AMB 176

11:15-12:15 MWThF and 10-11 T (or by appointment)

dana.ernst@nau.edu

928.523.6852

dcernst.github.io/teaching/mat136f15

MAT 125 or MAT 125H with a grade greater than or equal to C or satisfactory mathematics placement.

Calculus of one variable; basic concepts, interpretations, techniques, and applications of differentiation and integration. Letter grade only. Course fee required.

The required textbook is *Calculus I Lecture Notes* (Second Edition) by J. Neuberger, N. Sieben, and J. Swift. The book is available for purchase at the NAU Bookstore for a modest price. In addition, we will be making frequent use of Differential Calculus (PDF), which is a free set of inquiry-based learning (IBL) notes written by Brian Loft (Sam Houston State University) and published by the Journal of Inquiry-Based Learning. You are also welcome to utilize other books covering first semester calculus. Check out the Course Materials page for a list of free calculus textbooks and other resources.

I expect you to be *reading* the textbook and IBL notes. The only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the material in the textbook and notes whenever necessary by asking questions in class or posting questions to the course Google Group.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

The primary objective of this course is to aid students in becoming confident and competent in solving problems that require techniques developed in calculus. Successful completion of MAT 136 provides students with skills necessary for upper division mathematics courses, such as MAT 137: Calculus II. In general, calculus is a study of functions. The main tools are differentiation, which measures instantaneous change in a function, and integration, which gauges the cumulative effect of that change. The crowning achievement of first semester calculus is the Fundamental Theorem of Calculus, which explains how differentiation and integration are related. Students will have a working understanding of limits and continuity. Students will also be able to utilize various techniques to differentiate and integrate numerous functions including trigonometric functions. In addition, students will understand and be able to apply the Mean Value Theorem, the First and Second Derivative Tests, and the Fundamental Theorem of Calculus in both theoretical problems and applications. Also, the purpose of any mathematics class is to challenge and train the mind. Learning mathematics enhances critical thinking and problem solving skills.

Aside from the obvious goal of wanting you to learn calculus, one of my principal ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much “classroom knowledge” is fairly superﬁcial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are “getting it” comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.

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

- Demonstrate an understanding of the concepts of limit, derivative and integral in writing, and graphically.
- Calculate, or approximate as appropriate, the limit of a function using appropriate techniques including l’Hospital’s rule.
- Find the derivative of elementary polynomial, exponential, logarithmic and trigonometric functions.
- Use rules of differentiation including the power rule, product rule, quotient rule, chain rule, and implicit differentiation to compute the derivative of a function. Obtain expressions for higher order derivatives of a function.
- Interpret the derivative as the instantaneous rate of change and as the slope of the tangent line.
- Apply the derivative to find the line tangent to a function at a point and the linearization of a function at a point.
- Apply the derivative to analyze graphical behavior of a function, motion problems, other rate problems, and optimization problems.
- Construct a definite integral as the limit of a Riemann sum and use the sum to approximate a definite integral.
- Find the anti-derivative of elementary polynomial, exponential, logarithmic and trigonometric functions.
- Use substitution to find the anti-derivative of a composite function.
- Evaluate a definite integral and interpret an indefinite integral as a definite integral with variable limit(s) in order to evaluate it.
- Apply the definite integral to analyze the area under a curve and motion problems.
- Apply the Fundamental Theorem of Calculus.
- Apply differentiation and integration in setting up and critically evaluating hypotheses in the fields of science, engineering and technology.

We will make limited use of BbLearn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed here on our course webpage that lives outside of BbLearn. I suggest you bookmark this page. In addition, we will utilize a Google Group to facilitate out of class discussion. I will send the class an invite to our Google Group and briefly discuss its use. The only thing I will use BbLearn for is to communicate grades.

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand, with the readily available help of me, our Peer TA, and your classmates. Many of the concepts you learn and problems you work will be new to you and ask you to stretch your thinking. You will experience *frustration* and *failure* before you experience *understanding*. This is part of the normal learning process. **If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately.** Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work).

Perhaps this is sufficiently vague, but I believe that there are two essential elements to IBL. Students should as much as possible be responsible for:

- Guiding the acquisition of knowledge, and
- Validating the ideas presented. That is, students should not be looking to the instructor as the sole authority.

If you want to learn more about IBL, read my blog post titled What the Heck is IBL?

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the *Big Bang Theory* located here.

In this course, *everyone* will be required to

- read and interact with course notes and textbook on your own;
- write up quality solutions/proofs to assigned problems;
- present solutions/proofs on the board to the rest of the class;
- participate in discussions centered around a student’s presented solution/proof;
- call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar on first glance.

As the semester progresses, it should become clear to you what the expectations are. This will be new to many of you and there may be some growing pains associated with it.

There are two types of homework assignments: Daily and Weekly. Unless a student has a documented excused absence, late homework will not be accepted. There are many resources available to assist you with doing your homework (e.g., office hours, course Google Group, free tutoring at numerous places across campus).

**Daily Homework:** Homework will be assigned each class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. All assignments should be *carefully*, *clearly*, and *cleanly* written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.

The Daily Homework will generally consist of solving problems from the IBL course notes (PDF). On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the solutions/proofs that are due that day. Daily Homework will be graded on a $\checkmark$-system. Your Daily Homework score is worth 10% of your overall grade.

