MATH 251  Calculus and Analytic Geometry II4 Credit: (4 lecture, 0 lab, 0 clinical) 4 Contact Hours: [MATH 250 ] This course is the second course in a twosemester calculus sequence. Topics included are techniques of integration, applications of the definite integral, improper integrals, sequences and series, approximating functions and differential equations. This course stresses conceptual understanding and multiple ways of representing mathematical ideas. OFFERED: spring semesters
Course Goals/ Objectives/ Competencies: Goal 1: Students will apply techniques to reconstruct a function from its derivative.
Objectives: The student should be able to
 apply the graph of the derivative of some function to work backwards to reconstruct the original function + a constant
 apply the Fundamental Theorem of Calculus to determine points on F, where f = F ‘.
 describe the difference between a definite integral and an indefinite integral.
 identify functions which have elementary antiderivatives.
 determine the indefinite integrals of constants, and sums and differences of simple functions.
 determine general solutions to basic differential equations and unique solutions to initial value problems.
 apply the Second Fundamental Theorem of Calculus to define and analyze an antiderivative that is not elementary.
Goal 2: Students will apply methods for finding antiderivatives of different types of functions.
Objectives: The student should be able to
 determine antiderivatives using substitution.
 determine antiderivatives using integration by parts.
 determine antiderivatives using partial fractions.
 determine antiderivatives using trigonometric substitutions.
 transform integrals into a form necessary to enable the use of tables.
 determine antiderivatives using tables.
Goal 3: Students will apply several numerical techniques for approximating definite integrals.
Objectives: The student should be able to
 approximate definite integrals using a left hand sum, right hand sum, midpoint sum and trapezoid sum.
 apply calculator programs to approximate definite integrals using each of these techniques.
 sketch each of these approximations on a graph.
 determine whether the estimation using each of these techniques is an overestimation or an underestimation based particular characteristics of the function.
 show graphically the location of the errors for each technique.
 analyze the behavior of errors using numerical techniques to approximate definite integrals.
Goal 4: Students will analyze the convergence and divergence of improper integrals.
Objectives: The student should be able to
 identify an improper integral.
 test whether or not an improper integral converges or diverges numerically and graphically.
 determine the convergence or divergence of an improper integral.
 determine the convergence or divergence of certain classes of improper integrals.
 determine the convergence or divergence of an improper integral using the comparison test.
Goal 5: Students will apply several uses of the definite integral.
Objectives: The student should be able to
 apply the appropriate technique to set up a Riemann Sum to approximate an area or volume of a geometric figure or solid.
 determine the area or volume of the geometric figure or solid using a definite integral.
 compute the volume of a figure created by revolving a specified region around an axis using several techniques.
 determine the length of a curve.
 determine the volume of a region with a known cross section.
 determine the total population, mass, etc., given information about the density of the population or density of the object.
 determine the center of mass of a system with given density.
 determine the work done on an object.
 determine the force exerted by a liquid on a surface.
Goal 6: Students will analyze different types of series and explore the convergence of these series.
Objectives: The student should be able to
 identify finite and infinite geometric series.
 determine the sum of a geometric series.
 distinguish between a sequence and a series.
 determine the convergence or divergence of a sequence.
 describe properties that series must exhibit to be convergent.
 apply the integral test to determine the convergence or divergence of certain types of series.
 compare two series to determine convergence or divergence.
 apply the ratio test to determine convergence or divergence.
 determine the convergence or divergence of alternating series.
 determine the upper bound of the error in approximating an alternating series by a partial sum.
 determine the radius and interval of convergence of a power series.
Goal 7: Students will apply methods for approximating functions using polynomials.
Objectives: The student should be able to
 construct Taylor polynomials of degree n for approximating f(x) for x near 0 and for x near a.
 construct Taylor series for a function f(x) for x near 0 and x near a.
 determine the interval of convergence for a Taylor series.
 determine Taylor series using substitution, differentiation or integration.
 apply Taylor series for various situations.
 estimate the magnitude of the error when using an n^{th} degree Taylor polynomial to approximate f(x).
Goal 8: Students will apply different techniques for solving differential equations.
Objectives: The student should be able to
 determine the general solution to a differential equation.
 determine a solution to an initial value problem.
 describe the basic process behind constructing a slope field for a differential equation.
 visualize a solution to a first order differential equation by looking at a slope field.
 solve first order differential equations numerically using Euler’s method.
 solve first order differential equations analytically by separation of variables.
 apply differential equations to model exponential growth and decay and determine the equilibrium solution.
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