# Syllabus for Math 1552 "Calculus II"

## Text : Calculus, Early Transcendentals 7th Edition by James Stewart

This is a four hour Calculus course primarily designed for engineering majors and certain other technical majors. The text is Calculus: Early Transcendentals 7th Edition by James Stewart. The student is assumed to be capable in the standard Calculus I topics of taking limits, continuity, taking derivatives of fairly complicated functions, using derivatives, calculating the definite integral for basic functions, integration by substitution and the standard applications of the definite integral. Students who are not fully prepared for this course should review the chain rule, the basic integral formulas and integration by substitution, trigonometric equations and polar coordinates. Calculator policies vary by instructor. A set of WebAssign problems has been established and is available for interested faculty. Access to the WebAssign e-textbook is also available for students whose instructor has chosen to not use WebAssign. Contact George Cochran with any questions,

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### Basic skills the students should acquire during the course

- Techniques of Integration

- Approximate integrals using numerical integration
- Evaluate integrals using integration by parts
- Evaluate integrals of trigonometric forms
- Evaluate integrals by trigonometric substitution
- Evaluate integrals by the method of partial fractions
- Evaluate Improper Integrals

- Infinite Series

- Analysis of sequences and their convergence
- Use the definition of convergence for series
- Use the integral test, the comparison tests, the ratio test and the root test
- Determine power series and their intervals of convergence
- Form Taylor series for common functions and master simple applications of Taylor series

- Parametric Equations, Polar Coordinates and Conic Sections

- Draw parametric curves and calculate derivatives along parametric curves
- Calculate arc length and speed along parametric curves
- Draw polar curves and convert between rectangular and polar forms
- Calculate arc length and areas using polar coordinates
- Sketch conic sections and write the equations of conic sections

- Vectors

- Be able to draw two dimensional vectors and do simple arithmetic on vectors
- Be versant with three space and three space vectors
- Be able to calculate dot products, the angle between vectors and vector projections
- Calculate cross products and know the geometric interpretations of cross products
- Be able to write equations of planes meeting the usual conditions

- Calculus of Vector Valued Functions

- Recognize and sketch simple vector valued functions
- Compute limits and derivatives of vector valued functions
- Calculate arc length and speed for vector valued functions
- Calculate curvature, the unit normal and the osculating circle for simple parameterizations
- Be versant with uniform circular motion and ballistic motion

- Partial Derivatives

- Be able to compute partial derivatives of simple functions
- Understand Clairault's Theorem

A specific section by section syllabus and comments are shown below.

## Syllabus

### Chapter 7

**Section 7.1****Integration by Parts:**The presentation is fairly standard. Integration by parts is needed in later coursework. The instructor should be prepared for their students having difficulty with the idea of substitution. Some experienced instructors review the differentiation formulas and integration by substitution on the first day.**Section 7.2****Trigonometric Integrals:**The trigonometric forms are needed to do trigonometric substitutions in the next section. The material is fairly standard**Section 7.3****Trigonometric Substitutions:**The students have a great deal of trouble deciding which trig substitution to use. The instructor is urged to pay particular attention to explaining how to decide which substitution is appropriate and why. The author's summary is pretty good in helping with this.- Section 7.4
**Partial Fraction Decomposition:**This material is used in the differential equations courses. The instructor does not have to assign horribly complicated decompositions but the case with irreducible quadratic denominators should be covered. That case will show up when the students are computing inverse Laplace Transforms in D.E.**Section 7.5****Strategy for Integration:**This section gathers many different types of integration problems together. It is very useful in preparing students for an exam on techniques of integration where they are not told how to solve an integral. This section is somewhat challenging for the students as they have to use all of their techniques without guidance as to the appropriate approach.**Section 7.7****Approximate Integration:**The author introduces the Trapezoidal Rule and Simpson's Rule for approximating definite integrals. The error bounds should be discussed. This section is useful, as many students leave a calculus class thinking all functions have elementary antiderivatives.**Section 7.8****Improper Integrals:**This is a fairly standard topic. The instructor should carefully discuss the comparison tests, because that will set the stage for the use of comparisons in the infinite series chapter.

### Chapter 10

**Section 10.1 ****Parametric Equations:** The students have usually not seen parametric equations before this chapter. The students benefit from simple examples showing elimination of a parameter.

**Section 10.2 ****Calculus with Parametric Curves:** This is a standard presentation. When taking derivatives of parametric curves, the students should not confuse dx/dt, dy/dt and dy/dx. Please be aware that the integrals in this section can be challenging. Arc length and surface area of revolution integrals can easily become non-elementary nightmares. Asking students to set-up the integrals is a common solution to this issue.

**Section 10.3 ****Polar Coordinates:** This section is a useful re-introduction to polar coordinates. This is NOT optional. The students will not have seen polar coordinates in a while. Further, most students get a very superficial treatment in the lower courses. They benefit from going over this section. The tangent to polar curves is introduced in this section. The instructor should carefully explain this topic, as it confuses many students. Be advised that many students will have trouble solving trig equations.

**Section 10.4 ****Areas and Lengths in Polar Coordinates:** The students have difficulty in this section because they cannot identify the required regions. The limits of integration, frequently gotten by solving trig equations, are a perennial trouble spot.

