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Operate with and on space curves defined as vector functions of one variable
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
set up and/or compute the arc length integral of a vector function describing a space curve
set up and/or compute the curvature of a vector function describing a space curve
set up and/or compute the torsion of a vector function describing a space curve
set up and/or compute the unit tangent vector of a vector function describing a space curve
set up and/or compute the unit normal vector of a vector function describing a space curve
set up and/or compute the unit binomial vector of a vector function describing a space curve
compute the velocity vector of a position vector given as a function of time
compute the acceleration vector of a position vector given as a function of time
determine the normal and tangential components of the acceleration from a position vector given as a function of time
make appropriate connections between the kinematic description of a space curve and the unit tangent vector
make appropriate connections between the kinematic description of a space curve and the normal vector
make appropriate connections between the kinematic description of a space curve and the binomial vector
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Interpret three-dimensional coordinates
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
recognize rectangular coordinate systems
recognize cylindrical coordinate systems
recognize spherical coordinate systems
distinguish between rectangular, cylindrical and spherical coordinate systems
transform coordinates from any one of these coordinate systems to any other
make the connection between the equation of a three-dimensional space curve and its graph in any one of these coordinate systems
construct the graph of a three-dimensional space curve described by an equation in any one of these coordinate systems
use the appropriate coordinate system in setting up a problem
use the appropriate coordinate system in solving a problem
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Use and interpret Kepler's three laws of planetary motion
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
demonstrate the connection between Kepler's laws of planetary motion and an attractive central inverse square law force field
use Kepler's laws to solve applied problems in planetary motion
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Generate level curves and level surfaces
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
sketch the surfaces and level curves of a variety of functions of two variables
interpret properties about a function of two variables given the graph of its surface and/or level curves
connect the information found in the graph of the surface, the graphs of the level curves, and the formula for the surface
generate the surface and level curves of a function of two variables using numerical or graphical methods
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Compute limits of multivariate functions
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
compute the limits of multivariable functions
determine whether a multivariable function is continuous at a point
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Compute partial derivatives of multivariate functions
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
compute symbolically the partial derivatives of multivariable functions
relate this new process of differentiation to the earlier rules of differentiation
compute symbolically higher order partial derivatives of multivariable functions
compute symbolically mixed partial derivatives of multivariable functions
demonstrate the property of the total differential of a function
connect the property of the total differential of a function to the equation of a tangent plane
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Use the gradient of a multivariate function
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
compute the gradient of a multivariable function
relate the gradient of a multivariable function vector to the total differential
relate the gradient of a multivariable function vector to the equations of the tangent plane and normal line
relate the gradient of a multivariable function vector to the directional derivative
demonstrate the connection between the equation of the tangent plane and the linear approximation to a function
demonstrate the connections between the gradient vector and the geometry of the level curves of the function under study
demonstrate the connections between the gradient vector and the geometry of the level surfaces of the function under study
demonstrate the connections between the directional derivative and the geometry of the level curves of the function under study
demonstrate the connections between the directional derivative and the geometry of the level surfaces of the function under study
observe and record the properties about a function and its gradient given a graph of the level curves or level surfaces
compute derivatives using a multivariable form of the chain rule
connect the chain rule to the total differential and the gradient
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Locate the maximum and minimum values of a multivariate function
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
analyze multivariable functions for maxima
analyze multivariable functions for minima
analyze multivariable functions for saddle points
construct a Taylor series for a function of two variables to determine whether a critical point is an extremum or a saddle
locate the extremum of a function subject to constraints using Lagrange multipliers
demonstrate the connection between the method of Lagrange multipliers and the geometry of the problem
solve problems of finding the extremum in a variety of verbally stated applications
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Evaluate double and triple integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
recognize double integrals in rectangular coordinates
recognize double integrals in polar coordinates
formulate double integrals in rectangular coordinates
formulate double integrals in polar coordinates
evaluate double integrals in rectangular coordinates
evaluate double integrals in polar coordinates
recognize triple integrals in rectangular coordinates
recognize triple integrals in cylindrical coordinates
recognize triple integrals in spherical coordinates
formulate triple integrals in rectangular coordinates
formulate triple integrals in cylindrical coordinates
formulate triple integrals in spherical coordinates
evaluate triple integrals in rectangular coordinates
evaluate triple integrals in cylindrical coordinates
evaluate triple integrals in spherical coordinates
use Fubini's Theorem to simplify multiple integrals
use Fubini's Theorem to evaluate multiple integrals
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Transform double and triple integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
transform double integrals between rectangular and polar coordinates
transform triple integrals between rectangular, cylindrical and spherical coordinates
properly map the limits of integration when a transformation is used to evaluate a multiple integral
demonstrate the connection in two dimensions between the Jacobian determinant and the area of a parallelogram
demonstrate the connection in three dimensions between the Jacobian determinant and the volume of a parallelepiped
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Solve applications using double and triple integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
set up and/or evaluate the double integral for the surface area of a surface defined as an explicit function of two variables
set up multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc
evaluate multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc
interpret multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc
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Evaluate line integrals in two and three dimensions
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
recognize line integrals in two and three dimensions
formulate line integrals in two and three dimensions
evaluate line integrals in two and three dimensions
recognize line integrals representing the flow of vector fields in two dimensions
recognize line integrals representing the circulation of vector fields in two dimensions
recognize line integrals representing the flux of vector fields in two dimensions
formulate line integrals representing the flow of vector fields in two dimensions
formulate line integrals representing the circulation of vector fields in two dimensions
formulate line integrals representing the flux of vector fields in two dimensions
evaluate line integrals representing the flow of vector fields in two dimensions
evaluate line integrals representing the circulation of vector fields in two dimensions
evaluate line integrals representing the flux of vector fields in two dimensions
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Evaluate surface integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
produce parametric equations which generate a given surface
set up and/or evaluate the double integral for the surface area
set up and/or evaluate the double integral for the flux of a three dimensional vector field across a surface
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Use the curl and divergence of a vector field
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
compute the curl and divergence of three-dimensional vector fields
demonstrate the connections between the curl and the circulation of a vector field
demonstrate the connections between the divergence and the flux of a vector field
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Use Green's, Gauss's and Stokes' theorems
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
Green's Theorem to simplify circulation and flux integrals in two dimensions
use Green's Theorem to evaluate circulation and flux integrals in two dimensions
use Stokes' Theorem to simplify circulation integrals in three dimensions
use Stokes' Theorem to evaluate circulation integrals in three dimensions
uses auss's Theorem to simplify flux integrals in three dimensions
use Gauss's Theorem to evaluate flux integrals in three dimensions
demonstrate the connections between Green's Theorem in two dimensions, Stokes' Theorem and Gauss's Theorem in three dimensions and the Fundamental Theorem of calculus in one dimension
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Apply vector calculus
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
determine whether a vector field is conservative
you demonstrate the connections between conservative vector fields, the gradient operator, the curl operator, and Stokes' Theorem
set up line integrals which calculate mechanical work
evaluate line integrals which calculate mechanical work
interpret line integrals which calculate mechanical work
set up line and surface integrals in applications to fluid mechanics
set up line and surface integrals in applications to electromagnetic fields
evaluate line and surface integrals in applications to fluid mechanics
evaluate line and surface integrals in applications to electromagnetic fields
interpret line and surface integrals in applications to fluid mechanics
interpret line and surface integrals in applications to electromagnetic fields