Difference Between Right and Wrong

WRONG: $\frac{1}{(x^{2}+1)\sin x}\cdot\left((x^{2}+1)\cos x+(\sin x)\cdot2x\right)=\cos x+2x=2x\cos x$

You have to factor or distribute before you can cancel terms.

CORRECT: $\frac{1}{(x^{2}+1)\sin x}\cdot\left((x^{2}+1)\cos x+(\sin x)\cdot2x\right)=$
$\frac{(x^{2}+1)\cos x}{(x^{2}+1)\sin x}+\frac{2x\cdot\sin x}{(x^{2}+1)\sin x}=\frac{\cos x}{\sin x}+\frac{2x}{x^{2}+1}=\cot x+\frac{2x}{x^{2}+1}$

WRONG: $\frac{1}{2xyy^{\prime}+y^{2}}=\frac{1}{2xyy^{\prime}}+\frac{1}{y^{2}}$

CORRECT: Well... avoid that mistake.

WRONG: $2\sin4x^{2}=8\sin x^{2}$

CORRECT: Well... avoid that mistake.

WRONG: $\frac{\sin(2x^{2})(4x)}{x}=\sin(2x^{2})(3x)$

CORRECT: $\frac{\sin(2x^{2})(4x)}{x}=\sin(2x^{2})(4)=4\sin(2x^{2})$

WRONG: $\cos0=0$

CORRECT: $\cos0=1$

WRONG: $\cos=1$

CORRECT: Er... cosine is a function, so it should be $\cos x$ or
$\cos u$ etc...

(Most) Final Exam Solutions

\#2b)

Most people did chain rule:

$f^{\prime}(x)=\frac{1}{(x^{2}+1)\sin x}\left(2x\cdot\sin x+(x^{2}+1)\cos x\right)$

Most errors resulted in not having parentheses, not knowing how to
do chain rule, or perhaps as simple as not knowing the derivative
of $\sin x$ is $\cos x$.

A simpler way to do the problem is by writing

$f(x)=\ln((x^{2}+1)\sin x)=\ln(x^{2}+1)+\ln(\sin x)$

So that

$f^{\prime}(x)=\frac{2x}{x^{2}+1}+\frac{1}{\sin x}\cos x=\frac{2x}{x^{2}+1}+\cot x$.

Note that doing it this way part of the problem comes for free provided
you did \#2a correctly.

A Requested Explanation

In general: $F(x)=\int_{a}^{f(x)}g(t)dt$. Then $F^{\prime}(x)=g(f(x))\cdot f^{\prime}(x)$.

Or perhaps more generally: $F(x)=\int_{f(x)}^{g(x)}h(t)dt$. Then
to compute the derivate we should first rewrite $F(x)$.

$F(x)=\int_{f(x)}^{a}h(t)dt+\int_{a}^{g(x)}h(t)dt=-\int_{a}^{f(x)}h(t)dt+\int_{a}^{g(x)}h(t)dt$

$F^{\prime}(x)=-h(f(x))\cdot f^{\prime}(x)+h(g(x))\cdot g^{\prime}(x)$.

Last Minute Thoughts

$e^{2x}+e^{x}\neq e^{3x}$
$e^{2x}\cdot e^{x}=e^{2x+x}=e^{3x}$
In general, $a^b\cdot a^c=a^{b+c}$

It is important you remember
$\int \frac{1}{1+u^2}du=\tan^{-1}(u)$

Know the derivative of all the trigonometric functions: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x).

Some Requested Solutions and Help

1e) $\lim_{x\to0}e^{x\cos\left(e^{-1/x}\right)}$

It is enough to solve the following limit:

$\lim_{x\to0}x\cos\left(e^{-1/x}\right)$

Why is it enough to show this?

In any case, we observe that

$-|x|\leq x\cos\left(e^{-1/x}\right)\leq|x|$

Do you know why this is?

Please Ask Questions

Hi All,

With the final being cumulative, it's hard to give advice. The practice final seems to hit the major key points. Follow what I said in class regarding practice with the method of substitution and memorizing certain trigonometry identities and integrals/derivatives and even basic values. Less often but possible is determining if an integrand is an odd or even function to help in your computations. In any case, I hope you have learned good study habits by now and I wish you good luck, but do ask questions and I will help the best I can. This includes solution write-ups.