**Section 10.5 ****Conic Sections:** This is a common topic and the presentation is unremarkable. The reflective properties of conics, which are only mentioned in the exercises, are interesting and provide an example of conics used in real world settings.

### Chapter 11

**Section 11.1 ****Sequences:** The students have a hard time identifying a sequence as a function with a restricted domain. Thus, they do not see the limit of a sequence as the horizontal asymptote of a function. Since this is the chapter on series, the cornerstone of everyone's second calculus course, the instructor might want to spend more time on this chapter than on other chapters. It is my experience that students need to proceed slowly through the material on series.

**Section 11.2 ****Series:** This section includes the definition of convergence for series. The students have enormous difficulty with the Divergence Test. The students need to be aware that the terms of a series tending to zero is not sufficient to insure convergence. The geometric series and the telescoping series make their appearance in this chapter.

**Section 11.3 ****The Integral Test:** The section includes the integral test and a discussion on the remainder estimate for the integral test. The p-series and the logarithmic p-series are introduced.

**Section 11.4 ****The Comparison tests:** The instructor should provide numerous examples of using the comparison test. The students need to know how you choose the series you are using for comparison.

**Section 11.5**** Alternating Series:** This is a standard presentation. The students have a tendency to use the Alternating Series Test on series which do not alternate. The alternating series estimation result should be discussed.

**Section 11.6 ****Absolute Convergence and the Ratio and Root Tests:** The students need to be made aware that absolute convergence implies convergence. They will also claim a series is conditionally convergent without ever testing for absolute convergence. The Ratio Test is the more useful of the two tests in this section. The students will need an introduction to the notion of factorials.

**Section 11.7 ****Strategy for Testing Series:** This section gathers various series and lets the students try to determine convergence. This will provide a useful summary of the methods for analyzing the convergence of series of constants, before jumping to power series.

**Section 11.8 ****Power Series:** This section introduces the idea of the interval of convergence for a power series. The students frequently forget to check the endpoints of the interval of convergence so some attention to that detail is advised.

**Section 11.9 ****Representations of Functions as Power Series:** This material concerns constructing power series based on the geometric series. Term-by-term differentiation and integration of series also appear and are used to generate the familiar series for the logarithm and the arctangent.

**Section 11.10 ****Taylor and Maclauren Series:** The students should be able to produce some of the more simple Maclauren series from the definition. They should also be taught to use substitution to derive series representations from series they already know. It is important to cover the Taylor Inequality in this section. Many instructors view multiplication and division of series as optional.

**Section 11.11 ****Applications of Taylor Polynomials:** Please be advised that many of the exercises ask for the construction of graphs which are best handled with graphing technology. If your students do not have access to graphing devices, the exercises should be edited to remove the graphing component.

### Chapter 12

**Section 12.1 ****Three-Dimensional Coordinate Systems:** Many instructors cover this chapter after the material on techniques of integration. The reasoning is that students in physics would benefit from seeing vectors early. This section is basic and easy for most students. The distance formula in three-space, the equations of spheres and specifying regions of space are important but easily grasped.

.**Section 12.2 ****Vectors:** This section introduces two and three dimensional vectors, vector arithmetic, magnitude of vectors, unit vectors and normalizing vectors. This is pretty basic material for most students.

**Section 12.3 ****The Dot Product:** A standard presentation. Stress the uses of the dot product in calculating angles and projections. Some of the basic physics examples might be considered.

**Section 12.4 ****The Cross Product:** The author defines the cross product and then uses the determinant to help students recall the formula. He then proves the geometric aspects of the cross product. The volume of the rectangular parellelepiped can be calculated as the magnitude of the scalar triple product.

**Section 12.5 ****Equations of Lines and Planes:** The topic is standard and is normally not a problem for the students.

**Section 12.6 ****Cylinders and Quadric Surfaces:** This topic is mainly presented for the benefit of those students who will be going to Math 2057. The students do not normally draw very well. The instructor is encouraged to try to draw the figures carefully. The students will need to be able to draw figures well enough for them to use those figures in integral applications in Math 2057.

### Chapter 13

**Section 13.1 ****Vector Functions and Space Curves:** Students have problems with drawing or even visualizing three-space curves. The instructor is encouraged to try to sketch numerous examples. The questions involving parameterizing line segments is very important for students going on to Math 2057.

**Section 13.2 ****Derivatives and Integrals of Vector Functions:** This simple section includes limits, derivatives and integrals of vector valued functions.

**Section 13.3 ****Arc Length and Curvature:** The arc length integral is familiar to the students. The idea of parameterizing a curve using arc length is difficult for most students.The text introduces the various formulas for curvature and the students should be made familiar with all of them. The Unit Normal, the center of curvature and the osculating circle appear in this section.

**Section 13.4 ****Motion In Space:**The text introduces velocity, acceleration, uniform circular motion and ballistic motion. The instructor can treat the decomposition of acceleration into the normal and tangential vectors as an optional topic.

### Chapter 14

**Section 14.3 ****Partial Derivatives:** The instructor should present this topic for the benefit of those students moving to Math 2090. The material can be taught from a mechanical standpoint.