20804232Calculus and Analytic Geometry 2
Course Information
Description
Designed for students of mathematics, science, and engineering. Topics include the techniques of integration, analysis of infinite sequences and series, an introduction to first-order differential equations, parametric equations and derivatives of parametric curves, polar coordinates in the plane and integrals using polar coordinates, the analytic geometry of the conic sections, an introduction to vectors in two and three dimensions, scalar and vector cross products, and graphs of quadric surfaces.
Total Credits
5
Course Competencies
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Evaluate integrals using basic integration techniquesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaselect the basic integration formula to use for evaluating a given integralevaluate integrals using substitution
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Evaluate integrals using advanced integration techniquesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaevaluate integrals using integration by partsevaluate integrals involving powers of trigonometric functionsevaluate integrals involving sine-cosine products with different anglesevaluate integrals by trigonometric substitutionevaluate integrals by use partial fractions with linear factors
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Analyze improper integralsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriadetermine the convergence or divergence of improper integralsevaluate the improper integral if it is convergent
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Analyze infinite sequencesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaidentify the nth term of a sequencedetermine whether an infinite sequence is convergent or divergentfind the limit of a convergent sequenceapply the Monotone Sequence Theorem whenever appropriate
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Analyze infinite seriesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaidentify geometric seriesdetermine whether a geometric series is convergent or divergentfind the sum of a geometric series when it is convergentprove a series is divergent using the Divergence Testuse the Integral Test to determine the convergence or divergence of a series and to estimate the sumdetermine the convergence or divergence of p-seriesuse the Comparison Test and the Limit Comparison Test to discuss the convergence or divergence of a series with positive termsuse the Ratio Test and the Root Test to discuss the convergence or divergence of a series with positive termsuse the Alternating Series Test to determine whether a series is convergentprove a series is convergent by proving that it is absolutely convergentdetermine if a series is convergent conditionally or absolutely
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Analyze power seriesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriafind the radius of convergence of a power seriesfind the interval of convergence of a power seriesdifferentiate a series term-by-term within the interval of convergenceintegrate a series term-by-term within the interval of convergencefind Taylor series of elementary functionsfind Maclaurin series of elementary functionsfind the Taylor Series of an analytic function at a given pointfind the product of power seriesfind the quotient of power seriescalculate limits using Taylor series
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Analyze first-order differential equationsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaverify that a function satisfies a differential equationuse the direction field to sketch the graph of the solutions of a differential equationsolve separable first-order differential equationsset up the initial-value problem which models the exponential growth or decaysolve the initial-value problem which models the exponential growth or decaycompute integrating factors to solve linear first-order differential equationsuse integrating factors to solve linear first-order differential equationsset up an initial-value problem modeling the population growth
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Analyze plane curves using parametric equationsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriasketch the curve represented by parametric equationsgive direction of the motion represented by parametric equationsevaluate the first derivative with respect to xevaluate the second derivative with respect to xfind the equation of the tangent line to the curvefind all the points where the curve has a horizontal tangentfind all the points where the curve has a vertical tangentfind the arc length of a curve in parametric equation on a given intervalfind 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 coordinatesAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaplot a point in polar coordinatesfind the corresponding rectangular coordinates to a point in polar coordinatesfind the corresponding polar coordinates to a point in rectangular coordinatesconvert rectangular equation to a polar formfind dy/ds of a given polar equationfind all the points of horizontal or vertical tangent line to a polar curvesketch the graph of a polar equationuse integrals to find the area of the region bounded by the graphs of polar equationsfind points of intersection of the graphs of polar equationsfind the length of the graph of a given polar equation over a specified intervalfind the area of the surface formed by revolving a curve in polar equation about a given axis
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Analyze conic sectionsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriatranslate geometric information of conic sections into an analytic representation in rectangular coordinates, polar coordinates, or parametric formextract geometric information (eccentricity, foci, vertices, asymptotes, etc.) from the analytic representations of a conic sectionconstruct graphs of conic sections represented in rectangular, polar, or parametric forminterpret graphs of conic sections represented in rectangular, polar, or parametric formidentify geometric and analytic properties of a conic section by its graphinfer geometric and analytic properties of a conic section by its graph
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Explore three-dimensional coordinate systemsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriasketch points, lines, and planes in the rectangular coordinate systemfind the distance between two points in the space by using the distance formulafind the equation of a sphere with a given center and radiusplot points in cylindrical coordinate systemsplot points in spherical coordinate systemsinterchange between rectangular coordinates, cylindrical coordinates, and spherical coordinates of a pointinterchange the form of an equation in rectangular coordinates, cylindrical coordinates, and spherical coordinatessketch graphs in cylindrical coordinate systemssketch graphs in spherical coordinate systems
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Explore vectorsAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriafind the position vector of a pointfind the magnitude (length) of a vectorfind the direction of a vectordetermine whether vectors are equaladd vectorssubtract vectorsillustrate vector addition by using the Triangle Law and the Parallelogram Lawperform the operation of the multiplication of a vector by a scalarexpress any vector in terms of the standard basic vectors i, j, and kfind the dot product of two vectorsfind the angle between two vectorsdetermine whether two vectors are orthogonalfind the directional cosines of a vectorfind the scalar projection and the vector projection of a vector onto anotherfind the cross product of two vectors by using determinantuse cross product to find the area of the parallelogram determined by two vectorsuse cross product to determine whether two vectors are parallelcalculate the scalar triple product of three vectorsfind the volume of the parallelepiped determined by three vectors
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Analyze linear equations in three-dimensional spaceAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriafind the vector equation of a linefind the parametric equations of a linefind the symmetric equations of a linefind the vector equation of a planefind the scalar equation of a planefind the linear equation of a planecompute the angles between lines and planesfind distance between pointsfind distance between linesfind distance between planes
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Distinguish surfaces in three-dimensional spaceAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteriaidentify cylinderssketch cylindersidentify quadric surfacesuse traces to sketch quadric surfaces
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(Optional) Operate with and on space curves defined as vector functions of one variableAssessment StrategiesQuiz, Exam, Written Product, and/or ProjectsCriteria(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