Your TA,
Tim

Three Opinions and Two Facts

Partial credit is great. But there's no need to complain if partial credit isn't given. Why? Well, one day, it just might happen that partial credit won't be given and you won't be in a position to complain, so get used to it.

Example: You are working on a project, everybody does their part. Perhaps people check your work over. In any case, your project gets designed and then for some reason or another it fails. Well... you don't get partial credit. I suppose in this case, you wouldn't want to take any credit! (for the blame)

The previous example also lends itself to waiting for other people to check your work for you. Just because someone might check your work, doesn't mean you can skip checking your own work. Sometimes the wait is too late. Also it can just reflect badly. Granted, you don't always have to do such a thorough job, especially when you're tired, but when you can, check your work.

Speaking of checking your work, what about showing your work? Ideally, you should have a sufficient amount of work that will allow the reader (keeping your audience in mind, that is how familiar will the reader be in your work) to follow relatively easy and also, highlight (meaning to emphasize or summarize in some form) the answer. Sometimes clear and concise is better than showing an excessive amount of work, and sometimes being descriptive is better than being brief.

Example (in math):
Writing $\tan (x)+\sec ^2 (x)$ is a better alternative to writing $\frac{\sin(x)}{\cos(x)}+1/{\cos^2(x)}$

Example (in life):
Your boss requests a summary of the work you've been doing. Your write-up is long and will probably include a cover letter summarizing the contents. He or she will probably just read the cover letter, but it's possible he or she will glance inside or pass it on to someone who does understand. Alternatively, what's written inside should be understandable to the audience, in this case your boss.

And unrelated to the above, when integrating, area that is below the x-axis is negative! (This is of course for Calculus I)

Be consistent with your variable.
Example: $G(x)=\int _2^x \sqrt{3t}+\sin(t)dt$
Then $G^{\prime}(x)=\sqrt{3x}+\sin(x)$ for $x>0$
Don't write $G^{\prime}(x)=\sqrt{3t}+\sin(t)$ for $t>0$
You could if you want write $G^{\prime}(t)=\sqrt{3t}+\sin(t)$ for $t>0$
Or $G^{\prime}(u)=\sqrt{3u}+\sin(u)$ for $u>0$
Or $G^{\prime}(v)=\sqrt{3v}+\sin(v)$ for $v>0$
Or $G^{\prime}(%)=\sqrt{3%}+\sin(%)$ for $%>0$
And so on...

Don't Do This

Incorrect: $\lim_{x\to\infty}\frac{5}{x}=\frac{5}{\infty}=0$
Correct: $\lim_{x\to\infty}\frac{5}{x}=0$

Section 4.4 Exercise 10

$\begin{align*}\lim_{x\to 0}\frac{\sin 4x}{\tan 5x} & =\lim_{x\to 0}\frac{\sin 4x\cdot \cos 5x}{\sin 5x}\\
& =\lim_{x\to 0}\frac{4}{5}\frac{\sin 4x}{4}\frac{5}{\sin 5x}\cos 5x\\
& =\frac{4}{5}\cdot \lim_{x\to 0}\left(\frac{\sin 4x}{4}\right)\cdot \lim_{x\to 0}\left(\frac{5}{\sin 5x}\right)\cdot \lim_{x\to 0}\left(\cos 5x\right) \\
& =\frac{4}{5}\cdot 1 \cdot 1 \cdot 1=\frac{4}{5}
\end{align*}$

Alternative solution using L'Hospital's rule:
$\begin{align*}\lim_{x\to 0}\frac{\sin 4x}{\tan 5x} &=\lim_{x\to 0} \frac{4\cos 4x}{5\sec ^2 5x}\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \cdot \cos ^2 5x \right)\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \right) \cdot \lim_{x\to 0} \left( \cos ^2 5x \right)\\
& =\frac {4}{5}\cdot 1 \cdot 1=\frac{4}{5}\end{align*}$

Brief Study Suggestions

If when attempting these problems you have a question, there are many people you can ask. Your friends. TA's at the math help room. Or me, either by e-mail or commenting below.

