Math 242 Section 5&6

As per request, an example of error in Taylor expansions. Make sure to know how to do each of the types of problems that come up:

1) Finding error for a given approximation and interval (21 on 9.9 for example)

2) Finding an interval for a given approximation and given error (this problem)

Read instructions:  Many students lost a solid percentage of their grade for simply not answering the question. The top of HW #3 states for problem 1 and 2, write your answer in p(x) = q(x)d(x) + r(x) form. The next few questions required you to fully factor the given polynomial (a step which you do along the way, you just needed to make the point “here is the factorization.”)

Show your work: You must show work to justify your answer. It’s professor Myers’ policy that no attached paper be submitted with homework. Thus certain problems are harder to do (e.g. HW 4 problem 2). But you must show work. This means either write smaller, or if absolutely necessary, write on the back of the page. 

Use proper mathematical language: When writing an essay in english, you need to finish each of your sentences. It doesn’t matter if you have a profound thought: the reader can’t read your mind, only the words you write down. Or for those with coding experience, if you write improper code, your program won’t compile. Math language works the same way. Be logical and organized. Don’t turn in scratch work. The nature of such a mistake is very unique to the individual, but here is an example of “bad writing.”

You can see the issue here : if you put “=“ between all of your thoughts, you end up saying false statements. Everything you write needs to be true. I good self-check method would be to ask yourself: if a random person read this, could they follow it easily without any confusion? If the answer is no, it isn’t written well.

Completing the square: I’ve attached a sample completing the square problem, which addresses several of the types of mistakes made.

X-intercept and y-intercept notation: Remember, intercepts are values, not coordinates or equations. This might seem unnecessary or pedantic, but later in the course you will see why we need to be restrictive about our notation.  

Exponents don’t distribute over addition, only multiplication: It is very critical to identify this, and stop yourself from making this mistake. The majority of problems in this course require you to square a sum at some point. Also every time I see this mistake, I get sad. Don’t make your TA cry.   

So while preparing for class tomorrow, I realized when I did an example problem detailing one type commonly made mistake, I was guilty of making another one! So here is both the old problem showing the error, and then a new page explaining the issue. 

Overall, the issues were with problem 1, 3, and 4. My comments on 1 are related to the issues of rounding and logic of a proof. I’ll explain in detail on our class Thursday. Problems 3 and 4 are related, and I hope to clarify things with the example below. Also note that how I went about answering this question is the type of organization I’m looking for in homework. 

Here are some do’s and don’t’s for MAT 140. This is by no means an official syllabus, but rather a quick rundown of some of my expectations, mixed with some tips on how to succeed in this course. 

DO

Homework - Do the homework, and do it on time. The late policy is pretty forgiving, but not turning homework in on time has larger consequences. It is easy to fall behind, and then by the time an exam comes, you will not be prepared. 

Read the directions - I cannot emphasize this enough. There are several assignments where it is easy to accidentally lose many points over trivial stuff. For example, an assignment might have instructions which say “for each problem, solve it using method A, then solve it using method B.” If you didn’t read those instructions, you’d just solve the problems however you wanted, and then earn a max grade of a 50.

Professionalism of the document - Your homework is not scratchwork; it is a professional document, the purpose of which is to communicate your understanding the material. 

Show Work - If you don’t, I cannot show you where you made a mistake. Therefore, when you don’t show work, it is impossible for me to give partial credit. In addition, when you write your work down, you are more likely to catch a mistake. On homework, a missed mistake has little impact on your overall grade for the course. However you need to get in the habit of catching your own mistakes, because a mistake on an exam can dramatically affect your grade in the course. 

Legibility / Organization - A professional document must be neatly organized and easy to read. This means you should do scratchwork on the side, and then on your homework sheet, write the necessary and relevant calculations in an organized manner. This includes circling your answers. An excellent rule of thumb: as you finish your assignment, look at it and ask yourself, “If I gave this to my boss, would I be fired, or receive a warning?” If the answer is “yes” , then clean it up a bit. (NOTE: the only exception about professionalism is that homework must be done in pencil, not pen.) 

