Topics for Math 1550: “Differential and Integral Calculus”

Optional. This is a pre-calculus review chapter and may be briefly discussed or assigned at the discretion of the instructor. While incoming students should be familiar with the topics in this chapter, some may be ill-prepared for calculus. Some faculty members have begun to require students to take a diagnostic test based on this first chapter. The test may show the student the pre-calculus areas they need to improve. Many faculty members simply skip this chapter.

Section 2.1

The Tangent and Velocity Problems: The author employs the notions of tangents and velocity to motivate the idea of a limit. Many students feel better about limits if they are given some sort of rationale for their study.

Section 2.2

The Limit of a Function: This is the standard non-rigorous approach to limits. Most faculty members employ this sequential version of limits before giving the rigorous approach in Section 2.4. The students benefit from large numbers of examples, mainly graphical, but some numerical examples should be used as well.

Section 2.3

Calculating Limits Using the Limit Laws: The standard theorems on limits of sums, differences, products, etc. are stated in this section. The proofs must be deferred until the rigorous definition of limits is covered in Section 2.4.

Section 2.4

The Precise Definition of a Limit: The instructor should use their own discretion in this section. This material is more abstract than most in this course. If you are new to teaching, you may wish to see the course coordinator or consult with some of your colleagues about this topic. When presented, it should be carefully and slowly explained. The author postpones limits at infinity until Section 2.6. You may wish to present the precise definition of Limits at Infinity when you get to section 2.6.

Section 2.5

Continuity: Continuity is one of the most important ideas in the course. The instructor should explore the topic in detail, so that confusion might be avoided later. The text uses the sequential criteria for continuity at points and then expands the discussion to intervals. The students should be aware of continuity of the standard functions and how to combine them to make more complicated continuous functions. The students should be taught to evaluate limits of continuous functions with simple substitutions.

Section 2.6

Limits at Infinity and Horizontal Asymptotes: This is a standard section. Some students may actually be familiar with calculating horizontal asymptotes from their algebra classes.

Section 2.7 & 2.8

Derivatives and Rates of Change and the Derivative as a Function: This is the introduction to the derivative as limit of a difference quotient. The student should be made aware of the derivative as an instantaneous rate of change, a tangent line slope and the velocity of a particle. Too many students emerge from beginning calculus classes with an ability to compute difficult derivatives with little or no idea about the nature of the derivative. The instructor is advised to stress throughout the course the interpretation of what a derivative tells us. This is the section where that instruction begins. The students should be made familiar with the idea that differentiability means local linearity.

Section 3.1

Derivatives of Polynomials and Exponential Functions: The author verifies the power rule for natural numbers but then asserts it is true for reals. He then begins to use the power rule for reals after that. Just be aware of this fact. The standard beginning rules for the linearity of the derivative are presented and used to calculate derivatives for polynomials. The author then uses a naive limit approach to the derivative of the natural exponential function.

Section 3.2

The Product and Quotient Rules: This is fairly standard presentation using these two powerful differentiation rules. The instructor can verify the rules and present numerous examples. Most students do not have much trouble with this topic.

Section 3.3

Derivatives of Trigonometric Functions: The material is standard. The students may not recall as much trig as the instructor deems appropriate. On a somewhat related note, you will find that your students are largely unable to work with basic identities or solve trig equations.

Section 3.4

The Chain Rule: This topic is troublesome for the student. The problem begins with their algebra weakness with composition. They cannot differentiate composite functions very easily because they do not really understand composition itself. The instructor will have to present a large number of examples.

Section 3.5

Implicit Differentiation: The beginning instructor should be advised that this seemingly easy topic will present their students many challenges. Do not suspect that you will be able to breeze through this material and have your students master it quickly. Remember, they don't quite understand the chain rule yet.

Section 3.6

Derivatives of Logarithmic Functions: The author uses implicit differentiation to derive the formula for derivatives of logarithms. This section also incorporates logarithmic differentiation and the verification of the power rule for real numbers. Students will have difficulty in differentiating tower functions.

Section 3.7

Rates of Change in the Natural and Social Sciences: This section stresses the rate of change and velocity interpretation of the derivative. This section gives examples of derivatives in use in other fields. The students should know that derivatives have applications in courses other than math class. The section also continues the exposition about interpretation of the derivatives and reminds the students that a derivative is not simply a formula.

Section 3.8

Exponential Growth and Decay: Optional. This topic should really be covered in the differential equations classes.

Section 3.9

Related Rates: This topic gives the students some trouble because they do not understand the chain rule. The two basic guidelines for related rate problems are: 1 – differentiate with respect to time and 2 – never substitute numerical values until you have differentiated. These two rules should help most students.

Section 3.10

Linear Approximations and Differentials: This section revisits the tangent line and refers to the linearization of functions. The instructor may wish to mention this is the best first degree polynomial approximation for a function. The student should understand the geometric interpretations of $\mathrm{d}y$ and $\Delta y$.

Section 3.11

Hyperbolic Functions: The hyperbolic functions will be new to most students. The instructor should cover these as they provide further basic functions for examples and interesting limit problems. Inverse hyperbolic functions should be considered optional.