So I looked at the practice midterm and I would do the following.
Make sure I know all the basic differentiation rule. Including but not limited to product rule, chain rule, quotient rule. Knowing when to use which rule. Knowing some of the methods that will make a problem easier. So, how should this be done. Well I'd start with page 261 Concept Check. It asks to state various rules and derivative for basic functions. Correct me if I'm wrong, but I haven't seen any homework involving Concept Check 2o-2t so you can skip the derivative of hyperbolic trig functions and their inverses. But the concept questions are basic to the course. Your answers to 3-5 don't have to be on the dot, at least a close idea of the concept is sufficient. The true-false section on that page is optional if you have the time. Otherwise, go on to page 262 and do all the problems 1 through 50, skipping any that deal with hyperbolic trig functions. If you can do these all correctly, then you will likely get the first 30 points of the exam. If you can't do all of them correctly, keep practicing. Practice makes perfect.

Next, do all 5-64 on page 305. As the instructions indicate, try to avoid l'Hospital's Rule for a more elementary method if possible, and try to make sure you're allowed to use the rule when you apply it.
When possible, try to spot when a problem uses the squeeze theorem or $\lim_{\theta \to 0} \frac{\sin \theta}{\theta}=1$ such as problem #39. Doing this practice is another 20 points.

Optimization problems are slightly harder to practice. Doing many of them takes more time, and it's hard to generalize. This type of problem will often require a little extra thought in multiple ways. But strengthening the basics will allow you to try and think through these slightly more involved problems.
Remember:
1) Try drawing a picture
2) Try simplifying the problem by saying, what if I had a similar problem that asked this instead. What would I do in that case...
3) Try writing out some relationship between the information given.

The last 30 points hopefully will be fairly easy. Read through the first derivative test and the second derivative test, and all the other important boxes in section 4.1 and 4.2 and 4.3, such as the definition of a critical number, or even the definition of an absolute maximum for example can be handy to know. There are a lot of problems in those two sections to do for practice.

For any graphs, label and be detailed. Besides the typical labeling of axes, if you have time, add a few words justifying what you drew.

If you have time, read my homework 7 and homework 8 solutions, or at least glance for any tips that might prove useful.

General test taking tips:
Read through each problem. Writing down some ideas. This way the problems are in your mind throughout the test period and your mind will unconsciously think about them. This is another way of saying don't spend too much time on one problem.
Don't erase work that is correct. When in doubt, something is better than nothing.
Do show work.
Along the lines of the last two, if you get an answer, that is somewhat out of the blue compared to the work you did, you should explain how you got it. Otherwise, you're likely just to receive a zero for the correct answer which is unqualified by your work.
Don't look at other people's papers. Besides the fact that this is cheating and wrong, what if their answers are wrong?

HW8 Solutions

I skipped writing the solution for many problems. But if you need
any that I omitted, or even a problem that you tried apart from the
homework set, let me know. I'll get around to it.

Homework 8

4.3: 6, 8, 14, 20, 22, 30, 50, 68, 82

4.4: 2, 6, 12, 14, 18, 42, 52, 54, 64, 80

4.7: 24, 50

HW7 Solutions

Solutions to most problems are given. Please ask any questions which come up. Please let me know about any errors you find. You can leave a comment or send an email.

Homework 7:
3.10: 4, 14, 18, 20, 23, 28, 30, 32 a
4.1: 4, 34, 42, 53, 56, 68 b, 77
4.2: 14, 16, 18, 20, 24, 28, 32

What to Learn

Sometimes, perhaps what you need to learn, isn't what you think you should learn.
Maybe, instead of learning about calculus, you should be learning about methods and habits.

For example, when you do homework, do you do it in an organized fashion? I like to see homework that is done top-done. If you insist on saving space, then divide the page down the center and form two orderly columns. This is presentable. Clear, and understandable. When you go into the real world, that's how your boss would want your work.

Another thing is learning to break down big problems into little problems. Or taking problems you don't understand into problems you do understand. For example, a linear approximation is much easier to understand than a complicated function. All you have to do is take the derivative and form the tangent line at the point. If you ever learn about Taylor series, then you'll see that you can form better approximations, and yet still keep a certain about of simplicity. In any case, we are turning a complicated object into a simple object. In real life, this is a reasonable skill. If you can break down a big project into little tasks that are simple, then the problem is much easier to handle.

Drawing pictures. Who doesn't like pictures? The more you practice drawing diagrams that help you understand the problem, the more useful they'll be. But not only to you, but the people around you. Sometimes, words get complicated, and pictures get the job done.

