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# MAT 219 - Calculus III

Course Department: Mathematics
Last Date of Approval: Fall 2023

4 Credits
Total Lecture Hours: 60
Total Lab Hours: 0
Total Clinical Hours: 0
Total Work-Based Experience Hours: 0

Course Description:
This is the third course of the calculus sequence. It contains the study of vector-valued functions, functions of several variables, multiple integration, and vector analysis. This course will also help students gain mathematical literacy which will be of vital significance when making important life decisions. In addition, this course will help with any career that involves mathematics, decision making, or problem-solving.

Prerequisites: MAT 216 - Calculus II  or equivalent with C grade or better.
Mode(s) of Instruction: face-to-face, virtual

Credit for Prior Learning: There are no Credit for Prior Learning opportunities for this course.

Course Fees: ebook/Access Code: \$124.99 (charged once per term for all courses that use Cengage Unlimited)

Common Course Assessment(s): None

Student Learning Outcomes and Objectives:
Outcome 1: Solve calculus problems involving vector valued functions.
Task 1:    Interpret function values as vectors and curves as endpoints of vectors.
Task 2:    Differentiate vector valued functions.
Task 3:    Integrate vector valued functions.
Task 4:    Use differentiation and integration to find position, velocity, speed, acceleration, and direction of motion of vector valued functions.
Task 5:    Model ideal projectile motion in both vector and parametric forms.
Task 6:    Determine height, flight time, and range for ideal projectile motion.
Task 7:    Find curve length for vector valued functions.
Task 8:    Find the unit tangent and unit normal vectors to a curve at a point.
Task 9:    Evaluate and interpret the curvature of a vector valued function.
Task 10: Find the tangential and normal components of acceleration for a vector valued function.

Outcome 2: Solve algebra and calculus problems related to functions of two or more variables.
Task 1:    Discuss the definition of a function of two or more variables, including the domain and range.
Task 2:    Discuss the new terminology associated with discussions of domain and range, including interior points, boundary points, open regions, closed regions, bounded regions, and unbounded regions.
Task 3:    Be able to sketch a surface in space by utilizing its xy-trace, xz-trace, and yz-trace.
Task 4:    Use computer software or a graphics calculator to graph a surface in space.
Task 5:    Be able to sketch the level curves at a function in two variables.
Task 6:    Be able to sketch the contour lines of a function in two variables.
Task 7:    Be able to sketch the level surface for a function in three variables.
Task 8:    Use computer software to graph level curves and level surfaces.
Task 9:    Be able to calculate the limit of a function of two or more variables using the multiple path approach.
Task 10: Discuss the formal definition of the limit utilizing the terminology developed for multi-variable calculus.
Task 11: Apply the properties of the limit.
Task 12: Be able to apply the criteria for continuity of a function of two or more variables to determine whether a function is continuous at a point.
Task 13: Relate the limit of a function to the definition of the partial derivatives.
Task 14: Discuss the various notations used to express partial derivatives.
Task 15: Be able to calculate all partial derivatives for a function of two or more variables.
Task 16: Be able to calculate second-order partial derivatives.
Task 17: Be able to calculate mixed partial derivatives.
Task 18: Use the Chain Rule to calculate derivatives for functions of the form y=f(x(t), y(t)) or y=f(x(t), y(t), z(t)).
Task 19: Use the Chain Rule to calculate derivatives for functions of the form y=f(g(r,s), h(r,s), k(r,s)).

Outcome 3: Use the partial derivative to solve problems related to functions of two or more variables.
Task 1:    Use the notation associated with the directional derivative and the gradient.
Task 2:    Be able to calculate the directional derivation of a function at a point, in the direction of some unit vector u.
Task 3:    Be able to calculate the gradient of a function at some point.
Task 4:    Demonstrate that a function increases most rapidly at any point in its domain in the direction of its gradient.
Task 5:    Demonstrate that the gradient is always normal to the level curve of a function.
Task 6:    Be able to use the algebraic properties of the gradient.
Task 7:    Be able to calculate the equation of the tangent plane to a level surface.
Task 8:    Be able to calculate the parametric equations of the normal line to a surface.
Task 9:    Be able to calculate the equation of the tangent line to a level curve.
Task 10: Calculate the linearization of a function of two variables, and discuss possible error in this linearization.
Task 11: Calculate critical points for a function of two variables.
Task 12: Use the First Derivative Test to determine local or absolute extrema.
Task 13: Calculate the discriminant, fxxfyy - f2xy, for a function f(x,y).
Task 14: Use the Second Derivative Test to determine local extrema, or saddle points.
Task 15: Use the method of Lagrange multipliers to find extrema of constrained functions.

Outcome 4: Use multiple integrals to solve problems related to functions of two or more variables.
Task 1:    Discuss the development of the double integral.
Task 2:    Discuss the properties of the double integrals.
Task 3:    Calculate double integrals as iterated integrals.
Task 4:    Use computer software to calculate double integrals.
Task 5:    Use iterated integrals to calculate the area of a region.
Task 6:    Use iterated integrals to calculate the volume of a solid.
Task 7:    Use Fubini’s Theorem to show that changing the order of integration does not change the result of the iterated integral.
Task 8:    Be able to interchange the order of integration for a double integral.
Task 9:    Use the Jacobian to transform a double integral.
Task 10: Be able to calculate the area of a bounded region.
Task 11: Be able to calculate the first and second moments and centers of mass.

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