Calculus 2

Transcendental Functions | First Order Differential Equations |Integration Techniques | Applications of Integration | Sequences and Series | Conics, Parametric Equations, Polar Coordinates | Second Order Differential Equations

The Mathematics Department has a standard of performance expected of calculus students. The following pages describe in detail what you need to be able to do and what you need to understand to receive an A in Calculus 2. This provides an excellent checklist you can use each day to make sure you have learned the material well enough. Before an exam you can use the list of expectations as a checklist to make sure you have reviewed everything.

- Natural Logarithm
- Use ln x as an antiderivative for integration purposes, in particular, to integrate tan x, cot x, sec x, and csc x.
- Definition of e
- State the definition of e as a limit. For the final exam, also state the definition of e as the sum of an infinite series.
- Inverse Trig Functions
- Integrate using the inverse trig functions.

- Rudiments of Differential Equations
- Classify a differential equation with regard to order, linear/nonlinear, homogeneous/nonhomogeneous. Identify and interpret an initial value problem. Know how to draw solution curves and interpret a slope field. Do the iteration step in Euler's method.
- Exponential Growth and Decay
- State the differential equation defining exponential growth or decay and interpret what it says in terms of rate of change and proportionality. Solve this equation by separation of variables. Know the definition of half-life. Solve problems modeled as exponential growth or decay such as radioactive decay or population growth.
- Separation of Variables
- Use separation of variables to solve a first order differential equation or initial value problem.
- First Order Linear Differential Equations
- Find an integrating factor and use it to solve a first order linear differential equation.

- Area of a Region Between Two Curves
- Set up the area element rectangle (vertical or horizontal) for the area between two curves. Set up the integral for such an area and evaluate it.
- Volumes
- Set up the volume element (disc or washer) for a representative cross section of a solid. Set up the integral for the volume of such a solid and evaluate it. Do the preceding for other solids with known cross sections. Set up the volume element (cylindrical shell) for a solid of revolution. Set up the integral for the volume of such a solid and evaluate it.
- Arc Length and Surface Area
- Draw the diagram illustrating how the integral for arc length is derived. State the integral definition for arc length and evaluate (when possible). Set up the surface area element (frustrum of a cone) for a surface of revolution. Evaluate a surface area integral (when possible).
- Work
- State the definition of work done by a constant force. Set up the work element and limiting sum that defines the work done by a variable force as an integral. Use Hooke's law, Newton's law of universal gravitation, and Coulomb's law for modeling work problems. Solve them.
- Moments
- Determine the moment about a point and the center of mass for a finite set of point masses (arranged on a line). For the same point masses compute the torque about the origin and the center of gravity. Determine the moments about the x-axis, y-axis, and the center of mass (center of gravity) for a two-dimensional finite set of point masses. For a planar lamina of constant density, set up the moment elements and integrals for computing its moments about the x-axis and y-axis. Also compute its center of mass. State the definition for the centroid of a region in the plane and compute the centroid. State and apply the theorem of Pappus for the volume of a solid formed by revolving a plane region around a line.

- Integration by Parts
- Identify integrals that require integration by parts. Analyze the integrand to decide which part to differentiate and which part to integrate. Apply integration by parts. Apply integration by parts repeatedly to an integral that requires it and also use the tabular method for such integrals.
- Trigonometric Integrals
- Apply u-substitution and trig identities to evaluate integrals containing powers of sine and cosine, powers of secant and tangent, and sine-cosine products with different angles. Know Wallis's formulas.
- Trig Substitution
- Identify integrals that require trig substitution. Analyze the integrand of such integrals to identify the appropriate substitution and set up the substitution triangle. Evaluate the resulting trigonometric integrals.
- Partial Fractions
- Identify integrals that require partial fractions. Perform partial fraction decomposition of rational functions.
- Indeterminate Forms
- State the Cauchy (extended) mean value theorem. Identify indeterminate forms and determinate forms. Know when to apply L'Hôpital's rule and when not to use it. Apply L'Hôpital's rule. Apply L'Hôpital's rule more than once, if necessary, to evaluate one limit.
- Improper Integrals
- Identify an integral that is improper: one or both limits of integration are infinite or the integrand has an infinite discontinuity in the interval of integration. State the definition of each kind of improper integral. Apply integration techniques and L'Hôpital's rule to evaluate an improper integral. Know which of the 1/x
^{p}integrals converge.

