraints

TOPIC 5: MULTIPLE INTEGRATION

The double and triple integral are carefully developed, similarities with the single integral are mentioned. The student is expected to be able to evaluate these integrals by calculating the associated iterated integrals. Proficiency is gained by considering applications in the cylindrical and spherical as well as Cartesian coordinate systems.

Student Learning Outcomes:

The student will be able to:

5.1 Evaluate iterated integrals and use an iterated integral to find areas between two curves.

5.2 Set up and evaluate double integrals and use double integrals to find volumes.

5.3 Express double integrals in polar coordinates and evaluate them.

5.4 Use double integrals in either rectangular or polar form to solve application problems which

may include mass, moments, centers of mass, moments of inertia, and surface area.

5.5 Set up and evaluate triple integrals and use triple integrals to solve application problems which may include volumes, centers of mass and moments.

5.6 Set up and evaluate triple integrals in cylindrical and spherical coordinates.

TOPIC 6: MATHEMATICAL REASONING

One intention of the MA253 course is to give a student insight into mathematical proof. Understanding theorem statements, writing short proofs, and proving properties of vector operations are included.

Student Learning Outcomes:

The student will be able to:

6.1 Prove at least one property of vector operations and/or vector-valued functions.

6.2 Analyze the applicability of theorems, including the Second Partials Test.

MA 253 Teaching Guide

TITLE: Calculus 3

CATALOG NUMBER: MA253

CREDIT HOURS: 4

LAB HOURS: 0

PREREQUISITES: Calculus 2

CATALOG

DESCRIPTION: This is the third in a sequence of three courses in calculus. Topics include polar and space coordinates, parametric equations, the algebra and calculus of vectors, partial differentiation, and multiple integration, Applications are included. Prerequisite: MA152 Calculus 2.

TEXT: Contemporary Calculus, Dale Hoffman

Technology, including the use of graphing calculators and/or computer software, should be incorporated into the course when appropriate.

SOME PRELIMINARY COMMENTS:

The Calculus 1, 2 and 3 sequence is intended to give the student an insight into mathematical thought and proof. Although throughout the sequence there is an increasing expectation that the student acquires the discipline connected with systematic mathematical thinking, she/he is not expected to become a "theorem prover". The instructor should strive to achieve a reasonable balance between the manipulative and the theoretical aspects of the course.

TOPICS:

Chapter 9 Polar and Parametric Coordinates 12 hours

9.1 Polar Coordinates

9.2 Calculus with Polar Coordinates

9.3 Parametric Equations

9.4 Calculus with Parametric Equations

9.5 Conic Sections (Omit)

9.6 Properties of Conic Sections (Omit)

Chapter 11 Vectors, Lines and Planes in 3D 8 hours

11.1 Vectors in the Plane

11.2 Rectangular Coordinates in 3D

11.3 Vectors in 3D

11.4 Dot Product

11.5 Cross Product

11.6 Lines and Planes in 3D

Chapter 12 Vector-Valued Functions 6 hours

12.0 Introduction to Vector-Valued Functions

12.1 Vector-Valued Functions and Curves in Space

Note: The discussion of Bezier Curves in Three Dimensions is optional.

2. Derivatives and Antiderivatives of Vector-Valued Functions

3. Arc Length and Curvature of Space Curves

4. Cylindrical and Spherical Coordinate Systems in 3D

Note: This section may be covered after Section 14.6

Chapter 13 Functions of Several Variables 13 hours

13.0 Introduction to Functions of Several Variables

13.1 Functions of Two or More Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Tangent planes and Differentials

13.5 Directional Derivatives and the Gradient

13.6 Maximums and Minimums

13.7 Lagrange Multiplier Method

13.8 Chain Rule

Note: The process illustrated in Example 3 on P.4 is optional.

Chapter 14 Double and Triple Integrals 15 hours

14.0 Introduction to Double Integrals

14.1 Double Integrals over Rectangular Domains

14.2 Double Integrals over General Domains

14.3 Double Integrals in Polar Coordinates

14.4 Applications of Double Integrals (optional)

Note: It might be the case that the student has not been introduced to moments or center of mass.

14.5 Surface Area (Optional)

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Changing Coordinates in Double and Triple Integrals

Note: The development of the double integral in polar coordinates, and the triple integrals in cylindrical and spherical coordinates using the Jacobian is an appropriate alternative to the traditional method using Riemann sums.

Note: At the discretion of the instructor, fewer hours may be used for coverage of the material in the above chapters, with the additional time gained used to cover optional topics from Chapter 15 below.

Chapter 15 Vector Calculus (Optional)

15.0 Introduction to Vector Calculus

15.1 Vector Fields

15.2 Divergence, Curl and Del in 2D

15.3 Line Integrals

15.4 Fundamental Theorem of Line Integrals and Potential Functions

15.4.5 Theorems of Green, Stokes and Gauss: Discrete Introdutions

15.5 Green's Theorem

15.6 Divergence and Curl in 3D

15.7 Parametric Surfaces

15.8 Surface Integrals

15.9 Stokes' Theorem

15.10 Gauss/Divergence Theorem

Other Requirements

The teaching guide allows four additional hours for the in-class assessment of student learning. A two hour comprehensive final examination will also be given.

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