As the name suggests, this is an introductory class to calculus. Even though it sounds overly dramatic, it's not an exaggeration to say that calculus is probably the most powerful tool humans have ever discovered. In a single conceptual framework it allows us to understand the motion of the planets, how atomic particles behave, how fluids move, how chemical reactions progress, how populations grow, how to build bridges, how economies change, even how people behave in large groups. And that's just a tiny sample of what it can do! If one day we meet an advanced alien civilization from somewhere else in our universe, I guarantee that they'll know the fundamental principles of calculus.
Some of you will have been exposed to the basic ideas of calculus already, some of you won't. Let me stress that this is very much an introduction. You do not need to have any familiarity with calculus before starting the course. What almost all of you do have in common though is that this will be your first mathematics class at university. This is going to be a big change and I'm here to guide you through it.
The transition from high school to university-level mathematics is challenging. Of course the material becomes more complex, but there is more to it than that. The whole style of learning mathematics is different at university, and takes time to adjust to. So what are the major differences?
More Conceptual. The material will emphasize conceptual understanding rather than rote computation. While each course will build a toolkit of essential computational techniques, why and how they work is what really matters. You'll need to understand why specific methods are effective under certain conditions and how to adapt them when assumptions change. Some concepts may require weeks of concentrated thought to fully internalize.
Much Faster Pace. Nearly every lecture introduces a new concept or technique, assuming a solid grasp of previously covered foundational material. If you fall seriously behind it can be really difficult catching up.
More In-Depth Computations. Examples and problems often involve multiple parts that combine various techniques. Developing a strong intuition is crucial for progressing from point A to point B through a sequence of logical steps.
Serious Applications. Real-world problems frequently present complexities that necessitate sophisticated mathematical techniques for accurate modeling. Lower-division math courses are structured to swiftly bring students up to speed, balancing thoroughness with practicality.
I'm here to support you through this transition. In the tabs below I'll offer advice about how to approach the course and what pitfalls to avoid.
I hope you enjoy the class!
Alex
Lectures are where I will introduce and explore the material. Throughout the semester, I'll do my best to make you part of the process, giving you the chance to try problems yourself and ask questions. If there's something you feel unsure of, don't feel nervous about putting your hand up. If you are thinking it, then so are lots of other people in the room. I'm here to guide you towards proper understanding, not deliver dry uninterrupted monologues. The more like a back and forth conversation, the better!
LEC 001: 9.30am-11am TuTh, Wheeler Auditorium
LEC 002: 8.00am-9.30am TuTh, Wheeler Auditorium
Both lectures will be identical and you are welcome to attend either, although it would be best to attend your offical section for the first few weeks. As is tradition at Berkeley, the lecture will formally begin ten minutes after the official start time. That gives you enough time to get across campus if you need to. Attendance at lecture is not mandatory, although I strongly advise that you go.
All lectures will come with pre-recorded videos and lecture notes. You can find these in the Schedule tab below. In terms of content these will be identical to in-person lectures.
I strongly recommend against using the pre-videos as a total replacement for in-person lectures
In-person lectures offer a more focused learning experience, allowing you to ask and hear questions in real time. While videos are of course great for reviewing material at your own pace, it's challenging to stay fully engaged during an initial viewing. The temptation to skip through sections will lead to missing important details, something that's less likely to occur when we are all together in person.
Some advice I would offer to other people who have to take it is to make sure to go to lecture even if he posts his notes and lectures online later because it is helpful to be present.
Math 51 student, Fall 2019
Here's some advice about getting the most out of lectures and the pre-recorded videos.
Pay attention to the details. These lectures will likely be in more depth than what you are used to. Simply grasping the basic idea of a concept won't suffice. Your goal should be to comprehend each concept in granular detail. Every point covered will be crucial to the topic, so if something isn't clear, don't ignore it. Ask questions in class, or jot down your questions to discuss them during office hours or in the discussion section later on. Understanding these nuances is key to mastering the material.
Take your own notes. I'll be writing everything out in real-time during lecture. If you are following along with me doing the same, it'll really help you focus. A good tip is to highlight points that weren't clear to you so you can specifically revisit them later.