Practice makes perfect. You might take doing homework for granted, but when you are given something to learn without being able to practice it, or without seeing examples, you wish you had homework. In life, when you learn something just by reading about it or by hearing about it, you won't really have learned it. You need to make it your own and somehow put it to use or practice it, just like doing homework. Learning about how to ride about and learning to ride a bike are two different things. Once you learned, and haven't done it in a while, it can take a little readjustment, but if you actually learned it before, it's likely that you'll be able to pick up on it real quick the next time.

Listen. This is a good habit to practice. Here's a fact: a professor can hand students the answers in a lecture, and the people who normally get A's will likely get A's, the students who get B's will likely get B's, and so on. This is in large due to the correlation between grades and listening. It's true that sometimes, it's because people don't know how to process the information given to them. But there's usually help somewhere, wherever you are, you just have to know where to look. That's not always easy. But for example, maybe its a fear of taking advantage of help. At least in the setting of college, teaching assistants and professors hold office hours to help you. Schools have counselors. In real life, there's usually a support group for every problem you can think of. With the internet you can find all sorts of support and help.

The point is, sometimes it's the skills you learn that is important. Depending on your career choice, the content might not be that important.

All this applies to your other classes too.

Do you have any skills that you have learned in class that are useful for real life?

Some Main Ideas You Should Remember

The first part of calculus is about rates of change.
The derivative is just a function that gives the instantaneous rate of change of another function.
The instantaneous rate of change is the slope of the tangent line.
Thus, the derivative of a function gives us information about how the function behaves.
If you look very closely, a function can be approximated by a line.

What ideas do you think you should remember? What are other big ideas from the course?

HW6 Solutions Section 3.6

I let the mistake of forgetting to put parentheses when appropriate pass under the radar for this homework. But you might not be so lucky in future homeworks.

HW6 Solutions Section 3.5

This is the solutions to homework 6. For now there are only solutions. But I will slowly add errors to avoid.

Problems with a strike through them have not yet been answered.
3.5 #2a,2b,2c, 7,12,24,26,52
3.6 #6,11,12,16,20,24,30,40,48,50
3.9 #4,30

How Grades Tend to Work

In a math class, most instructors assign grades something like this:
At the beginning of the semester, percentages for homework, midterms, quizzes, and final are determined.
Throughout the semester, all grades are kept as raw scores. Sometimes an instructor/professor will give ranges that serve as an indicator for how well you are doing in the course. Such ranges only serve as indicators, and do not actually shift the score in any way.
At the end of the semester, all the grades, homework, midterms, quizzes, and final are added together (using the appropriate percentages), and compared in bulk.
The instructor looks at these raw totals and assigns grades appropriately. Students who sit on the edge of two possible grades, are moved up or down depending on underlying factors such as improvement, final exam score, participation/effort in class, etc.

Example:
Say I had a class and Homework is 20%, Midterm is 30%, Final is 50%.
I take the midterm and get a 30 out of 100 points. But the average on the exam was 25, so the instructor sends out an email that says my grade falls between an A/B. It's the end of the semester and I got a 80 out of 100 on the final. I've gotten an 8/10 on all the homeworks, so I have 80% of my homework score.
Thus my raw total is .8*.2+.3*.3+.8*.5=.16+.09+.40=.65
This is low, but other students will have low raw scores too.
Perhaps I fall into the range deemed A's. Perhaps everybody got 90's on the final and 100% homework, so that I fall into the range of B's or C+'s.

I like this person's post with regards to curving and grading exams, it's titled "How to curve an exam and assign grades."

A list of things you can't do...

$\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$

Example:$\sqrt{4+16}=2\sqrt{5}\neq6=2+4=\sqrt{4}+\sqrt{16}$

~

$\frac{a+b}{a+c} \neq \frac{1+b}{1+c}$

Example:$\frac{4+8}{4+2}=\frac{12}{6}=2 \neq 3 = \frac{9}{3}=\frac{1+8}{1+2}$

~

$(a+b)^2 \neq a^2 + b^2$

Example:$(2+3)^2 = 25 \neq 13 = 2^2 + 3^2$

~

$\frac{0}{0} \neq 0$

$\frac{0}{0}$ is undefined!

