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...
Tuesday, November 30, 2010
Wednesday, November 10, 2010
Don't Do This
Incorrect: $\lim_{x\to\infty}\frac{5}{x}=\frac{5}{\infty}=0$
Correct: $\lim_{x\to\infty}\frac{5}{x}=0$
Correct: $\lim_{x\to\infty}\frac{5}{x}=0$
Tuesday, November 9, 2010
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:
$\begin{align*}\lim_{x\to 0}\frac{\sin 4x}{\tan 5x} &=\lim_{x\to 0} \frac{4\cos 4x}{5\sec ^2 5x}\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \cdot \cos ^2 5x \right)\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \right) \cdot \lim_{x\to 0} \left( \cos ^2 5x \right)\\
& =\frac {4}{5}\cdot 1 \cdot 1=\frac{4}{5}\end{align*}$
& =\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:
$\begin{align*}\lim_{x\to 0}\frac{\sin 4x}{\tan 5x} &=\lim_{x\to 0} \frac{4\cos 4x}{5\sec ^2 5x}\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \cdot \cos ^2 5x \right)\\
& =\frac {4}{5}\lim_{x\to 0} \left( \cos 4x \right) \cdot \lim_{x\to 0} \left( \cos ^2 5x \right)\\
& =\frac {4}{5}\cdot 1 \cdot 1=\frac{4}{5}\end{align*}$
Saturday, November 6, 2010
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?
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
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
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
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