Students are allowed (in fact, encouraged!) to modify their written solutions in light of presentations made in class; however, **you are required to use the colored marker pens provided in class**. I will provide more guidance with respect to this during the first couple weeks of the semester.

**Weekly Homework:** In addition to the Daily Homework, we will also have Weekly Homework assignments. The majority of the Weekly Homework assignments are to be completed via WeBWorK, which is an online homework system. You should log in with your NAU credentials. There will likely be some growing pains associated with getting used to the online homework system, so we should all plan to be patient with each other as we get used to the system. Your Weekly Homework score is worth 10% of your overall grade.

Your combined homework score is worth 20% of your overall grade. You are allowed and encouraged to work together on homework. However, each student is expected to turn in his or her own work.

Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken seriously since they spearhead the work of the class. Here are some of my expectations:

- The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the solution clear to the other students.
- Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
- Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of his or her ability.
- Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Presentations will be graded using the rubric below.

Grade | Criteria |
---|---|

4 | Completely correct and clear solution/proof. Yay! |

3 | Solution/proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct. |

2 | A partial explanation or solution is provided but a significant gap still exists. |

1 | Minimal progress has been made that includes relevant information & could lead to a solution/proof. |

0 | You were completely unprepared. |

However, you should not let the rubric deter you from presenting if you have an idea about a solution/proof that you’d like to present, but you are worried that your solution/proof is incomplete or you are not confident your solution/proof is correct. You will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

I will often ask for volunteers to present solutions, but when no volunteers come forward, I will call on someone to present. If more than one student volunteers, the student with the fewest number of presentations has priority. The problems chosen for presentation will come from the Daily Homework assignments. Each student in the audience is expected to be engaged during another student’s presentation.

In order to receive a **passing grade** on the presentation portion of your grade, you must:

- present at least once prior to each of the four midterm exams and at least once after the fourth midterm exam, and
- present at least 5 times during semester.

Your overall performance during presentations, as well as your level of interaction/participation during class, will be worth 10% of your overall grade.

There will be 4 midterm exams, which are *tentatively* scheduled for the following dates: **Friday, September 25**, **Friday, October 16**, **Monday, November 9**, and **Friday, December 4**. Each exam will be worth 12.5% of your overall grade. There will also be a *cumulative* final exam, which will be on **Monday, December 14** at **10:00-12:00**. The final exam is worth 20% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.

As per university policy, attendance is *mandatory* in all 100-level courses, and in particular, I am required to record attendance each class session. Since a significant part of your grade is class participation, and since you cannot participate if you are not present, it will be impossible to succeed (i.e., not fail the course) if class is missed. You are responsible for all material covered in class. You can find more information about NAU’s attendance policy on the Academic Policies page.

In summary, your final grade will be determined by your scores in the following categories.

Category | Weight | Notes |
---|---|---|

Homework | 20% | A combination of Daily and Weekly Homework. |

Presentations and Participation | 10% | Students will present proposed solutions to the class. |

4 Midterm Exams | 50% | Each exam is worth 12.5% of your overall grade. |

Final Exam | 20% | The final exam is cumulative. |

In general, you should expect the grades to adhere to the standard letter-grade cutoffs:

You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Office of Student Life.

As per Department Policy, cell phones, MP3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.

Students are expected to treat each other with respect. Students are also expected to promote a healthy learning environment, as well as minimize distracting behaviors. In particular, you should be supportive of other students while they are making presentations. Moreover, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class.

Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.

I am a huge fan of technology and believe that when it is used appropriately, it can greatly enhance one’s learning experience. However, when learning, technology should never replace one’s own amazing cognitive abilities. When we are discussing concepts in class or when you are doing homework, you should feel free to use whatever resources you feel will help you understand the concepts better. So, feel free to use things like Sage, Wolfram|Alpha, Mathematica, your graphing calculator, etc. when doing homework. However, on exams **you will not be allowed to use a calculator**.

Moreover, be warned that I am much more interested in the process by which you arrived at your answer than the answer itself. An answer to a homework or exam question that is correct but lacks justification may be worth little to no points. If you understand a concept, then barring a silly computational error, the correct answer comes along for the ride. Yet, getting the correct answer does not imply that you understand anything!

There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible. Second, you are strongly encouraged to ask questions in the course Google Group, as I (and hopefully other members of the class) will post comments there for all to benefit from. You are also encouraged to stop by during my office hours and you can always email me. Lastly, *free* tutoring is available in AMB 137 through the Math Achievement Program.

Here are some important dates:

**Monday, September 7**: Labor Day, no classes**Thursday, September 10**: Last day to drop/add (no W appears on transcript)**Friday, September 25**: Exam 1**Friday, October 16**: Exam 2**Friday, November 6**: Last day to withdraw from a course (W appears on transcript)**Monday, November 9**: Exam 3**Wednesday, November 11:**Veterans Day, no classes**Thursday, November 26**and**Friday, November 27**: Thanksgiving Holiday, no classes**Friday, December 4**: Exam 4**Monday, December 14**at**10:00-12:00**: Final Exam

Any changes in this syllabus made during the term will be properly communicated to the class.

Calculus is one of the great achievements in human history, and I am happy that I get to share it with you.

Portions of "Purpose and Goals" and "Class Presentations and Participation" are adapted from Carol Schumacher's *Chapter Zero Instructor Resource Manual*. The first paragraph of "An Inquiry-Based Approach" is borrowed from Robert Talbert. Lastly, I've borrowed a few phrases here and there from Bret Benesh.

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

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