~

$\frac{a}{b} \neq \frac{\log (a)}{\log (b)}$

Example:$\frac{100}{10}=10 \neq 2=\frac{\log (100)}{\log (10)}$

~

$\frac{a}{b} \neq \frac{\sqrt{a}}{\sqrt{b}}$

Example:$\frac{4}{9} \neq \frac{2}{3} = \frac{\sqrt{4}}{\sqrt{9}}$

~

$\frac{d}{dx}\frac{f(x)}{g(x)}\neq\frac{\frac{d}{dx}f(x)}{\frac{d}{dx}g(x)}$

Example:$\frac{d}{dx}\frac{x^3}{x^2}=\frac{d}{dx}x=1\neq\frac{3x}{2}=\frac{3x^2}{2x}=\frac{\frac{d}{dx}x^3}{\frac{d}{dx}x^2}$

In general, use the quotient rule:
$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f^{\prime}g-fg^{\prime}}{g^2}$

Midterm Exam Solutions

Problems 1d) and 4) were considered difficult. A common mistake was missing the second horizontal asymptote y=-2 for problem 2 (myself included). Points were deducted for forgetting to include the domain of f inverse. By the way, my practice midterm 1 solutions explicitly advise to include the domain of the inverse function.

Practice Midterm 1 Solutions

Like my homework 4 solutions, what you'll find here aren't straightforward solutions, but can be helpful. And I might occasionally throw in other random pitfalls to avoid.

1a) This type of problem is straightforward. The individual functions are continuous at x=0 so their sum is continuous at x=0, and so by definition, the limit of a function f continuous at x=a as x tends to a, is f(a). Problems similar to this will use pg 99 Limit Laws, pg 101 other Limit Laws, pg 102, pg 119, pg 122, pg 124, pg 125.

1b) Read examples about quotients of polynomials. pg 101 example 2. pg 102 example 3. pg 133 theorem 5. pg 133 example 3. pg 134 example 4. pg 135 example 5.

Important: pg 133 states that most of the limit laws in section 2.3 also hold for limits at infinity. Read this paragraph.

It's important to read other examples in section 2.3 and other examples in section 2.6 as well.

1c) This definitely uses theorem 8 on page 125. So first find the limit of the expression inside arcsin. One can rationalize the numerator, or factor the denominator as done on page 125 example 8.

1d) This is, like 1c, similar to an example from the book. Check out page 135 example 5.

Important: read examples from section 2.3 and examples from 2.6.

2) Know the definition of horizontal and vertical asymptotes (pages 132 and page 95, respectively).

Note: Understanding the definition of a vertical asymptote, it is enough to have either a left-hand limit at x=a or a right-hand limit at x=a go to infinity, to say x=a is a vertical asymptote of a function f(x).

TYPO: I didn't catch this earlier, but this problem has an error.
The domain is incorrectly written

It should somehow indicate f(x) does not exist at x = 1.
or


3a) Explain why this function is not one-to-one and hence is not invertible. Read page 61, the definition of an inverse function carefully.

3b) Go through the process of finding the inverse function, but remember to indicate the domain of the inverse function. Again, read the definition of an inverse function on page 61 carefully. If f has domain A and range B, then its inverse has domain B and range A.

Note, page 61 warns the reader to not confuse notation. Do not mistake
and

If for some reason you get confused on the exam, feel free to clarify with whoever is proctoring.

4) This problem has multiple parts for which you can get credit.
First, you should be able to identify that the slope of the tangent line must be 3/2.
Second, you should take the derivative of f(x) (here use the power rule, but the problem can give you any function you've learned how to derivate using material covered in weeks 1 through 5.
Third, solve for x's for which f'(x)=3/2.
Fourth, the answers to the first question are of the form (x,f(x)). In this problem, there is only x for which f'(x)=3/2 and so there is one point.
Fifth, to write the equation of the normal line through this point, you must realize that you need to take -1/m where m is the slope of the tangent line (3/2). Hence the normal line has slope -2/3.
Sixth, use the point-slope formula to determine the equation of the normal line, using the slope -2/3 and the point (x,f(x)).

Note: Typically, even if you got a value wrong, if you correctly carry the value over to the next part of the problem, you will get credit. If you cannot obtain values, you should still write down the formula you would have used. In this case, the slope of a normal line is -1/m and the point-slope formula are relevant formulae that might help you get partial credit.

On that note, you should have an intuitive idea of what you can and can't argue credit for. If you felt like you sufficiently showed you knew what you were doing and didn't get enough credit for a problem, then by all means ask. More often than not, it doesn't hurt to ask. A popular reason that credit won't be awarded to you is when everybody else who put the same work you put into the problem, got the same amount of credit you did. Made-up example: Nobody who wrote the point-slope formula got a point for writing the point-slope formula. Nevertheless, I recommend showing you know what you're talking about when possible (as well as hiding what you don't know you're talking about when possible.)

