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?
Tuesday, October 26, 2010
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?
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?
Tuesday, October 19, 2010
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
Problems with a strike through them have not yet been answered.
3.5 #2a,
3.6 #6,11,12,16,20,24,30,40,48,50
3.9 #
Tuesday, October 12, 2010
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."
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."
Friday, October 8, 2010
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:
$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f^{\prime}g-fg^{\prime}}{g^2}$
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.
Thursday, October 7, 2010
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
]^{-1} = \frac{1}{f(x)})
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".
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.
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
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.
Friday, October 1, 2010
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
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.
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.
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