After completing this section, students should be able to do the following.State the definition of a function.Find the domain and range of a function.Distinguish between functions by considering their domains.Determine where a function is positive or negative.Plot basic functions.Perform basic operations and compositions on functions.Work with piecewise defined functions.Determine if a function is one-to-one.Recognize different representations of the same function.Define and work with inverse functions.Plot inverses of basic functions.Find inverse functions (algebraically and graphically).Find the largest interval containing a given point where the function is invertible.Determine the intervals on which a function has an inverse.

After completing this section, students should be able to do the following.Recognize when a limit is indicating there is a vertical asymptote.Evaluate the limit as xx approaches a point where there is a vertical asymptote.Match graphs of functions with their equations based on vertical asymptotes.Discuss what it means for a limit to equal ∞∞.Define a vertical asymptote.Find horizontal asymptotes using limits.Produce a function with given asymptotic behavior.Recognize that a curve can cross a horizontal asymptote.Understand the relationship between limits and vertical asymptotes.Calculate the limit as xx approaches ±∞±∞ of common functions algebraically.Find the limit as xx approaches ±∞±∞ from a graph.Define a horizontal asymptote.Compute limits at infinity of famous functions.Find vertical asymptotes of famous functions.Identify horizontal asymptotes by looking at a graph.Identify vertical asymptotes by looking at a graph.

After completing this section, students should be able to do the following.Consider values of a function at inputs approaching a given point.Understand the concept of a limit.Use limits to understand local behavior of functions.Calculate limits from a graph (or state that the limit does not exist).Understand possible issues when estimating limits using nearby values.Define a one-sided limit.Explain the relationship between one-sided and two-sided limits.Distinguish between limit values and function values.Identify when a limit does not exist.Define continuity in terms of limits.Use the continuity of famous functions (on their domains) when computing limits.

After completing this section, students should be able to do the following.Determine when a function is a composition of two or more functions.Calculate indefinite and definite integrals requiring complicated substitutions.Recognize common patterns in substitutions.Evaluate indefinite and definite integrals through a change of variables.

After completing this section, students should be able to do the following.Determine if a series converges absolutely or conditionally.Answer conceptual questions about absolute convergence 

After completing this section, students should be able to do the following.Apply the procedure of “Slice, Approximate, Integrate” to derive a formula for volume of solids with known cross-section areasSet up an integral or sum of integrals that gives the volume of a solid whose cross sections are familiar geometric shapes.

After completing this section, students should be able to do the following.Determine if a series converges using the alternating series test.Determine if a series converges absolutely.Determine if a series converges conditionally.Determine if an alternating series diverges.

After completing this section, students should be able to do the following.Understand linear density and its connection to mass.Calculate the mass of an objection with varying density.Understand work and how it is computed.Calculate work when force is varying.Know when to integrate a cross-section to solve a physics problem.Calculate work when distance is varying.

After completing this section, students should be able to do the following.Compute Taylor polynomials.Use Taylor’s theorem to estimate the error of a Taylor polynomial.Determine the maximum error between a function and a given Taylor polynomial.

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 area bounded by given curves.Understand the difference between net and total area.Find the area bounded by several curves.Set up an integral or sum of integrals with respect to xx that gives the area bounded by several curves.Set up an integral or sum of integrals with respect to yy that gives the area bounded by several curves.Decide whether to integrate with respect to xx or yy.

After completing this section, students should be able to do the following.Compute derivatives of common functions.Compute antiderivatives of common functions.Understand the relationship between derivatives and antiderivatives.Use algebra to manipulate the integrand.Evaluate indefinite and definite integrals through a change of variables.Evaluate integrals that require complicated substitutions.Recognize common patterns in substitutions.

After completing this section, students should be able to do the following.Use Taylor series to read-off derivatives of a function.Use Taylor series to solve differential equations.Use Taylor series to compute integrals.

With one input, and vector outputs, we work component-wise.A question I’ve often asked myself is: “How do you know when you are doing a calculus problem?” The answer, I think, is that you are doing a calculus problem when you are computing: a limit, a derivative, or an integral. Now we are going to do calculus with vector-valued functions. To build a theory of calculus for vector-valued functions, we simply treat each component of a vector-valued function as a regular, single-variable function. Since we are currently thinking about vector-valued functions that only have a single input, we can work component-wise. Let’s see this in action.

After completing this section, students should be able to do the following.Use the comparison test to determine if a series diverges or converges.Use the limit comparison test to determine if a series diverges or converges.

After completing this section, students should be able to do the following.Define the cross product.Compute cross products.Use cross products in appled settings.

After completing this section, students should be able to do the following.Compute derivatives of polar curves.Determine where the derivative of a polar curve is undefined.Find the equation of a tangent lines to a polar curve.

After completing this section, students should be able to do the following.Identify a differential equation.Verify a solution to a differential equation.Compute a general solution to a differential equation via integration.Solve initial value problems.Determine the order of a differential equation. 

After completing this section, students should be able to do the following.Compute dot products.Use dot products to compute the angle between vectors.Find orthogonal projections.Find scalar projections.Use the dot product in applied settings.Find orthogonal decompositions.

After completing this section, students should be able to do the following.Compute derivatives of common functions.Compute antiderivatives of common functions.Understand the relationship between derivatives and antiderivatives.Use algebra to manipulate the integrand.Evaluate indefinite and definite integrals through a change of variables.Evaluate integrals that require complicated substitutions.Recognize common patterns in substitutions.

After completing this section, students should be able to do the following.Identify an improper integral.Determine if an improper integral converges or diverges.Compute integrals over infinite intervals.Compute integrals of functions with vertical asymptotes.