Finally, show work and/or explanation even when a problem doesn't explicitly ask for it. While you may be marked off for extra information which is incorrect, you won't be marked off for extraneous relevant and correct information. You don't have to quote theorem number and pages, but when possible, state relevant steps. Examples: "because the function is continuous" "the limit of a sum is the sum of the limits" "by definition of a vertical asymptote".

A Bad Idea

It is a bad idea to study so much that you burn out and fall asleep and miss the exam. If you do choose to study excessively, make sure you have someone to wake you up. Personally I wouldn't recommend this, because then you're brain will be too fried to take the exam. But you do what you think you have to do I guess.

HW4 Solutions Part 2

Again, not your typical homework solutions, but what's written here should be useful.

I last left off classifying my thoughts about finding the limit to problems 3-20 for the Chapter 2 Review on page 167.

Notation:
\sqrt{f} means the square root of f
\frac{f}{g} means the quotient f/g
a^b means a to the power b

Finding Old Exams

Well, there's really no difference between an old calculus exam that you can find on any course website from another school and an old calculus exam that you can find on a course website from Hopkins. Exams differ from professor/lecturer to professor/lecturer. Sometimes if a professor/lecturer is teaching the same course, he/she will remove his/her old exams from his/her website. This allows a professor/lecturer to recycle good exam questions.

In any case, I can't take the fun out of your search for old exams. But if you're new at looking for old exams, the first place to look would be at old course websites. If the number of exams you can find there isn't enough, you can use your favorite search engine to look for more. Or, go to your second favorite university's website and seek out their math course webpages.

It helps to plan ahead.

HW4 Solutions Part 1

Not your typical homework solutions, but it should be useful.

Information

Practice exam will be posted online on Friday.

The exam will cover material from week 1 to 5 (except epsilon-delta).

I left the textbook at the office, so while I planned on posting solutions to HW 4 today, I was unable to write up the solutions. Mostly likely I can get them up tomorrow.

In the meantime, you should review the limit laws and derivation rules. Do lots of problems! Ask questions. You can e-mail me any questions you have, post a comment on this blog, or go to help room and ask questions (remember they're open from 9am to 9pm Mondays through Friday). Also you can search around for old exams.

HW3 Solutions (Select Problems)

[20100924]
Here are some solutions to the third homework set. Page references in my solutions should be looked at. Theorems referenced, especially those with names, should be understood or memorized. For example, you should if the limit of two functions f and g at a point a, then the limit of the sum is the sum of the limits.

You can click on the pictures to get a larger view:

20100804 Wed

Covered Final Exam Topics:
1) Inequality with absolute values
2) Equation with exponents
3) Finding roots of a third degree polynomial
4) Determining an equation of a line
5) Definition of a function
6) Find difference quotient
7) Is this an even function? odd function?
8) Finding asymptotes
9) Amount formula (Compounded periodically, compounded continuously)
10) Equation with logarithms
11) Evaluate trigonometric functions
12) Composition of trigonometric functions, for example: find sin(arccos(x)).
13) Using identities
14) More using identities
15) Finally more using identities

Note: I forgot to mention #9 in class. This is simply plugging the right numbers into a formula.

Homework:
Review for the exam, but at the same time, get a good night's sleep. There will be no make-up exam for a missed final.

[20100805]
The above topics have been posted since Monday, 20100802, perhaps with a small change. They did not change, however, between the review on Wednesday, 20100804, and the exam on Thursday, 20100805. Sometimes, its not about how hard you study, but knowing what to study. Sometimes, the material is just plain difficult. That's life. You need to pick yourself up and move on.

20100803 Tue

Went over problems , , ,

Covered sections 6.5, 6.6

Key Concepts:
Section 6.5
pg 547 Product-to-Sum Identities
Section 6.6
pg 558 Summary: Solving cos x = a
pg 559 Summary: Solving sin x = a
pg 560 Summary: Solving tan x = a
pg 564 Strategy: Solving Trigonometric Equations

Homework:
Section 6.5 1-15 (odd)
Section 6.6 1-25 (every 8)
Not due.

