Trying to figure out a way to organize Euler's method, I thought the following might do a decent job. I saw several nice tables similar to this in submitted homework. Definitely play around and do whatever works best for you.
Tuesday, February 28, 2012
Thursday, February 23, 2012
Test Your Knowledge 20120223
More random practice problems. If you hesitate on the answer, strongly consider reviewing the relevant material.
Wednesday, February 22, 2012
Tuesday, February 21, 2012
Class Discussion due 20120228
Do some method recognition on your own. Perhaps we'll do odd in class and you can do evens on your own. If you do evens now and if we exhaust the odd ones in class, then by the time we do evens as a class, it'll be half-new. Note that there's also the review section of chapter seven for which we can go through method recognition.
Any feedback on things you find distracting or that can be improved on?
What is something you learned from reading the book?
I can't be sure I can cover all the bases, but at the minimum, try testing your knowledge here (20120209) and here (20120221).
Any feedback on things you find distracting or that can be improved on?
What is something you learned from reading the book?
I can't be sure I can cover all the bases, but at the minimum, try testing your knowledge here (20120209) and here (20120221).
Class Summary 20120221
[Section 1]
9.2.23 About three iterations.
9.2.22 About three iterations.
Emphasized being able to do calculations by hand.
Drew a picture, but didn't have time to explain the equation in full. Thought perhaps it was too conceptual to be repeated in class.
Was trying to just help with problem 9.3.42 but basically did it only without determining the constants.
pg 499 15-25 odd method recognition.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
25 Allowed to discuss the degree and needing to simplify by using partial fractions. While there, I was able to sneak in my blurb about simplifying common factors and Heaviside cover-up method.
[Section 2]
I started with
pg 499 15-25 odd method recognition.
I gave each one it's own column making the work a little more neat.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
A student for 19 made an important simplification before the u-substitution that I overlooked.
$e^{x+e^x}=e^xe^{e^x}$
I couldn't figure out on the spot if the u-substitution $x+e^x$ really works or not, but I didn't linger, as the purpose of the exercise is to test recognition.
I was able to be more thorough about Heaviside cover-up method, throwing in an example of a repeated factor and using an easy to compute $x$ to compute any lingering variables.
After that I went to discus 9.2.23 up to $y_3$ but without computing $y_3$. I spend a moment to explain the meaning of the formula and drew a little picture.
A student asked about multiple solutions and what that means. I first refer to Figure 8. Then I explain there are infinitely many solutions. If you give an initial condition, then you get a particular solution.
Then I talked about level sets but with #5 instead of #3. (I did #3 in Section 1)
After class a student asked about finishing problem 7.5.25, that is after we have divided to make sure the numerator is less than the denominator. I wanted to hand back homework to those who were waiting for it, while he had to leave.
Another student asked about $y(5)$. I suppose the confusion was why we stop at $y_5$. After some discussion and drawing a picture, it was made apparent that it depends on the step size $h$.
Comments:
There's a lot to cover and there's only so much I can make sure students pay attention to. Ultimately it's up to them to ask the questions and do whatever they can to secure their understanding. For example, by reading the book, going to math help room, going to the professor's or TA's office hours, and comparing solutions with a friend.
9.2.23 About three iterations.
9.2.22 About three iterations.
Emphasized being able to do calculations by hand.
Drew a picture, but didn't have time to explain the equation in full. Thought perhaps it was too conceptual to be repeated in class.
Was trying to just help with problem 9.3.42 but basically did it only without determining the constants.
pg 499 15-25 odd method recognition.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
25 Allowed to discuss the degree and needing to simplify by using partial fractions. While there, I was able to sneak in my blurb about simplifying common factors and Heaviside cover-up method.
[Section 2]
I started with
pg 499 15-25 odd method recognition.
I gave each one it's own column making the work a little more neat.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
A student for 19 made an important simplification before the u-substitution that I overlooked.
$e^{x+e^x}=e^xe^{e^x}$
I couldn't figure out on the spot if the u-substitution $x+e^x$ really works or not, but I didn't linger, as the purpose of the exercise is to test recognition.
I was able to be more thorough about Heaviside cover-up method, throwing in an example of a repeated factor and using an easy to compute $x$ to compute any lingering variables.
After that I went to discus 9.2.23 up to $y_3$ but without computing $y_3$. I spend a moment to explain the meaning of the formula and drew a little picture.