Come back to the videos after the in-person lecture. Video lectures offer great flexibility for learning. Being able to revisit them allows you to reinforce your understanding of complex topics or clarify any confusion you may have. This ability to review at your own pace can significantly enhance comprehension and retention of the material. Make use of it, but don't let it completely replace in-person lectures.
Be very cautious of other math resources online. This course is structured to be as self-contained as possible. If you find yourself struggling with a concept, revisit the corresponding video lecture rather than turning to random online sources. Such sources almost always lack sufficient detail and can lead to a false sense of understanding. Additionally, avoid relying on tools like ChatGPT. They won't contribute to your learning, especially since AI is still terrible at mathematics.
Make sure you understand the examples. The exercises in the worksheets and homework are closely related to examples I go through in the lecture. Really make sure you really understand the approach I'm taking to solving problems in the lectures.
Don't skip any lectures. Even if the topic is familiar to you, there will be subtleties that you have likely not seen before. Think of it like reading a story. If you skip a paragraph you'll get lost very quickly.
There is a key difference between LEC001 and LEC002. Section LEC001 is high-touch, and is for students who require the most possible support. Section LEC002 is low-touch, and is for students who are more confident and require less support. Practically speaking the main difference between the two is how discussion sections and quizzes are handled.
If you are in LEC001 (High-Touch)...
Every week you will have three hours of discussion session led by a GSI or UGSI (graduate or undergraduate student instructor).
These sessions provide an excellent opportunity to explore the material as a group, with each section consisting of around 20-30 students. In these interactive discussions, you will collaborate with your peers to tackle challenging problems, enabling you to truly grasp and excel in the subject matter. Here are some tips to get the most out of these sessions.
Mastering Mathematics is a challenge and most challenges are best approached as a group. Regard your discussion section not only as an opportunity to understand the course material better but also as a place to connect with classmates for collaborative work outside of class. Having other people to bounce ideas off is incredibly valuable. The more perspectives on a difficult problem the better.
Don't feel apprehensive about participating. Many of us get nervous speaking in public, doubly so when it comes to mathematics. You may think that only you are struggling with a concept, but that is never ever true. If something's confusing, ask questions. If the instructor asks a question, try answering it. It can be scary at first but over time it won't be and the payoff will be enormous in the long run
Be active in group work. Your will frequently be split into smaller groups where you'll work on problems more collaboratively. Be an active participant in these. Even if you're not completely confident in your approach, still share it.
Be there. This is probably the most important of all. Even if you feel that you aren't getting much out discussion section, you are. It's three whole hours a weeks where you are completely focused on the class. Ask yourself the following question: Am I really going to spend three hours a week going through the worksheets on my own? For me the answer would be no! That's a lot of time to lose and it adds up over the course of a semester. Not attending section is almost always a direct indicator of heading for a bad grade.
You can only attend the section you are officially enrolled in. If you have enrollment questions contact: enrollment@math.berkeley.edu
You'll be following fixed worksheets in your discussions. After you've had section, they'll be posted in the Schedule and Course Materials tab along with solutions.
Attendance at discussion section is mandatory, and will count towards your final grade . For more details see the Grading and Course Policy tab below.
Take advantage of discussion and office hours to practice more problems. You really do most of the learning in homework and discussion, just attending lectures isn't enough!
In general, if you have any queries or issues relating to the course, the first person to contact directly is your GSI.
If you are in LEC002 (Low-Touch)...
Each week you will have access to a large lab space where you will be able to work at your own pace (solo or in groups) on problems of your choice (worksheets, homework, etc). There will be GSIs in these sessions providing support. Attendance is not mandatory and will not count towards your grade. The lab will take place at the following times and location:
In addition to lectures and discussion section/lab, you will have access to daily office hours.
Office hours are your chance to talk to me or a GSI/UGSI. They are a really good way to get to know your instructors better and get help with any aspect of the course. These are also a space for you to work in small study groups at your own pace if you'd like.
In office hours you can talk about any aspect of the course (and beyond). If you've spent time on a homework problem and have stalled, come to office hours; If you're unsure about your academic trajectory, come to office hours; If you want to learn what mathematics research is about, come to office hours; If you're struggling with the course and don't know what to do, come to office hours; If you think you might need a reference from a professor in the future, come to office hours; If you just want a chat, come to office hours.