20100802 Mon

Went over two problems

Covered sections 6.1, 6.3, 6.4, 3.6

Key Concepts:
Section 6.1
pg 510 Pythagorean Identities [in particular sin^2(x) + cos^2(x) = 1]
pg 513 Odd and Even Identities
Section 6.3
pg 528 Identity: Cosine of a Sum
pg 529 Identity: Cosine of a Difference
pg 530 Cofunction Identities
pg 531 Identities: Sine of a Sum or Difference
Section 6.4
pg 537 Double-Angle Identities
pg 540 Half-Angle Identities
Section 3.6
pg 330 Definition: Rational Function
pg 334 Summary: Finding Asymptotes for a Rational Function
pg 337 Example 7: A graph with a hole in it

Homework:
Section 6.1 1-57 (every 8), also 11, 13, 15, 29-36, 87-97 (odd)
Section 6.3 1-49 (every 8)
Section 6.4 1-49 (every 8)
Due Thursday

The due date on the extra credit assignment has been extended to tomorrow.

20100729 Thu

Covered sections 5.4, 5.5, 5.6. Also discussed a little bit of asymptotes again, as well as emphasized determining the domain and range of functions. In particular, I discussed the domain and range of inverse functions.

Key Concepts:
Section 5.4
pg 465 Definition: Tangent, Cotangent, Secant, and Cosecant Functions
pg 465 Identities from the Definitions
Section 5.5
pg 477 Definition: The Inverse Sine Function
pg 479 Definition: The Inverse Cosine Function
pg 480 Inverses of Tangent, Cotangent, Secant, and Cosecant
pg 481 Function Gallery
pg 482 Identities for the Inverse Functions
Section 5.6
pg 490 Thoerem: Trigonometric Functions of an Acute Angle of a Right Triangle

Homework:
Section 5.4 1-89, every 8; Also 2, 3-28 ("Undefined" is a possible answer),
Section 5.5 1-97, every 8; Skip 49, 57, 81, 89; Also 65-79 Odd, 93-98
Section 5.6 1-57, every 8
Due Monday

[20100805]
I might have learned on this day that the signs of Sin(x), Cos(x), and Tan(x) in quadrants I, II, III, and IV, can be given by ASTC (All Students Take Calculus). That is "All" are positive, only "Sine" is positive, only "Tangent" is positive, and finally only "Cosine" is positive, in quadrants I, II, III, and IV, respectively.

20100728 Wed

Covered sections 5.1, 5.2, 5.3
Also reviewed solving inequalities

Key Concepts:
Section 5.1
pg 422 Degree-Radian Conversion
pg 425 Theorem: Length of an Arc
pg 427 Theorem: Linear Velocity in Terms of Angular Velocity
Section 5.2
pg 434 Definition: Sine and Cosine
pg 441 The Fundamental Identity of Trigonometry
Section 5.3
pg 448 Definition: Periodic Function
pg 449 Definition: Amplitude
pg 453 Theorem: Period of y = sin(Bx) and y = cos(Bx)
pg 453 The General Sine Wave
pg 456 Definition: Frequency

Homework:
Section 5.1 1-97, every 8 (i.e. 1, 9, 17, ...)
Section 5.2 1-89, every 8
Section 5.3 1-73, every 8
Due Monday

20100727 Tues

Covered sections 4.3, 4.4

Key Concepts:
Section 4.3
pg 387 Rules of Logarithms with Base a
pg 389 Base-Change Formula
Section 4.4
pg 399 STRATEGY: Solving Exponential and Logarithmic Equations

Homework:
Section 4.3 1-89, every 8
Section 4.4 1-81, every 8
Due Monday

Extra Credit Homework
1) Copy all brown and green boxes, 2) Copy and work through examples, 3) Try This for Sections P.1, P.2, P.3, P.5, 1.1, 2.4, 2.5, 3.6.
1) Copy all brown and green boxes for Sections 1.6, 2.1, pg 220

[20100728]

20100726 Mon

Covered sections 4.1, 4.2

Key Concepts:
Section 4.1
pg 354 Definition: Exponential Function
pg 358 Properties of Exponential Functions
pg 360 One-to-One Property of Exponential Functions
pg 362 Compound Interest Formula
pg 363 Continuous Compounding Formula
Section 4.2
pg 370 Definition: Logarithmic Function
pg 372 Properties of Logarithmic Functions [compare this with pg 358 Properties of Exponential Functions]
pg 374 One-to-One Property of Logarithms

