Math 1550 Calculus I
Topics with Section Comments
This is an introductory calculus course designed primarily for students pursuing an engineering major or certain
other technical majors. The student is assumed to be well versed in the standard pre-calculus topics of functions,
graphing, solving equations and the exponential, logarithmic and trigonometric functions.
Basic skills the students should acquire during the course
1. Limits and Continuity
a. Evaluate limits from a graph
b. Evaluate limits at points of continuity
c. Evaluate limits of indeterminate forms using algebraic simplifications and l’Hôpital’s rule
d. Know what continuity implies about a graph and behavior of a function
e. Determine points of discontinuity for functions defined as formulas or graphs
2. Differentiation
a. Know the various interpretations of the derivative (velocity, rate of change, slope of tangent line)
b. Evaluate the derivatives of simple functions using a difference quotient
c. Evaluate the derivatives of combinations of the basic elementary functions
d. Take the derivative using implicit and logarithmic differentiation
e. Find tangent lines and be able to use them as linear approximations
f. Find critical values, local extrema and the intervals of concavity for differentiable functions
g. Find absolute extrema of constrained functions
h. Solve problems involving related rates
i. Solve basic optimization problems
j. Understand the Mean Value Theorem for Derivatives
3. Integration
a. Understand anti-derivatives and know the basic anti-derivative formulas
b. Have an understanding of the Riemann Integral as a limit of Riemann sums
c. Be able to use both parts of the Fundamental Theorem
d. Evaluate definite integrals using substitution
e. Find the area between two curves and the volumes of solids of revolution
f. Find arc lengths of plane curves
g. Understand the Mean Value Theorem for Integrals.
Chapter 1 Optional. This is a pre-calculus review chapter and may be briefly discussed or assigned at the
discretion of the instructor. Incoming students should be familiar with the topics in this chapter. Some faculty
members have begun to require students to take a diagnostic test based on this first chapter but most instructors
skip this chapter.
Section 2.1 The Idea of Limits: The author employs the notions of tangents, average velocity 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 Definitions of Limits: 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 2.7. The students benefit
from large numbers of examples, mainly graphical, but some numerical examples should be used as well.
Section 2.3 Techniques for Computing Limits: The standard theorems on limits of sums, differences,
products, etc. are stated in this section. The proofs of these theorems, if desired, must be deferred until the
rigorous definition of limits is covered in Section 2.7.
Section 2.4 Infinite Limits: This is a standard section. Some students may be familiar with calculating vertical
asymptotes from their algebra classes.
Section 2.5 Limits at Infinity: As in the previous section, the student may be familiar with the notion of
horizontal asymptotes. The treatment is standard and seems to give students little trouble.
Section 2.6 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 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 substitutions.
Section 2.7 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 consult with
some of your colleagues about this topic. When presented, it should be carefully and slowly explained.
Sections 3.1 Introducing the Derivative: The author uses the standard tangent line motivation for the
derivative and ties it very loosely to the notion of a rate of change. He moves quickly from the derivative at a
point to the idea of the derivative function. The instructor may want to further explore the derivative as a
velocity in this section because the author largely postpones the physical interpretations of the derivative until
section 3.6.
Section 3.2 Working with Derivatives: This section introduces some of the key ideas of graphing a function
using a derivative, by considering the reverse issue of graphing a derivative based on a given function. The
author also considers the relationship between differentiability and continuity in this section. Examples of nondifferentiable functions are also discussed.
Section 3.3 Rules of Differentiation: The material is standard. Derivatives of constants, the power rule, the
linearity properties and higher order derivatives are covered. The author differentiates the natural exponential
function by defining e as a limit. This is a non-rigorous but typical way to introduce the exponential function
early.
Section 3.4 The Product and Quotient Rules: The students benefit from numerous examples when teaching
this material. Many students have trouble getting the numerator correct in the quotient rule. Starting with this
section, students can improve their algebra skills by trying to simplify their answer to match the ones in the text.
Section 3.5 Derivatives of Trigonometric Functions: This material is presented in the usual manner. Students
can easily follow the proof of the limit for sin(x)/x and it is a useful exercise to go over. You may find some of
your students have difficulty working with basic identities or solving trig equations.
Section 3.6 Derivatives as Rates of Change: The author finally begins to firmly link the derivative with its
physical interpretations. The instructor should have already made mention of this in prior lectures. The
economics and business applications are usually omitted since the thrust of this class is towards the engineering
and physics students.
Section 3.7 The Chain Rule: This topic is troublesome for the students. The problem begins with their algebra
weakness with composition. Many have trouble differentiating composite functions because they do not really
understand composition itself. The instructor will have to present a large number of examples in this section.
Section 3.8 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 3.9 Derivatives of Logarithmic and Exponential 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 of the form f(x)^g(x).
Section 3.10 Derivatives of Inverse Trigonometric Functions: The material is presented in the standard
manner but the students have a bit of trouble with these derivatives. The formulas are a bit more complicated
and harder to memorize. The instructor should decide how much to tell the student about the inverse secant
function and the two different interpretations of its negative branch. The section also includes the derivatives of
general inverse functions.
Section 3.11 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 4.1 Maxima and Minima: Finding critical values seems to give students trouble. They can usually
find the derivatives easily enough but solving the resulting equations is the part with which they struggle.
Trigonometric examples are a must since many students have forgotten how to solve trigonometric equations.
Section 4.2 What Derivatives Tell Us: 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 4.3 Graphing Functions: This section summarizes graphing techniques. Many of the problems
involving trigonometric and exponential functions are challenging but should not be avoided. These problems
will build the students manipulative skills.
Section 4.4 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 4.5 Linear Approximation 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 dy and Δy.
Section 4.6 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 4.7 L'Hospital's Rule: This is a fairly standard presentation. Some instructors spend two days on this
section.
Section 4.8 Newton’s Method *Optional*: 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 standard presentation that is routinely grouped with the Chapter 5 material by
many instructors.
Section 5.1 Approximating Areas under Curves: 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 and midpoint rule approximations. The author also uses distance traveled over time to further
motivate Riemann Sums.
Section 5.2 Definite Integrals: 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 notion of net
area 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. 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 Working with Integrals: This section presents the symmetry rules for integrals, along with the
average value of a function and the mean value theorem for derivatives. These topics present the student with
little difficulty.
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 Velocity and Net Change: The author discusses 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. This section also has simple differential equation problems involving construction of position
functions from velocity or acceleration functions.
Section 6.2 Regions Between Curves: This is a fairly standard presentation. Be sure to demonstrate some
problems involving integration with respect to y.
Section 6.3 Volume by Slicing: Many students have a great deal of difficulty with volume problems. The
instructor should illustrate the examples with pictures (to the best of their 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 6.4 Volume by 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 6.5 Length of Curves: Be advised that many of the integrals associated with arc length are too difficult
for the students and many are simply non-elementary. The safest thing to do is to prepare your examples to be
worked out beforehand and give most examples as “Set-up” the integral type problems.
Section 6.6 Surface Area *Optional*: This topic is often covered by instructors but it is listed as optional.
Some of these integrals will be too difficult for the students. 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 6.7 Physical Applications *Optional*: These standard physic topics 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 and be sure to explain the difference between weight density and mass density
if you intend on doing things in the English system.
Section 6.10 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.