The registrar schedules the final exam.

The exam will cover the entire semester, but there will be a slight emphasis on the material since the last midterm (the material on Problem Sets 28 - 35).

Books, calculators, and notes will not be permitted on the exam.

As this
recent article highlights, two of the best study strategies are
**testing yourself** (with flashcards, practice problems,
and so on) and **spacing your studying over
a period of time**.
So, we encourage you to start your studying early: use these review questions to help you
recall the key skills and concepts we've learned this semester, and go
back to your notes,
worksheets, and homework to help you fill in details you've
forgotten. Make a list of important ideas and strategies,
and think about how different ideas connect. Do some practice
problems on each topic. Here is some material to help you
with this process:

- A compilation of the learning objectives for the semester
- Review questions - Some questions to think about as you look back at the material we've covered.
- Practice problems by chapter
- If there are topics from the first 2 exams that you had trouble with, we encourage you to go back and try some of the practice midterms again.

As always, we strongly encourage you to **complete several
practice exams without looking at the solutions or using extra
aids**. After you've finished a practice exam, go to office hours with any questions you have!

- Practice Exam Reflection: To get the most out of a practice exam, you don't want to just do it; you also want to use it to figure out what to study next, as well as what errors you tend to make and how to avoid those errors in the future. This practice exam reflection is meant to help you do that; do it right after you've taken a practice exam (before looking at the solutions).
- Practice Exam 1 - solutions
- Practice Exam 2 will be given as the proctored practice final, so we'll post it online afterwards. If you'd like to take the proctored practice final, please fill out this anonymous poll by Thursday 12/7 so that we know how many copies to make.
- Practice Exam 3 - solutions
- Practice Exam 4 - solutions
- Practice Exam 5 - solutions
- Practice Exam 6 - solutions

A full schedule is posted here.

- We have asked for the review sessions to be taped. Videos of the review sessions will be posted here as soon they're available.

Exam 2 will be **Tuesday 11/14, 5:30 - 7:30 pm in Science
Center C**. If you have a valid conflict (see the top of this
page), please fill out this exam
conflict form by Wednesday 11/8 to arrange an earlier exam time.
Please note that most sections and labs can be rearranged to accomodate
an exam, and we expect you to contact your section organizers right away
to arrange to attend a different section if your section conflicts with
the 21b exam.

The exam will focus on the material on Problem Sets 12 - 27, but it is cumulative in the sense that this material relies on the ideas and techniques from Problem Sets 0 - 11, so you will still need to be comfortable with those. Remember that Math 21b emphasizes deep and flexible understanding of the material. As such, you should expect the exam to include problems you haven't previously encountered, so that you can demonstrate your ability to apply the fundamental concepts to new problems.

Books, calculators, and notes will not be permitted on the exam.

We encourage you to start studying early by reviewing your notes, worksheets, and homework to help you recall the key definitions, skills, and concepts we've covered so far. Make a list of important definitions, ideas, and strategies for yourself, and explain to yourself why the strategies work. Here is some material that may help you with this review:

- A compilation of the learning objectives from Problem Sets 12 - 27
- Review questions - Here are some key questions to check your understanding of the material we've covered. (We won't post answers, but we're happy to talk about these with you in office hours.)
- Interactive review:

The single best thing you can do after this is to **complete a
few practice
exams without looking at the solutions or using extra aids**.
(It is very tempting to peek at the solutions when you get stuck, but
we have found repeatedly that this is not a useful study strategy. Recent
research backs this up!) After you've
finished a practice exam, don't look at the solutions yet! Instead, go
back to any problems you weren't able to do, and try them again, using
any source you like except the solutions (so, for example, look for
related problems from the worksheets and homework). Finally, after
you've done that,
look at the solutions and grade your work,
and go to office hours with any questions you have!

- Practice Exam Reflection: To get the most out of a practice exam, you don't want to just do it; you also want to use it to figure out what to study next, as well as what errors you tend to make and how to avoid those errors in the future. This practice exam reflection is meant to help you do that; do it right after you've taken a practice exam (before looking at the solutions).
- Practice Exam 1 -
solutions
**This practice exam only covers the material through Problem Set 23, so it's a good one to do if you're starting your studying early.** - Practice Exam 2 - solutions
- Practice Exam 3 - solutions
- Practice Exam 4 -
solutions
- Updated 11/12 to fix a typo in #7(a) (the last entry in the given vector should be -5); thanks, Alexandra!

- Practice Exam 5 - solutions
- Fall 2009 exam -
solutions
**This exam only covers the material on Problem Sets 15 - 24, so it's another good one to do if you're starting your studying early.**- Solution updated 11/8 to fix an arrow in the phase plane analysis on pg. 17. (Thanks, Sofia!)

- Recommended practice problems by topic (mostly from Bretscher) - includes lots of true-false questions!

- Review session problems and brief answers to the ones we didn't talk about
- Video of the review session will be posted here as soon as it is available.

Exam 1 will be **Wednesday 10/4, 5:30 - 7:30 pm in Science
Center B**. If you have a valid conflict (see the top of this
page), please fill out this exam
conflict form by Wednesday 9/27 to arrange an earlier exam time.
Please note that most *section meetings* and *lab
sections* can be rearranged to accomodate an exam, and we expect you
to contact your section organizers right away to arrange to attend a
different section if your section conflicts with the 21b exam.

The exam will cover the material on Problem Sets 0 - 11. Remember that Math 21b emphasizes deep and flexible understanding of the material. As such, you should expect the exam to include problems you haven't previously encountered, so that you can demonstrate your ability to apply the fundamental concepts to new problems.

Books, calculators, and notes will not be permitted on the exam.

We encourage you to start by reviewing your notes, worksheets, and homework to help you recall the key definitions, skills, and concepts we've covered so far. Make a list of important definitions, ideas, and strategies for yourself, and talk over your list with other students.

The single best thing you can do after this is to **complete a
few practice
exams without looking at the solutions or using extra aids**.
(It is very tempting to peek at the solutions when you get stuck, but
we have found repeatedly that this is not a useful study strategy. Recent
research backs this up!) After you've
finished a practice exam, don't look at the solutions yet! Instead, go
back to any problems you weren't able to do, and try them again, using
any source you like except the solutions (so, for example, look for
related problems from the worksheets and homework). Finally, after
you've done that,
look at the solutions and grade your work,
and go to office hours with any questions you have!

Here is review material; the 4 practice exams are formatted in the same style as your exam will be, but the past exams are also excellent sources of problems.

- A compilation of the learning objectives from Problem Sets 0 - 11
- Recommended practice problems from Bretscher - These include lots of true/false questions, which are a great way to practice thinking more flexibly about the material!
- Practice Exam Reflection: To get the most out of a practice exam, you don't want to just do it; you also want to use it to figure out what to study next, as well as what errors you tend to make and how to avoid those errors in the future. This practice exam reflection is meant to help you do that; do it right after you've taken a practice exam (before looking at the solutions).
- Practice Exam 1 - solutions
- Practice Exam 2 - solutions
- Practice Exam 3 - solutions
- Practice Exam 4 - solutions
- Fall 2007 exam -
solutions
- Omit T/F #17, as we have not discussed similar matrices.

- Fall 2009 exam
(shortened to omit topics we have not yet covered) - solutions
- In #2, skip the last column of the table, as we have not discussed orthogonal matrices.

- Here are the review session problems. Full solutions will not be posted, but you are welcome to discuss them with a TF in office hours.
- Video of the review session will be posted here as soon as it is available..