A student asked about multiple solutions and what that means. I first refer to Figure 8. Then I explain there are infinitely many solutions. If you give an initial condition, then you get a particular solution.
Then I talked about level sets but with #5 instead of #3. (I did #3 in Section 1)
After class a student asked about finishing problem 7.5.25, that is after we have divided to make sure the numerator is less than the denominator. I wanted to hand back homework to those who were waiting for it, while he had to leave.
Another student asked about $y(5)$. I suppose the confusion was why we stop at $y_5$. After some discussion and drawing a picture, it was made apparent that it depends on the step size $h$.
Comments:
There's a lot to cover and there's only so much I can make sure students pay attention to. Ultimately it's up to them to ask the questions and do whatever they can to secure their understanding. For example, by reading the book, going to math help room, going to the professor's or TA's office hours, and comparing solutions with a friend.
Test Your Knowledge 20120221
What is $\tan^{-1}(1)$?
What is $\ln(1)$?
Set up the partial fractions $$\frac{1}{(x-1)^2(x^2+2x+2)^2}$$
Set up and solve the partial fractions, try using the Heaviside cover-up method. $$\frac{x}{(x-2)(x-3)}$$
What is $\int \sec(\theta) d\theta$? Note: The easiest way to do this is just to recite the formula from the book found in section 7.2.
What is $\int \frac{1}{\sqrt{1+x^2}} dx$ Note: Some students get this confused with another important integral. Don't get confused.
What is $\sin(\pi/2)$?
What is $\ln(1)$?
Set up the partial fractions $$\frac{1}{(x-1)^2(x^2+2x+2)^2}$$
Set up and solve the partial fractions, try using the Heaviside cover-up method. $$\frac{x}{(x-2)(x-3)}$$
What is $\int \sec(\theta) d\theta$? Note: The easiest way to do this is just to recite the formula from the book found in section 7.2.
What is $\int \frac{1}{\sqrt{1+x^2}} dx$ Note: Some students get this confused with another important integral. Don't get confused.
What is $\sin(\pi/2)$?
Thursday, February 16, 2012
7.3.18, 7.3.25, and help on 7.3.30
7.3.18
For simplicity, I will assume $a$ and $b$ or positive real numbers.
$$\int \frac{dx}{[(ax)^2-b^2]^{3/2}}$$
We identify that we should use secant.
For simplicity, I will assume $a$ and $b$ or positive real numbers.
$$\int \frac{dx}{[(ax)^2-b^2]^{3/2}}$$
We identify that we should use secant.
Tuesday, February 14, 2012
Class Summary 20120214
[Section 1]
Partial fractions. Covered examples from blog.
7.4.50 and 7.4.44.
method recognition pg 499
1) u = sin
3) try simplifying [leave how to integrate $\frac{1}{\sin(x)}$]
5) $u=t^2$ [maybe partial fraction]
7) $u = \tan^{-1}(x)$
9) by parts
11) u = denom (LEAVE IT OPEN)
13) trig 7.2 (We stopped with this one)
trig recognition pg 483
5-13 odd
we ended up finishing 5 through 11 odd.
[Section 2]
trig recognition pg 483
5-11 odd
Partial fractions. Covered examples from blog.
method recognition pg 499
see above.
In this class we stopped at #11.
[Comments]
I wanted to, but didn't have time to do 18, 25, or a modification of 30.
In section 2, I realized that many people didn't have their books during the method recognition exercise and so I wrote the problems on the board.
Partial fractions. Covered examples from blog.
7.4.50 and 7.4.44.
method recognition pg 499
1) u = sin
3) try simplifying [leave how to integrate $\frac{1}{\sin(x)}$]
5) $u=t^2$ [maybe partial fraction]
7) $u = \tan^{-1}(x)$
9) by parts
11) u = denom (LEAVE IT OPEN)
13) trig 7.2 (We stopped with this one)
trig recognition pg 483
5-13 odd
we ended up finishing 5 through 11 odd.
[Section 2]
trig recognition pg 483
5-11 odd
Partial fractions. Covered examples from blog.
method recognition pg 499
see above.
In this class we stopped at #11.
[Comments]
I wanted to, but didn't have time to do 18, 25, or a modification of 30.
In section 2, I realized that many people didn't have their books during the method recognition exercise and so I wrote the problems on the board.
Class Discussion due 20120221
Next time bring your books. It'll help the class move along more smoothly and allow for better group discussions.
