CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2415 CALCULUS III Semester Hours Credit: 4 I. INTRODUCTION A. Calculus III is a continuation course from Calculus II, which includes advanced topics in calculus, including: vectors and vector-valued functions, partial differentiation, Lagrange multipliers, multiple integrals, and Jacobians; application of the line integral, including Green's Theorem, the Divergence Theorem, and Stokes' Theorem. B. This is a required course for an Associate in Science with a major in Mathematics. C. Prerequisite: A grade of C or higher in Math 2414. II. LEARNING OUTCOMES Upon successful completion of this course, students will: A. Perform calculus operations on vector-valued functions, including derivatives, integrals, curvature, displacement, velocity, acceleration, or torsion. (F1, F3, F4, F7, F9, F10, F11) B. Perform calculus operations on functions of several variables, including partial derivatives, directional derivatives, and multiple integrals. (F1, F3, F4, F7, F9, F10, F11) C. Find extrema and tangent planes. (F1, F3, F4, F10, F11) D. Solve problems using the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem, and Stokes' Theorem. (F1, F3, F4, F7, F9, F10, F12) E. Apply the computational and conceptual principles of calculus to the solutions of real-world problems. (F1, F3, F4, F7, F9, F10, F12) III. INSTRUCTIONAL MATERIALS The Instructional materials identified for this course are viewable through www.ctcd.edu/books IV. COURSE REQUIREMENTS A. Assignments are given in WebAssign and are due as scheduled by your instructor. The instructor will monitor students progress in completing the assignments.
B. Students are expected to attend every class, to arrive at each class on time, and remain in class for the entire period. Instructors may choose to lower a student's grade because of tardiness. V. EXAMINATIONS A. Examinations will be given at appropriate points during the semester. Each examination will be announced in class in advance. There will be two unit exams and a comprehensive final exam. B. Students who miss an exam should discuss with the instructor the circumstances surrounding the absence. The instructor will determine whether a make-up exam is to be given. It is necessary to make an appointment with the instructor for a make-up exam. VI. SEMESTER GRADE COMPUTATIONS A. The semester average is derived from the homework, quizzes, unit exams, and REQUIRED comprehensive final exam in MyMathLab. You must take the final exam and score at least 50% to pass the course. Final grades will follow the grade designation below: Grade Performance A 90-100% and 50% or better on the Final Exam. B 80-89% and 50% or better on the Final Exam. C 70-79% and 50% or better on the Final Exam. D 60-69% and 50% or better on the Final Exam. F 0-59% and 50% or better on the Final Exam. VII. NOTES AND ADDITIONAL INSTRUCTIONS Withdrawal from Course: It is the student's responsibility to officially drop a class if circumstances prevent attendance. Any student who desires to, or must, officially withdraw from a course after the first scheduled class meeting must
file an Application for Withdrawal or an Application for Refund. The withdrawal form must be signed by the student. An Application for withdrawal will be accepted at any time prior to Friday of the 12th of classes during the 16- fall and spring semesters. The deadline for sessions of other lengths is as follows. Session 12- session 10- session 8- session 6- session 5- session Deadline for Withdrawal Friday of the9 th Friday of the 7 th Friday of the 6 th Friday of the 4 th Friday of the 3 rd The equivalent date (75% of the semester) will be used for sessions of other lengths. The specific last day to withdraw is published each semester in the Schedule Bulletin. Students who officially withdraw will be awarded the grade of "W" provided the student's attendance and academic performance are satisfactory at the time of official withdrawal. Students must file a withdrawal application with the college before they may be considered for withdrawal. A student may not withdraw from a class for which the instructor has previously issued the student a grade of "F". B. An Incomplete Grade: The College catalog states, "An incomplete grade may be given in those cases where the student has completed the majority of the course work but, because of personal illness, death in the immediate family, or military orders, the student is unable to complete the requirements for a course..." Prior approval from the instructor is required before the grade of "I" is recorded. A student who merely fails to show for the final examination will receive a zero for the final and an "F" for the course. C. Cellular Phones and Beepers: Cellular phones and beepers will be turned off while the student is in the classroom or laboratory. D. Americans With Disabilities Act (ADA): Disability Support Services provide services to students who have appropriate documentation of a disability. Students requiring accommodations for class are responsible for contacting the Office of