Go to Paulin's OH and your GSI's OH at least once in the first month of classes, just to see what they are like (and you'll see how helpful they are!).
Math 51 student, Fall 2019
I'll be having office hours every TWThF. As a default GSI/UGSI office hours are in-person, although some may be conducted online via Zoom. See the full schedule for more details.
Here is a link with the full office hour schedule.
Mathematics is an extremely constructive subject. Every new concept builds on the ones that came before it. If there are gaps in your foundational knowledge this is very likely to make things challenging fast.
This class assumes that you've got a solid grasp of the main topics in precalculus and the subjects that come before it. That said, we will be spending the first three weeks reviewing key precalculus topics to make sure everyone is on the same page. I don't want anyone falling behind at the start.
Some of you may want a more detailed review of these topics. Here are three resources I've designed to support you.
Precalculus Cheatsheet This is the most basic of all. A lightning quick review of the key topics required before starting a first calculus class.
Precalculus Essentials This is a comprehensive online precalculus course I designed several years ago. It explains everything in greater detail than other online resources and is a good place to turn if you're struggling with specific topics.
Math 1: Foundations of Lower Division Mathematics. This is the option if you want the most robust support with your foundational mathematics skills. It's a course I designed to sit somewhere between precalculus and calculus. It's a 2-unit class that runs in the Fall semester and can be taken in parallel to Math 51. There will be significant overlap between Math 1 and the material in the first five weeks of Math 51.
The textbook for this course is:
Stewart, Single Variable Calculus: Early Transcendentals , 9th edition.
Even though it's probably something you aren't used to doing, I strongly recommend reading the textbook throughout the semester. It provides more detail than can possibly be covered in lectures (either in-person or recorded), and is a habit you want to get into now. Future courses will not have videos to fall back on and reading the textbook will be absolutely vital. It's also where the homework exercises are taken from. If you read the examples in the main text it'll be much more obvious how to approach them.
Reading the textbook alongside the lectures really helps. When you're stuck on a homework problem try to find a similar example in the book.
Math 52 student, Spring 2018
There will be weekly homework assignments, each covering topics from the previous week. These will be due each Wednesday at 11.59pm on Gradescope (an online platform to submit assignments). You can access Gradescope from this link. Here are instructions for how to upload your work. If you have issues submitting your work, contact your GSI/UGSI.
The homework exercises will be from the textbook.
Each assignment will be graded on completion (and not correctness). More specifically you will get three points for submitting solutions to all exercises. This means you can make mistakes in the learning process and it won't negatively affect your grade. Your two lowest scores will be dropped.
You may discuss the homework problems
with your classmates, but you must write your solutions on your own.
Make use of discussion sections and office hours if
you need assistance, but in the end, you should still write up your own
solutions.
Lectures are where new concepts are introduced, but homework is where much of the real learning happens. It's where you internalize the abstract ideas and discover for yourself how they can be used to solve problems. From my experience the main distinction between those who succeed in mathematics classes and those who don't is how they treat the homework. Here's some advice about how to approach it.
Be organized. Don't leave things for the
last moment. You may struggle to complete the homework assignment if you start
on the night before it is due. Work consistently in small installments.
As you progress through the homework you'll notice it increasing in difficulty. The more difficult problems are often the most important, giving you the opportunity to really master the material. Make sure you attempt them. Almost every problem will be very similar to one done in the video lectures or the worksheets. Consult those if you're unsure of where to start.
Seek help, but only after seriously thinking about a problem on your own. If you're struggling with a challenging problem you should spend at least 30 minutes on your own thinking about it carefully. Even if you fail to make a breakthrough, this is still more worthwhile than giving up after a couple of minutes and talking to a peer or looking up a random source on the internet. Knowing what doesn't work is just as important as knowing what does. The questions will be very closely related to the lectures and textbook. Those should be where you look for help first. In future courses you'll be solving problems that require days (or weeks) of dedicated thought. Now is the time to hone this skill and get comfortable with not being able to immediately solve a problem.
Homework Solutions will be made available shortly after submission.
Actually do the weekly homework! Always attempt to do it on your own and do not immediately look at solutions.