- Sequences
- State the definition of a sequence. Represent a sequence as ordered pairs in the plane or points on the real line. State and apply the definition for the limit of a sequence. Know the hypotheses and apply the algebraic properties (sum, product, etc.) of limits of sequences. Evaluate the limit of a sequence using an appropriate technique such as the squeeze theorem. Know |a
_{n}| —> 0 implies a_{n}—> 0. State the definition for a sequence to be monotonic or bounded. Apply the theorem that a bounded and monotonic sequence is convergent. - Series
- Understand that a series is a sequence of partial sums. For a series, state the definition of partial sum, convergent, divergent, tail end of the series. Apply the basic algebraic properties (constant multiple, addition, and subtraction) of series. Identify and compute the sum of a telescoping series. Identify and compute the sum of a geometric series. Apply the fact that a series and any of its tail ends both converge or both diverge. Recognize a series diverges if its sequence of terms does not tend to 0 (
*n*th-term test for divergence). Identify the harmonic series and know that it diverges to infinity. - Integral Test
- Know the hypotheses for when to apply the integral test. Draw the diagrams on which the integral test is based. Draw the diagram and use an integral to approximate the remainder
*R*when a series is truncated. Apply the integral test to determine whether a_{n}*p*-series converges or diverges. - Comparison Tests
- Identify series to which the direct comparison or limit comparison tests can be applied. Apply the direct comparison or limit comparison tests to determine convergence or divergence of a series. Know which series are useful for comparisons.
- Alternating Series
- Identify an alternating series. Know the hypotheses that permit using the alternating series test. Approximate the sum of an alternating series remainder (when the series is truncated). State the definitions of absolute and conditional convergence. Know that absolute convergence implies conditional convergence. Determine whether a series is absolutely or conditionally convergent.
- Ratio and Root Tests
- Identify series for which the ratio or root test is needed. Apply the ratio and root tests. Use Stirling's formula with the root test. Know that the root test is more powerful than the ratio test.
- Taylor Polynomials
- Understand that a Taylor polynomial is an approximate representation for a function. Compute the nth degree Taylor or Maclaurin polynomial for a function. State and approximate the size of the remainder term in Taylor's theorem (with remainder). Recognize that the mean value theorem is a special case of Taylor's theorem.
- Power Series
- Know the difference between a power series and a series of constant terms. Compute the radius of convergence and determine the interval of convergence for a power series. Within the interval of convergence perform term-wise differentiation and integration of a power series.
- Power Series Representation of Functions
- Use the algebraic properties of power series. Determine a power series representation for certain functions using geometric series, partial fractions, or integration
- Taylor and Maclaurin Series
- Compute the Taylor (or Maclaurin) series of a function. Understand that if f is a power series, then it
*is*a Taylor series. But if f is not a power series, then its Taylor series may or may not converge to f. State the power series for elementary functions (such as e^{x}, ln, cos, sin, arctan, and the binomial series).

- Conic Sections
- State the equation for a general second-degree polynomial in two variables. Define and (given sufficient information) find the standard equation for a parabola, ellipse, and hyperbola. Given its standard equation, sketch the graph of a parabola, ellipse, and hyperbola. For a parabola, find its vertex, directrix, focus, axis, latus rectum, and state its reflective property. For an ellipse, find its vertices, center, major axis, minor axis, foci, and eccentricity. For a hyperbola, find its transverse axis, center, vertices, asymptotes, and eccentricity.
- Plane Curves and Parametric Equations
- Determine parametric equations and orientation for elementary plane curves. Eliminate the parameter from a set of parametric equations using trig identities and by solving for the parameter. State the definitions for a cycloid, smooth, and piece-wise smooth curves. State the tautochrone and brachistochrone problems and their solution.
- Calculus and Parametric Equations
- For a plane curve given by parametric equations, compute the slope of a tangent line to the curve, find an equation for the tangent line, and find any points at which the curve has horizontal or vertical tangents. Compute arc length for a parametrized curve. Compute the surface area of a surface of revolution generated by a parametrized plane curve.
- Polar Coordinates and Graphs
- Plot points and sketch graphs in polar coordinates. Convert polar coordinates to rectangular coordinates and vice versa. Identify the graphs of elementary polar equations. Compute the slope of a tangent line to the graph of a polar equation.
- Area and Arc length in Polar Coordinates
- Compute the area of a polar region. Find the points of intersection of polar graphs. Compute arc length for a graph in polar form. Compute surface area for a surface of revolution generated by the graph of an equation in polar coordinates.
- Polar Equations of Conics
- Classify conic sections by their eccentricity. Identify and sketch a conic section given its polar equation. State Kepler's three laws of planetary motion.

- Homogeneous Second Order Differential Equations
- Know the definition of linear combination and determine linear independence/dependence of two functions. Identify a homogeneous linear second order differential equation with constant coefficients and solve its characteristic equation. In the cases of distinct real zeros, repeated real zeros, and complex zeros, specify the general solution of the corresponding differential equation or initial value problem.
- Nonhomogeneous Second Order Differential Equations
- Identify a nonhomogeneous linear second order differential equation with constant coefficients. Determine a trial solution and, using the method of undetermined coefficients, find a particular solution of a nonhomogeneous linear second order differential equation with constant coefficients. Determine when to apply variation of parameters and use it together with Cramer's rule to solve a nonhomogeneous linear second order differential equation with constant coefficients.