Disability Support Services (DSS) located on the central campus. This service is available to all students, regardless of location. Explore the website at www.ctcd.edu/disability-support for further information. Reasonable accommodations will be given in accordance with the federal and state laws through the DSS office. E. Civility: Individuals are expected to be cognizant of what a constructive educational experience is and respectful of those participating in a learning environment. Failure to do so can result in disciplinary action up to and including expulsion. F. Advanced Math Lab: The Math Department operates an Advanced Mathematics Lab in Building 152, Room 145. All courses offered by the Math Department are supported in the lab with appropriate tutorial software. Calculators are available for student use in the lab. Students are encouraged to take advantage of these opportunities. See posted hours for the Advanced Math Lab. G. Office Hours: Full-time instructors post office hours outside the door of the Mathematics Department (Building 152, Room 223). Part-time instructors may be available by appointment. If you have difficulty with the course work, please consult your instructor. VIII. COURSE OUTLINE A. Lesson One: Vectors and the Geometry of Space a. Write the component form of a vector. b. Perform vector operations and interpret the results geometrically. c. Write a vector as a linear combination of standard unit vectors. d. Understand the three-dimensional rectangular coordinate system. e. Analyze vectors in space. f. Use properties of the dot product of two vectors. g. Find the angle between two vectors using the dot product. h. Find the direction cosines of a vector in space. i. Find the projection of a vector onto another vector. j. Use vectors to find the work done by a constant force. k. Find the cross product of two vectors in space. l. Use the triple scalar product of three vectors in space. m. Write a set of parametric equations for a line in space. n. Write a linear equation to represent a plane in space. o. Sketch the plane given by a linear equation. p. Find the distances between points, planes, and lines in space. q. Recognize and write equations of cylindrical surfaces. r. Recognize and write equations of quadratic surfaces.
s. Recognize and write equations of surfaces of revolution. t. Use cylindrical coordinates to represent surfaces in space.
u. Use spherical coordinates to represent surfaces in space. a. Read Chapter 11. (F1) a. 11.1 Vectors in the Plane b. 11.2 Spaces Coordinates and Vectors in Space c. 11.3 The Dot Product of Two Vectors d. 11.4 The Cross Product of Two Vectors in Space e. 11.5 Lines and Planes in Space f. 11.6 Surfaces in Space g. 11.7 Cylindrical and Spherical Coordinates B. Lesson Two: Vector-Valued Functions a. Analyze and sketch a space curve given by a vector-valued function. b. Extend the concepts of limits and continuity to vector-valued functions. c. Differentiate a vector-valued function. d. Integrate a vector-valued function. e. Describe the velocity and acceleration associated with a vector-valued function. f. Use a vector-valued function to analyze projectile motion. g. Find a unit tangent vector and a principal unit normal vector at a point on a space curve. h. Find the tangential and normal components of acceleration. i. Find the arc length of a space curve. j. Use the arc length parameter to describe a plane curve or space curve. k. Find the curvature of a curve at a point on the curve. l. Use a vector-valued function to find frictional force. a. Read Chapter 12. (F1) a. 12.1 Vector-Valued Functions b. 12.2 Differentiation and Integration of Vector-Valued Functions c. 12.3 Velocity and Acceleration d. 12.4 Tangent Vectors and Normal Vectors e. 12.5 Arc Length and Curvature C. Lesson Three: Functions of Several Variables a. Understand the notation for a function of several variables.