Homework:
Section 4.1 1-85, every 4
Section 4.2 1-101, every 4
Due Monday

[20100728]

20100722 Thu

Midterm Exam - covers preliminaries, Chapter 1, Chapter 2, and Chapter 3

Covered section 4.1

Homework:
Read Chapter 4

[20100728]

20100721 Wed

July 21, 2010
Review for Midterm

Went over
CHP Test 12, 15, 19, 29, 32
CH1 Test 4, 11, 17
Section 2.1 Example 6
CH2 Test 6, 7, 10, 24
CH2 Review 69
Section 3.1 75 c,d
CH3 Test 2, 7, 12, 14, 20, 21, 23, 25, 29

20100720 Tue

July 20, 2010
Went over
3.2 61, 71
3.3 31, 33, 58, 59, 61, 83, 84, 86, 87, 88
3.4 17, 45

20100719 Mon

July 19, 2010
Went over
3.1 9, 15, 29, 57. 75g was silly.
3.2 25, 53, 57. Didn't have time for 71.

Homework 9 due Wednesday
Chapter 3 test, page 351, 1-28

20100715 Thu

July 15, 2010
Went over
2.2 69
2.3 65 (Yesterday)
2.4 3, 12, 13, 14, 33, 41, 45
2.5 27, 41, 45, 47, 51, 67, 69

On the test, I will ask you the definition of a function, page 186.

Homework 7 due Tuesday
3.3 1-9 odd, 19, 27, 39-47 odd, 49-55 odd, 57-71 odd, 83-88
3.4 11-25 odd, 35-47 odd

Homework 8 due Wednesday
Chapter P test, page 85, 1-36
Chapter 1 test, page 181, 1-26
Chapter 2 test, page 265, 1-24

20100714 Wed

July 14, 2010
Went over
2.1 21
2.2 59, 63
2.3 31, 33, 55, 57, 65

Homework 6 due Monday
3.1 1-12, 13, 15, 21, 25, 29, 33-43 odd, 57, 61, 65, 75, 76, 77, 81, 85, 89, 93
3.2 1-71 odd

20100713 Tue

July 13, 2010
Went over
1.2 63
1.3 39, 41
1.4 79

Homework 5 due Thursday
2.4 1-12, 21-28, 41-44, 49-84 odd.
2.5 19-33 odd, 39-60 odd, 63-78 odd

20100712 Mon

July 12, 2010
Went over
1.2 9, 32, 40
1.3 29, 79
1.4 93
1.6 35, 39, 77, 96, 99, 100, 115
1.7 21, 25, 33, 71, 75, 77, 85

Homework 4 due Thursday
2.1 1-10, 11, 15, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 59, 63, 67, 75, 79, 83, 87, 91
2.2 1, 9, 17, 25, 33, 41, 47, 49, 51, 53, 55, 59, 63, 67, 69, 71
2.3 17-24, 25-69 odd, 71-78

20100708 Thu

July 8, 2010
Went over
1.2 13, 17, 18, 63
1.3 56, 94

KEY CONCEPT:
Quadratic Formula

Homework 3 due Monday
1.6 1, 5, 11, 15, 25, 29, 31, 35, 39, 41, 49, 53, 61, 67, 77, 85, 93, 95-100, 115
1.7 9, 13, 17, 21, 25, 29, 33, 37, 39, 43, 47, 53, 57, 61, 65, 71-78, 85

20100707 Wed

July 7, 2010
Went over
P.5 8, 50
P.6 18
P.7 22, 46, 74
1.1 20, 30, 40, 42

Homework 2 due Monday
1.2 1, 5, 9, 13, 17, 23, 31, 39, 47, 55, 63
1.3 1, 11, 19, 25, 29, 33, 37, 41, 45, 55, 59, 71, 75, 79, 91, 95
1.4 1, 11, 19, 23, 27, 37, 47, 57, 63, 67, 71, 75, 79, 85, 89, 93

20100706 Tue

July 6, 2010
Homework 1 due Thursday
P.4 10, 20, 28, 44, 52, 62, 76, 80, 84
P.5 2, 8, 12, 18, 22, 32, 42, 50, 60, 70, 78, 84, 92
P.6 2, 10, 18, 32, 42, 52, 60, 76, 80
P.7 2, 10, 22, 34, 40, 46, 64, 74, 83
1.1 2, 6, 20, 30, 42, 56, 66, 74