## Course Syllabus for "MA101: Single-Variable Calculus I"

**Please note: a fully-supported alternate version of this course is available at https://learn.saylor.org/course/ma101/.
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This course is designed to introduce you to the study of calculus. You
will learn concrete applications of how calculus is used and, more
importantly, why it works. Calculus is not a new discipline; it has
been around since the days of Archimedes. However, Isaac Newton and
Gottfried Leibniz, two 17^{th} century European mathematicians
concurrently working on the same intellectual discovery hundreds of
miles apart, were responsible for developing the field as we know it
today. This brings us to our first question, what is calculus today?
In its simplest terms, calculus is the study of functions, rates of
change, and continuity. While you may have cultivated a basic
understanding of functions in previous math courses, in this course you
will come to a more advanced understanding of their complexity, learning
to take a closer look at their behaviors and nuances. In this course, we
will address three major topics: limits, derivatives, and integrals, as
well as study their respective foundations and applications. By the end
of this course, you will have a solid understanding of the behavior of
functions and graphs. Whether you are entirely new to calculus or just
looking for a refresher on a particular topic, this course has something
to offer, balancing computational proficiency with conceptual depth.

### Learning Outcomes

Upon successful completion of this course, the student will be able to:

- Define and identify functions.
- Define and identify the domain, range, and graph of a function.
- Define and identify one-to-one, onto, and linear functions.
- Analyze and graph transformations of functions, such as shifts, dilations, and compositions of functions.
- Characterize, compute, and graph inverse functions.
- Graph and describe exponential and logarithmic functions.
- Define and calculate limits and one-sided limits.
- Identify vertical asymptotes.
- Define continuity and determine whether a function is continuous.
- State and apply the Intermediate Value Theorem.
- State the Squeeze Theorem, and use it to calculate limits.
- Calculate limits at infinity and identify horizontal asymptotes.
- Calculate limits of rational and radical functions.
- State the epsilon-delta definition of a limit, and use it in simple situations to show a limit exists.
- Draw a diagram to explain the tangent-line problem.
- State several different versions of the limit definition of the derivative, and use multiple notations for the derivative.
- Describe the derivative as a rate of change, and give some examples of its application, such as velocity.
- Calculate simple derivatives using the limit definition.
- Use the power, product, quotient, and chain rules to calculate derivatives.
- Use implicit differentiation to find derivatives.
- Find derivatives of inverse functions.
- Find derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions.
- Solve problems involving rectilinear motion using derivatives.
- Solve problems involving related rates.
- Define local and absolute extrema.
- Use critical points to find local extrema.
- Use the first and second derivative tests to find intervals of increase and decrease and to find information about concavity and inflection points.
- Sketch functions using information from the first and second derivative tests.
- Use the first and second derivative tests to solve optimization (maximum/minimum value) problems.
- State and apply Rolle’s Theorem and the Mean Value Theorem.
- Explain the meaning of linear approximations and differentials with a sketch.
- Use linear approximation to solve problems in applications.
- State and apply L’Hopital’s Rule for indeterminate forms.
- Explain Newton’s method using an illustration.
- Execute several steps of Newton’s method and use it to approximate solutions to a root-finding problem.
- Define antiderivatives and the indefinite integral.
- State the properties of the indefinite integral.
- Relate the definite integral to the initial value problem and the area problem.
- Set up and calculate a Riemann sum.
- Estimate the area under a curve numerically using the Midpoint Rule.
- State the Fundamental Theorem of Calculus and use it to calculate definite integrals.
- State and apply basic properties of the definite integral.
- Use substitution to compute definite integrals.

### Course Requirements

In order to take this course, you must:

√ Have access to a computer

√ Have continuous broadband Internet access

√ Have the ability/permission to install plug-ins or software (Adobe Reader, Flash, etc.)

√ Have the ability to download and save files and documents to a computer

√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.)

√ Be competent in the English language

√ Have read the Saylor Student Handbook.

### Course Information

Welcome to **MA101 Single-Variable Calculus**. Below, please find
general information on this course and its requirements.

**Course Designer:** Clare Wickman

**Primary Resources:** This course is comprised of a range of different
free, online materials. However, the course makes primary use of the
following:

- Whitman College: David Guichard’s
*Calculus* - Massachusetts Institute of Technology: David Jerison’s
*Single Variable Calculus* - Temple University: Gerardo Mendoza’s and Dan Reich’s
*Calculus on the Web*

**Requirements for Completion:** In order to complete this course, you
will need to work through each unit and all of its assigned materials.
Pay close attention to Units 1 and 2, as these lay the groundwork for
understanding the more advanced, exploratory material presented in the
latter units. Note that for many students Unit 1 will mostly be a
review.

Note that you will only receive an official grade on your final exam.
However, in order to adequately prepare for this exam, you will need to
work through the quizzes and problem sets listed above.

In order to pass this course and earn your Saylor Foundation
certificate, you will need to earn a 70% or higher on the final exam.
Your score on the exam will be tabulated as soon as you complete it. If
you do not pass the exam, you may take it again.

**Time Commitment:** This course should take you approximately **93.75
hours** to complete. At the beginning of each unit, there is a detailed
list of time advisories for each subunit. These estimates factor in the
time required to watch each lecture, work through each reading
thoughtfully, and complete each assignment. However, these should be
seen as guidelines, not goals; each learner is different, and you may
find that your pace changes throughout the course. Mastery of the
material, rather than strict adherence to the time estimates, is the
measure of success in this course. It may be useful to take a look at
these time advisories and to determine how much time you have over the
next few weeks to complete each unit, and then to set a schedule for
yourself. For example, Unit 1 should take you 13 hours. Perhaps you
can sit down with your calendar and decide to complete subunit 1.1 (a
total of 2 hours) on Monday night; subunit 1.2 (a total of 3 hours) on
Tuesday night; etc.

**Table of Contents:** You can find the course's units at the links below.