b. Sketch the graph of a function to two variables. c. Sketch level curves for a function to two variables. d. Sketch level surfaces for a function of three variables. e. Use computer graphics to graph a function of two variables. f. Understand the definition of a neighborhood in the plane. g. Understand and use the definition of the limit of a function of two variables. h. Extend the concept of continuity to a function of two variables. i. Extend the concept of continuity to a function of three variables. j. Find and use partial derivatives of a function of two variables. k. Find and use partial derivatives of a function of three or more variables. l. Find higher-order partial derivatives of a function of two or three variables. m. Understand the concepts of increments and differentials. n. Extend the concept of differentiability to a function of two variables. o. Use a differential as an approximation. p. Use the Chain Rules for functions of several variables. q. Find partial derivatives implicitly. r. Find and use directional derivatives of a function of two variables. s. Find the gradient of a function of two variables. t. Use the gradient of a function of two variables in applications. u. Find directional derivatives and gradients of functions of three variables. v. Find equations of tangent planes and normal lines to surfaces. w. Find the angle of inclination of a plane in space. x. Compare the gradients grad(f(x,y)) and grad(f(x,y,z)). y. Find absolute and relative extrema of a function of two variables. z. Use the Second Partials Test to find relative extrema of a function of two variables. aa. Solve optimization problems involving functions of several variables. bb. Use the method of least squares. cc. Understand the Method of Lagrange Multipliers. dd. Use Lagrange multipliers to solve constrained optimization problems. ee. Use the Method of Lagrange Multipliers with two constraints. a. Read Chapter 13. (F3) a. 13.1 Introduction to Functions of Several Variables b. 13.2 Limits and Continuity c. 13.3 Partial Derivatives d. 13.4 Differentials e. 13.5 Chain Rules for Functions of Several Variables f. 13.6 Directional Derivatives and Gradients
g. 13.7 Tangent Planes and Normal Lines h. 13.8 Extrema of Functions of Two Variables i. 13.9 Applications of Extrema j. 13.10 Lagrange Multipliers D. Lesson Four: Multiple Integration a. Evaluate an iterated integral. b. Use an iterated integral to find the area of a plane region. c. Use a double integral to represent the volume of a solid region and use properties of double integrals. d. Evaluate a double integral as an iterated integral. e. Find the average value of a function over a region. f. Write and evaluate double integrals in polar coordinates. g. Find the mass of a planar lamina using a double integral. h. Find the center of mass of a planar lamina using double integrals. i. Find moments of inertia using double integrals. j. Use a double integral to find the area of a surface. k. Use a triple integral to find the volume of a solid region. l. Find the center of mass and moments of inertia of a solid region. m. Write and evaluate a triple integral in cylindrical coordinates. n. Write and evaluate a triple integral in spherical coordinates. o. Understand the concept of a Jacobian. p. Use a Jacobian to change variables in a double integral. a. Read Chapter 14. (F3) a. 14.1 Iterated Integrals and Area in the Plane b. 14.2 Double Integrals and Volume c. 14.3 Change of Variables: Polar Coordinates d. 14.4 Center of Mass and Moments of Inertia e. 14.5 Surface Area f. 14.6 Triple Integrals and Coordinates g. 14.7 Triple Integrals in Other Coordinates h. 14.8 Change of Variables: Jacobians E. Lesson Five: Vector Analysis a. Understand the concept of a vector field. b. Determine whether a vector field is conservative. c. Find the curl of a vector field.
d. Find the divergence of a vector field. e. Understand and use the concept of a piecewise smooth curve. f. Write and evaluate a line integral. g. Write and evaluate a line integral of a vector field. h. Write and evaluate a line integral in differential form. i. Understand and use the Fundamental Theorem of Line Integrals. j. Understand the concept of independence of path. k. Understand the concept of conservation of energy. l. Use Green's Theorem to evaluate a line integral. m. Use alternative forms of Green's Theorem. n. Understand the definition of a parametric surface, and sketch the surface. o. Find a set of parametric equations to represent a surface. p. Find a normal vector and a tangent plane to a parametric surface. q. Find the area of a parametric surface. r. Evaluate a surface integral as a double integral. s. Evaluate a surface integral for a parametric surface. t. Determine the orientation of a surface. u. Understand the concept of a flux integral. v. Understand and use the Divergence Theorem. w. Use the Divergence Theorem to calculate flux. x. Understand and use Stokes's Theorem. y. Use curl to analyze the motion of a rotating liquid. a. Read Chapter 14. (F3) a. 15.1 Vector Fields b. 15.2 Line Integrals c. 15.3 Conservative Vector Fields and Independence of Path d. 15.4 Green's Theorem e. 15.5 Parametric Surfaces f. 15.6 Surface Integrals g. 15.7 Divergence Theorem h. 15.8 Stokes's Theorem