Collaboration - Working with others is strongly encouraged. However, do not let this become a crutch. The purpose of collaboration with another student is to ensure you both understand 100% of the material, not to make you have 50% of the work. Homework prepares you for an exam, and if you aren’t understanding the homework you turn it, you won’t understand the exam. If you cannot do a certain question, ask me. I am absolutely willing to help, inside and outside of my office hours. If you don’t have time, write on the assignment as much as you can, including what is confusing to you. I’ll give appropriate partial credit and detailed feedback.

Read my feedback - I take grading very seriously. This means not only do I grade harshly, it means I give very detailed feedback. Further below is detailed explanation of why to read the feedback carefully.

Prepare for each class - 

Practice problems - Look at the lecture online ahead of time (On the website, they are called “Practice Quizzes”.) Try doing it yourself. This way, If you get stuck on something, you will know exactly when to ask me a question as I go over the problem.

Group classwork - You have to bring in your own classwork sheet. Start before class. If you get stuck, then you know what to ask me during class. If you bring in the classwork already completed, all I have to do is grade it (NOTE: you receive the lowest grade in your group. If you bring a complete classwork, make sure to help the rest of your group.) If you meet your group before class and all finish together, its possible for you to leave class an entire 30 minutes early!

Monitor your own deadlines - I’ll try to give you guys reminders when I can, but all your deadlines are clearly given on the website. The responsibility falls on you to make sure you hand in assignments on time.

Take advantage of a free tutor: me! - Any tutor you pay for won’t know exactly what your grader is expecting, or might get stuck because they haven’t seen the material in years. Well I AM your grader, so I know exactly what I expect of you, and I have to deal with this all the time! So come to my office hours, or schedule an appointment—I’m super flexible. 

Have fun - My secret agenda when I teach this class is to get people to like math. 

DON’T

Take a low grade the wrong way - I have very high standards for grading. In the first few weeks of class, you will get back a few grades you will be upset about. It is critical to NOT interpret this the wrong way. Don’t get back the bad grade and think to yourself “wow, I suck at math. I can’t do this.” The low grade means there is something you need to work on. Recall that I give detailed feedback. Read the feedback, make sure you understand what the issue is (if you don’t, ask me) and then incorporate this into future assignments. Most of the time, the mistake is in the process of how a problem is done. You will see that in this (and every math course) all the material cumulates, so it is imperative to understand everything every step along the way. For example, say you do an assignment which involves “completing the square” , and you consistently make the same methodical error. This causes every answer to be wrong, and you earn a 40 on the assignment. You see the grade, become upset, and do not want to even look at any of the comments. Then later in the course, when we get to equations for ellipses, you will get every problem wrong again because you don’t know how to complete the square (which is necessary for every ellipse problem.) Every homework assignment is connected to multiple others, so neglecting to understand your errors will guarantee more later.

Use phones or computers in class - I will learn all of your names. And I will call you out on using your cell phone in really embarrassing ways. 

Let yourself fall behind - If you find yourself falling behind in this class, do something about it as soon as possible. Mathematical education is cumulative; if you fall behind, you cannot move forward until you catch up. This means you have to move faster, which makes it harder to pay attention to current material, and the cycle repeats. Don’t get caught in it. If you feel overwhelmed, come to my office or send me an email. 

Use pen - This is really a matter of the “Do be professional” bullet above. When you write an essay, lab report, or even an email, do you ever use the backspace key? Of course you do. So use a pencil for this class. 

If any of this is unclear to you, please do not hesitate to email me with questions. 

I understand how to get C1 and B1, but what do they mean by C2 and B2?

When you’re given two sides and an angle that is not between them, it is possible to get two possible answers for one angle. One answer for C will be theta and the other will be 180-theta. The picture below is a visual explanation. The large acute triangle (A, and the red B and C) and the smaller obtuse triangle (A and the blue B and C) will both satisfy the given information. (Note: This only occurs when the side opposite the given angle is shorter than the side adjacent)