Math 51 student, Fall 2018
In addition to homework, roughly every week, you will have either a quiz or project.
Projects will give you the opportunity to explore certain topics in more depth, perhaps exploring interesting real-world applications, or more subtle questions that we don't have time to cover in lectures. They will be due on Friday, roughly every two weeks on Gradescope. Here are instructions on how to upload your work. Your lowest score will be dropped from your grade.
If you are in LEC001 (High-Touch)...
Quizzes will take place on Friday, roughly every two weeks in discussion section (see the schedule below for the exact dates). They will last about 20 minutes and be closely related to the homework submitted that week. Your lowest score will be dropped from your grade.
If you are in LEC002 (Low-Touch)...
You'll take the same basic quizzes, but it will be done in the Computer Based Test Facility (CBTF). During quiz weeks, you'll be given specific slots, over several days (see schedule below), that you'll be able to sign up to take your quizzes there. Details will be sent directly to you. Here is some general information about the CBTF.
There will be no make-up quizzes, unless there are exceptional circumstances.
There will be two (1-hour) midterm exams and one (2-hour) final exam.
Because of the grading scheme, you can miss one midterm, for whatever reason, without penalty.
On the other hand, missing both midterms will seriously harm your grade and make it much more difficult to pass the course.
As per university policy, if you do not sit the final exam, you automatically fail the course.
Please check the dates now to make sure
that you have no unavoidable conflicts!
Here is the schedule:
Midterm 1: Thursday October 1, in class (Wheeler Auditorium)
Midterm 2: Thursday November 5, in class (Wheeler Auditorium)
Final Exam: LEC 001 Tuesday December 15, 3pm(sharp)-5pm ;
LEC 002 Wednesday December 16, 3pm(sharp)-5pm.
Calculators are NOT be allowed for the exams. That's the policy for all university math classes. That may seem scary, but it means the exams are designed not to require challenging mental arithmetic. That's a good thing!
Cheatsheets are not be allowed for the exams. Making one yourself is a very good way of preparing, but it can't be taken into the test.
To obtain full credit for an exam question, you must obtain the
correct answer and give a correct and readable
derivation or justification of the answer. Unjustified correct
answers will be regarded very suspiciously and will receive little or
no credit. The graders are looking for demonstration that you
understand the material. To maximize credit, cross out incorrect
work. We will be scanning all exams so you will get them back electronically.
After each midterm, there will be a brief window when you can request a regrade.
If you are unsure about making a regrade request consult myself or your GSI beforehand. Regrade requests may result in a lowering of your grade.
As per university policy, final exams cannot be regraded.
All special exam accomodations for DSP students will be arranged by us and not the DSP proctoring service. If you are a DSP student with exam accomodations we will contact you directly the week before an exam.
It goes without saying that cheating is unacceptable. Any student caught cheating will be
reported to higher authorities for disciplinary action.
There will be no make-up exams, unless there are truly exceptional circumstances.
Here is my basic advice on how to do well in exams.
Take a moment to look through the whole exam at the start. We can all agree that exams are no fun. I have always been very anxious taking exams and something that always helped was looking through the exam for a few minutes at the start. Find a question you know how to approach and tackle that first. It'll help settle your nerves.
Success is in the details. It's not enough that you have a broad superficial understanding of a topic. You must appreciate the subtleties and how they turn up in problems.
Diagnose and solve systematically. Once you've identified what type of problem you are dealing with, think about what's in your toolkit. How does one approach such a problem in a methodical way? You want to be familiar and comfortable with the standard procedures to deal with different problems. This is a skill for life and is the essence of analytic problem solving.
The practice exams are the single most important component of exam preparation. I'm going to provide you with three practice exams for each test. These will be pretty challenging. You should be aiming for mastery of every question in the practice exams before the real thing. The main exam will be close variations of these questions and if you've not spent serious time with them it is going to be much harder. Ideally, by the time you take the main exam you want to be able to write perfect solutions to the practice questions unprompted. You should be looking at these at least one week before the main exam.
Spend a lot of time studying the practice exams. Don't leave it till the day before to look at them. They are hard, but the real exams are basically the same.
Math 52 student, Fall 2021
Lastly, I know that many of you find exams acutely stressful (I know I do). If you are struggling (before or during the exam) do come and speak to me or a GSI. We are here to help.