Section 4.1

Maximum and Minimum Values: This section includes the definitions of absolute extrema, local extrema and critical values. This sections covers finding the extrema of continuous functions on closed intervals. The beginning instructor should stress the algebraic aspects of finding critical values. Sections 4.1 to 4.3 are very important.

Section 4.2

The Mean Value Theorem: This section includes Rolle's Theorem and the Mean Value Theorem. The MVT is a building block for the proofs of many other theorems and should be carefully presented.

Section 4.3

How Derivatives Affect the Shape of a Graph: This section begins with the increasing and decreasing test. This is possibly the most important thing students can learn in calculus. The first derivative test follows. The author introduces the notion of concavity and its indication by the second derivative. This section finishes off with the second derivative test. This section takes more than one day to cover well. The algebra and trigonometry in this section can be intense. A warning to the students might be appropriate.

Section 4.4

Indeterminate Forms and L'Hospital's Rule: This is a fairly standard presentation.

Section 4.5

Summary of Curve Sketching: This section summarizes graphing techniques. Many of the problems involving trigonometric and exponential functions are challenging. The beginning instructor may wish to be careful about putting some of these problems on exams.

Section 4.6

Graphing With Calculus and Calculators: OMIT.

Section 4.7

Optimization Problems: The students will encounter some difficulty in this section. Many will have forgotten geometry, needed for many of the optimization problems. You should expect to spend more than one day on this section.

Section 4.8

Newton’s Method: This method of finding roots, while not robust, does offer the students reinforcements of the geometric interpretation of the derivative and the ideas of convergence. Demonstrations of cases where Newton’s Method fails to converge should be presented.

Section 4.9

Antiderivatives: A fairly standard presentation.

Section 5.1

Areas and Distances: This section motivates the formal definition of the Riemann Integral in Section 5.2. This material is important, as it establishes the presence of sigma notation and left and right hand sums. He also uses distance traveled over time to further motivate Riemann Sums.

Section 5.2

The Definite Integral: The instructor should carefully develop the definition of the Riemann Integral. This section also includes many of the basic properties of the definite integral. The instructor should try to ground much of this section in geometric terms. The author introduces the midpoint approximation in this section.

Section 5.3

The Fundamental Theorem of Calculus: The author introduces the idea of area functions. The student will not have seen these before. Be sure to explain the notion of positive and negative area at this time. The proof of the FTC is necessary for every calculus class. Spend enough time in class to do it right. Differentiation of the area function is harder for the students to remember than the integral evaluation result.

Section 5.4

Indefinite Integrals and the Net Change Theorem: The students should memorize the first small table of integrals in section 5.4. The author brings up displacement and total distance traveled as integrals. The students will have trouble with total distance traveled because of the absolute value being applied to velocity.

Section 5.5

The Substitution Rule: Students have trouble with this topic. Since the idea of substitution is so important in Calculus II, the instructor should be very sure that his students receive adequate practice in this topic. The instructor should stress the need to change the limits of integration when evaluating a definite integral using substitution. You might seek out more problems from other sources to augment those in the book.

Section 6.1

Areas Between Curves: This is a fairly standard presentation. Be sure to demonstrate some problems involving integration with respect to $y$.

Section 6.2

Volumes: Many students have a great deal of difficulty with volume problems. The instructor should illustrate his examples with pictures (to the best of his ability). Be advised that even with marvelously drawn pictures the volumes by slicing problems represent some of the most challenging problems in Calculus I. The students do not “see” well in three dimensions. The instructor should be careful when assigning homework, as some of these slicing problems are difficult for the beginning student. In addition to slicing problems, this section introduces volumes of solids of revolution using washers and disks. Remind the students that Riemann Rectangles rotated around an axis perpendicular to the rectangle form disks or washers.

Section 6.3

Volumes by Cylindrical Shells: Similar to the above, many illustrations may make this topic easier for your students. Remind the students that when a Riemann Rectangle is rotated around an axis parallel to the rectangle a shell is formed.

Section 6.4

Work: This fairly standard physic topic may actually take two days to cover. The students have more trouble with the fluid pumping problems than with the other problems in this section. Be careful with the units.

Section 6.5

Average Value of a Function: The average value is an easy topic. Students understand the average value well enough. The Mean Value Theorem for Integrals is presented but not proven. The instructor should consider showing the proof, since it is another nice application of the MVT for derivatives.

Section 8.1

Arc Length: Be advised that some of these integrals will be too difficult for the students since we are skipping Chapter 7. Of course, even if we covered Chapter 7 some arc length integrals are non-elementary.

Section 8.2

Area of a Surface of Revolution: Again, some of these integrals will be too difficult for the students since we are skipping Chapter 7. The revolutions about the $y$-axis should also be discussed. Gabriel's horn, while it is an improper integral, is an interesting example with which to tease the students.

Section 8.3

Fluid Pressure and Center of Mass: These are standard topics from physics. The warning about troublesome integrals is again appropriate. Be certain of units. Remember, there is a difference between weight-density and mass-density.