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Evaluate integrals using basic integration techniques
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
select the basic integration formula to use for evaluating a given integral
evaluate integrals using substitution
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Evaluate integrals using advanced integration techniques
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
evaluate integrals using integration by parts
evaluate integrals involving powers of trigonometric functions
evaluate integrals involving sine-cosine products with different angles
evaluate integrals by trigonometric substitution
evaluate integrals by use partial fractions with linear factors
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Analyze improper integrals
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
determine the convergence or divergence of improper integrals
evaluate the improper integral if it is convergent
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Analyze infinite sequences
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
identify the nth term of a sequence
determine whether an infinite sequence is convergent or divergent
find the limit of a convergent sequence
apply the Monotone Sequence Theorem whenever appropriate
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Analyze infinite series
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
identify geometric series
determine whether a geometric series is convergent or divergent
find the sum of a geometric series when it is convergent
prove a series is divergent using the Divergence Test
use the Integral Test to determine the convergence or divergence of a series and to estimate the sum
determine the convergence or divergence of p-series
use the Comparison Test and the Limit Comparison Test to discuss the convergence or divergence of a series with positive terms
use the Ratio Test and the Root Test to discuss the convergence or divergence of a series with positive terms
use the Alternating Series Test to determine whether a series is convergent
prove a series is convergent by proving that it is absolutely convergent
determine if a series is convergent conditionally or absolutely
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Analyze power series
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
find the radius of convergence of a power series
find the interval of convergence of a power series
differentiate a series term-by-term within the interval of convergence
integrate a series term-by-term within the interval of convergence
find Taylor series of elementary functions
find Maclaurin series of elementary functions
find the Taylor Series of an analytic function at a given point
find the product of power series
find the quotient of power series
calculate limits using Taylor series
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Analyze first-order differential equations
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
verify that a function satisfies a differential equation
use the direction field to sketch the graph of the solutions of a differential equation
solve separable first-order differential equations
set up the initial-value problem which models the exponential growth or decay
solve the initial-value problem which models the exponential growth or decay
compute integrating factors to solve linear first-order differential equations
use integrating factors to solve linear first-order differential equations
set up an initial-value problem modeling the population growth
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Analyze plane curves using parametric equations
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
sketch the curve represented by parametric equations
give direction of the motion represented by parametric equations
evaluate the first derivative with respect to x
evaluate the second derivative with respect to x
find the equation of the tangent line to the curve
find all the points where the curve has a horizontal tangent
find all the points where the curve has a vertical tangent
find the arc length of a curve in parametric equation on a given interval
find the surface area of the solid generated by rotating a curve in parametric equation about a given axis
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Analyze graphs using polar coordinates
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
plot a point in polar coordinates
find the corresponding rectangular coordinates to a point in polar coordinates
find the corresponding polar coordinates to a point in rectangular coordinates
convert rectangular equation to a polar form
find dy/ds of a given polar equation
find all the points of horizontal or vertical tangent line to a polar curve
sketch the graph of a polar equation
use integrals to find the area of the region bounded by the graphs of polar equations
find points of intersection of the graphs of polar equations
find the length of the graph of a given polar equation over a specified interval
find the area of the surface formed by revolving a curve in polar equation about a given axis
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Analyze conic sections
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
translate geometric information of conic sections into an analytic representation in rectangular coordinates, polar coordinates, or parametric form
extract geometric information (eccentricity, foci, vertices, asymptotes, etc.) from the analytic representations of a conic section
construct graphs of conic sections represented in rectangular, polar, or parametric form
interpret graphs of conic sections represented in rectangular, polar, or parametric form
identify geometric and analytic properties of a conic section by its graph
infer geometric and analytic properties of a conic section by its graph
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Explore three-dimensional coordinate systems
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
sketch points, lines, and planes in the rectangular coordinate system
find the distance between two points in the space by using the distance formula
find the equation of a sphere with a given center and radius
plot points in cylindrical coordinate systems
plot points in spherical coordinate systems
interchange between rectangular coordinates, cylindrical coordinates, and spherical coordinates of a point
interchange the form of an equation in rectangular coordinates, cylindrical coordinates, and spherical coordinates
sketch graphs in cylindrical coordinate systems
sketch graphs in spherical coordinate systems
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Explore vectors
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
find the position vector of a point
find the magnitude (length) of a vector
find the direction of a vector
determine whether vectors are equal
add vectors
subtract vectors
illustrate vector addition by using the Triangle Law and the Parallelogram Law
perform the operation of the multiplication of a vector by a scalar
express any vector in terms of the standard basic vectors i, j, and k
find the dot product of two vectors
find the angle between two vectors
determine whether two vectors are orthogonal
find the directional cosines of a vector
find the scalar projection and the vector projection of a vector onto another
find the cross product of two vectors by using determinant
use cross product to find the area of the parallelogram determined by two vectors
use cross product to determine whether two vectors are parallel
calculate the scalar triple product of three vectors
find the volume of the parallelepiped determined by three vectors
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Analyze linear equations in three-dimensional space
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
find the vector equation of a line
find the parametric equations of a line
find the symmetric equations of a line
find the vector equation of a plane
find the scalar equation of a plane
find the linear equation of a plane
compute the angles between lines and planes
find distance between points
find distance between lines
find distance between planes
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Distinguish surfaces in three-dimensional space
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
identify cylinders
sketch cylinders
identify quadric surfaces
use traces to sketch quadric surfaces
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(Optional) Operate with and on space curves defined as vector functions of one variable
Assessment Strategies
Quiz, Exam, Written Product, and/or Projects
Criteria
(optional) compute the velocity vector of a position vector given as a function of time
(optional) compute the acceleration vector of a position vector given as a function of time
(optional) set up and/or compute the arc length integral of a vector function describing a space curve
(optional) set up and/or compute the unit tangent vector of a vector function describing a space curve
(optional) you set up and/or compute the curvature of a vector function describing a space curve
(optional) you set up and/or compute the unit binomial vector of a vector function describing a space curve
(optional) you set up and/or compute the unit normal vector of a vector function describing a space curve
(optional) you set up and/or compute the torsion of a vector function describing a space curve
(optional) you make appropriate connections between the kinematic description of a space curve and the binomial vector
(optional) you determine the normal and tangential components of the acceleration from a position vector given as a function of time
(optional) you make appropriate connections between the kinematic description of a space curve and the unit tangent vector
(optional) you make appropriate connections between the kinematic description of a space curve and the normal vector