This is your one-stop-shop for all the course materials: videos, notes, worksheets, etc. As the semester progresses each of the links below will go live. For example, video lectures will be available directly after the LEC002 in-person lecture is complete.
This is a warm up video to make sure you are comfortable with three important pieces of notation: the equals symbol, the implies symbol, and the if and only if symbol. Using these correctly is critical for this and all future mathematics classes.
The final will cover the whole course. Exam dates and times are set by the Registrar and will be confirmed here.
Grades are calculated as follows. The exact breakdown depends on which type of section you are enrolled in, high-touch (LEC001) or low-touch (LEC002).
If you are in LEC001 (High-Touch)...
Category
Percentage of Grade
Notes
Discussion Attendance
10%
Three discussions can be missed without penalty.
Homework Assignments (13)
5%
The lowest 2 scores will be dropped.
Projects (5)
5%
The lowest score will be dropped.
Quizzes (5)
15%
The lowest quiz score will be dropped.
Midterms (2)
30%
We'll replace the lowest midterm score with the Final Exam score if it is higher.
Final Exam
35%
Covers the whole course.
If you are in LEC002 (Low-Touch)...
Category
Percentage of Grade
Notes
Homework Assignments (13)
10%
The lowest 2 scores will be dropped.
Projects (5)
10%
The lowest score will be dropped.
Quizzes (5)
15%
The lowest quiz score will be dropped.
Midterms (2)
30%
We'll replace the lowest midterm score with the Final Exam score if it is higher.
Final Exam
35%
Covers the whole course.
In both cases, if your final exam score is higher than your lowest midterm exam score then it will replace that midterm score. This means you can miss a midterm for any reason whatsoever and it will not necessarily adversely affect your grade.
Your final letter grade will ultimately be decided by your ability to demonstrate a crisp understanding of the material and the ability to apply it to a diverse set of problems.
Broadly speaking I will be looking for the following criteria for each letter grade:
A-/A/A+: A clear demonstration that the central concepts have been fully understood;
Computational techniques (and their many subtleties) have been mastered and can be applied accurately to a diverse problem set;
A strong understanding of how the abstract concepts can be applied to many real world applications.
B-/B/B+: Demonstration that the central concepts have been reasonably understood, but perhaps with minor misunderstandings;
Core computational techniques have been reaonably understood (but generally not key subtleties) and can be applied fairly accurately to a fairly large problem set;
Reasonable understanding of how the abstract concepts can be applied to some real world applications.
C-/C/C+: Demonstration that the central concepts have been vaguely understood, but with major misunderstandings; Core computational techniques have been poorly understood and can be applied accurately only in the most standard examples; Weak understanding of how the abstract concepts can be applied to even basic real world applications.
This is not high school. In general, letter grades will no longer directly correspond to the familiar percentages (70-80 for a C, 80-90 for a B, 90-100 for an A). University exams are more challenging than highschool exams and it would not be fair to impose a system this rigid. The eventual grade bins will be decided once the course is complete, although I'll let you know roughly how you are doing after each midterm. To be as fair as possible, I will also take into account the historic average of the class. This means that if I set an exam which is very difficult it will be taken into account in the final letter grades.
I know that grades are important, and how tempting it is to fixate on getting specific percentage scores. However, do remember that every moment spent trying to micromanage your grade is time spent away from actually mastering the material. The most effective way to do well is to focus all your attention on the mathematics.
Please note that incomplete grades, according to university policy, can be given
only if unanticipated events beyond your control (e.g. a
medical emergency) make it impossible for you to complete the course,
and if you are otherwise passing (with a C- or above).
I have no control over enrollment issues. For question about enrollment contact Marsha Snow, or email enrollment@math.berkeley.edu.
The Student Learning
Center provides support for this class, including a full adjunct course,
review sessions for exams, and drop-in tutoring. This is a truly fantastic resource. I definitely recommend you take advantage of it.
I know almost all of you have used AI to help with academics at some point. I'm not going to tell you that you shouldn't use AI, indeed, it can really enhance the learning process. However, there are good and bad ways to use it. You don't need to take my word for it. Here's what Gemini correctly has to say about the matter:
The High Cost of the Shortcut
Using AI to bypass the "struggle" of math homework might feel like a life hack, but from a cognitive and academic perspective, it is a recipe for long-term failure.
