After completing this section, students should be able to do the following.Use the integral test to determine that a series diverges.Use integrals to estimate the value of a series within an error.Use the divergence test to determine that a series diverges.Manipulate series using the properties of convergent series.Recognize known convergent or divergent series.

After completing this section, students should be able to do the following.Find the area of a region bound by a polar curve.Find the intersection points of two polar curves.Find the area of a region bound by two polar curves.

After completing this section, students should be able to do the following.Describe the relationship between the product rule and integration by parts.Identify the functions required to transform the integral via integration by parts.Evaluate integrals using integration by parts, including multiple iterations.Use integration by parts to derive reduction formulas.

After completing this section, students should be able to do the following.Give the definition of a Maclaurin series.Give the definition of a Taylor series.Find the Maclaurin/Taylor series of a function.Use given Maclaurin/Taylor series to find new power series.Find the interval and radius of convergence of a Maclaurin/Taylor series. 

After completing this section, students should be able to do the following.Convert between polar and Cartesian coordinates.Plot basic curves in polar coordinates.Use the Cartesian to polar method to plot polar graphs. 

After completing this section, students should be able to do the following.Apply the procedure of “Slice, Approximate, Integrate” to derive a formula for the length of a curve segment.Understand the connection between the Pythagorean Theorem and the length of curves.Set up an integral or sum of integrals with respect to xx and yy that gives the length of a curve segment.Evaluate integrals that involve square roots.

After completing this section, students should be able to do the following.Use Taylor series to approximate constants.Use Taylor series to compute limits.Use Taylor series to sum series. 

After completing this section, students should be able to do the following.Understand what a slope field is.Understand what a slope field tells us about a differential equation.Define an autonomous differential equation.Understand and use Euler’s method.Sketch slope fields.Find equilibrium solutions.

After completing this section, students should be able to do the following.Sketch a parametric curve.Eliminate a parameters of a parametric equation.Represent a graph with parametric equations.Find the derivative of a parametric curve. 

After completing this section, students should be able to do the following.Understand how to break up fractions.Recognize integrals that are good candidates for the method of partial fractions.Use long-division to simplify rational expressions.Find the coefficients of partial fraction decomposition.Use partial fractions to integrate functions.

After completing this section, students should be able to do the following.Give the definition of a power series.Find the interval and radius of convergence of a power series.Express functions as power series.Express power series as closed-form functions.Differentiate and integrate power series.

The various concepts associated with sequences and series are reviewed.All of the series convergence tests we have used require that the underlying sequence (an)(an) be a positive sequence. We can actually relax this and state that there must be an N>0N>0 such that an>0an>0 for all n>Nn>N; that is, (an)(an) is positive for all but a finite number of values of nn. We’ve also stated this by saying that the tail of the sequence must have positive terms. In this section we explore series whose summation includes negative terms.

After completing this section, students should be able to do the following.Use the ratio test to determine if a series diverges or converges.Use the root test to determine if a series diverges or converges.Determine when to use the ratio or root test.

After completing this section, students should be able to do the following.Identify separable differential equations.Solve separable differential equations.Find implicit solutions for separable differential equations.Identify steady-state (or equilibrium) solutions.

After completing this section, students should be able to do the following.Define a sequence.Write the first several terms of a sequence using an explicit formula.Write the first several terms of a sequence using a recurrence relation.Find an explicit formula for a sequence given recursively.Find a recurrence relation for a sequence given explicitly.

After completing this section, students should be able to do the following.Recognize sequences can be generated by functions.Compute limits of sequences.Understand growth rates of basic sequences.Introduce important terminology for sequences.Apply the monotone convergence theorem 

After completing this section, students should be able to do the following.Understand what a solid of revolution is and the two ways to generate one.Use the procedure of “Slice, Approximate, Integrate” to derive the washer method formula.Use the procedure of “Slice, Approximate, Integrate” to derive the shell method formula.Set up an integral or sum of integrals using the washer method.Set up an integral or sum of integrals using the shell method.Compute volumes using the washer method.Compute volumes using the shell method.Determine whether to use washer or shell method given the variable of integration.Determine the variable of integration given the method.Determine if washer method or shell method is more convenient to set up a volume.

After completing this section, students should be able to do the following.Define a series.Recognize a geometric series.Recognize a telescoping series.Compute the sum of a geometric series.Compute the sum of a telescoping series.

After completing this section, students should be able to do the following.Know the definition of a frustum.Know how to compute the surface area of a frustum.Set-up an integral to compute the area of a surface of revolution in terms of xx.Set-up an integral to compute the area of a surface of revolution in terms of yy.Compute the area of a surface of revolution.