Math is less about "finding the answer" and more about building the mental machinery to solve problems you haven't seen before.
Why relying on AI can be disastrous...
1. The "Cognitive Atrophy" Effect
Learning math requires a specific type of mental effort called germane cognitive load. This is the energy your brain uses to build "schemas", permanent mental maps of how numbers and logic work.
Without AI: You struggle, your brain creates new neural pathways, and the concept moves into long-term memory.
With AI: You skip the struggle. Research shows this leads to "cognitive atrophy," where your critical thinking skills shrink because those "muscles" aren't being used.
2. The Illusion of Competence
When you see an AI generate a solution, you experience passive comprehension. However, there is a massive gap between recognizing a solution and generating one. (The same goes for looking at practice exam solutions before spending serious time working on the questions.)
The Data: A 2024 study found that students using AI to complete math practice scored 48% higher on homework but 17% lower on actual exams where the AI was unavailable (and remember, in this course homework is worth only 10% of your grade, while exams are 80%)
Immediately feeding your homework and project problems directly into AI is a guaranteed recipe for disaster in the long term.
3. Missing the "Hidden" Skills
Math homework is a training ground for:
Pattern Recognition: Relating new problems to old ones.
Resilience: Sitting with a problem without giving up.
Precision: Noticing small details like minus signs.
By offloading this to an AI, you do not develop these critical skills.
4. The Risk of "Hallucinations"
AI models are "stochastic parrots", they predict the next likely word but don't "calculate" like a calculator. They can confidently provide incorrect proofs, and without your own knowledge, you won't realize the error until it's graded.
How to Use AI productively...
Shift from using AI as a ghostwriter to using it as a tutor.
Feature
Disastrous Use (The Crutch)
Productive Use (The Coach)
Goal
Get the answer to turn in.
Understand the logic to learn.
Prompt
"Solve this problem for me."
"Explain the first step of this problem."
Verification
Copy the output exactly.
Try to solve it first, then compare.
Result
High homework grade, failed exam.
Solid understanding, long-term skill.
To turn AI from a cheating tool into a high-level tutor, you must use Scaffolded Prompting. This forces the AI to guide your thinking rather than doing the heavy lifting for you.
1. The Master Template
Copy and paste this prompt to set the "rules of engagement" with the AI:
"I am working on [Topic]. I want you to act as a supportive math tutor. Do not give me the final answer. Instead, look at the problem I provide and ask me what I think the first step should be. If I am stuck, give me a small hint or a similar example to help me figure it out myself. Here is the problem: [Insert Problem]"
2. Calculus Case Study: The Power Rule
In a First-Year Calculus course, the Power Rule is the foundation for derivatives. Here is how a productive interaction looks:
Problem: Find the derivative of f(x) = 3x4 - 5x + 2
User: How do I find the derivative of this function?
AI Tutor: I'd be happy to help. The Power Rule states that for any term xn, the derivative is nxn-1. Looking at the first term, 3x4, what do you think happens to that exponent of 4 when we apply the rule?
Note: The AI identifies the rule but leaves the calculation to the student.
User: Does the 4 move to the front and multiply by the 3 to get 12?
AI Tutor: Exactly! Now that we have 12 as the coefficient, what happens to the power of x? Does it stay as a 4, or does it change?
Note: The AI confirms the correct logic and prompts the student for the next specific step.
User: It becomes a 3. So the first part is 12x3.
AI Tutor: Perfect. Now let's look at the second term: -5x. Keep in mind that x is the same as x1. What is the derivative of a linear term like that?
3. Why This Works
Active Recall: You are forced to remember rules (like x having an invisible exponent of 1).
Procedural Memory: By typing the steps, you "code" the logic into your brain.
Confidence Building: You finish the problem knowing why the answer is correct, making you ready for the exam.
Long Story Short: Do not use AI to write your homework and project solutions. You'll get crushed when it comes to exams, which is where your grade is really determined. Not to mention the fact that you'll be underwater in later classes. If you do use it, let it act as a tutor, as outlined above.