Do you like working in partners? Do you like being forced to work with someone or would you rather partner with someone on your own?
What don't you like about the discussion section?
What do you like about discussion section?
People did a good job of stapling their papers, I would like to remind that failure to do so will result in a loss of a point on homework.
Do you like working in partners? Do you like being forced to work with someone or would you rather partner with someone on your own?
What don't you like about the discussion section?
What do you like about discussion section?
People did a good job of stapling their papers, I would like to remind that failure to do so will result in a loss of a point on homework.
5 Solutions to Problem 7.2.10
Problem 7.2.10
This took at least two hours to put together and write up. Just letting
you know the effort I'm willing to put in to help you do well.
This took at least two hours to put together and write up. Just letting
you know the effort I'm willing to put in to help you do well.
Thursday, February 9, 2012
Test Your Knowledge 20120209
Look through and see if you can do these. Try to work them out. You may submit questions here and qualify for participation of Class Discussion due 20120214.
If you finished these and would like more, let me know with a comment or e-mail! I need positive feedback that students are making use of such a post. Such a comment will not satisfy participation requirements, unless it provides constructive criticism, such as comments regarding the type of problems which appear.
If you finished these and would like more, let me know with a comment or e-mail! I need positive feedback that students are making use of such a post. Such a comment will not satisfy participation requirements, unless it provides constructive criticism, such as comments regarding the type of problems which appear.
Tuesday, February 7, 2012
Class Discussion due 20120214
Check out the discussion that's already happened here: Class Discussion Due 20120207
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
Thursday, February 2, 2012
Integral of x*exp(x)*cos(x) with respect to x
This post is a follow-up to Useful Mnemonic: DETAIL where the problem was posed.
Determine $f(x)=\int x e^x \cos (x)dx$.
CHOICE A (Suggested by DETAIL):
Let $u=x\cos(x)$ and $dv= e^x dx$.
Then $du=\cos(x)-x\sin(x)dx$ and $v=e^x$.
Then $f(x)=x\cos(x)e^x - \int e^x \left[ \cos(x)-x\sin(x) \right] dx$
CHOICE B:
Let $u=x e^x$ and $dv=\cos(x)dx$.
Then $du = e^x+x e^x dx$ and $v=\sin(x)$.
Then $f(x)=x e^x \sin(x) - \int \sin(x) \left[ e^x +x e^x\right] dx$
CHOICE C:
Let $u=e^x$ and $dv=x \cos(x)dx$.
Then $du = e^x dx$ and $v=\int x\cos(x)dx$
Let $\hat{u}=x$ and $d\hat{v}=\cos(x)dx$.
Then $\hat{v}=\sin(x)$
Then $\int x\cos(x)dx = x \sin(x)- \int\sin(x)dx = x\sin (x)+\cos(x) +\hat{C}$
We only concern ourself with a particular $v$, so we assume $\hat{C}=0$.
Then $f(x) = e^x \left[ x\sin(x) + \cos (x) \right] -\int \left[x\sin(x)+\cos(x) \right] e^x dx$
What remarks can we make? For this specific problem, regardless of the above three choices for $u$ and $dv$, we will have to do integration by parts again. DETAIL recommended choice A, but choice B was just as short. Though as a personal opinion, I find it easier to make sign errors when determining antiderivatives to trigonometric functions than when determining antiderivatives to exponential functions.
Recall that $\frac{d}{dt} \sin (t) = \cos (t)$, while $\frac {d}{dt} \cos (t)=-\sin(t)$.
Determine $f(x)=\int x e^x \cos (x)dx$.
CHOICE A (Suggested by DETAIL):
Let $u=x\cos(x)$ and $dv= e^x dx$.
Then $du=\cos(x)-x\sin(x)dx$ and $v=e^x$.
Then $f(x)=x\cos(x)e^x - \int e^x \left[ \cos(x)-x\sin(x) \right] dx$
CHOICE B:
Let $u=x e^x$ and $dv=\cos(x)dx$.
Then $du = e^x+x e^x dx$ and $v=\sin(x)$.
Then $f(x)=x e^x \sin(x) - \int \sin(x) \left[ e^x +x e^x\right] dx$
CHOICE C:
Let $u=e^x$ and $dv=x \cos(x)dx$.
Then $du = e^x dx$ and $v=\int x\cos(x)dx$
Let $\hat{u}=x$ and $d\hat{v}=\cos(x)dx$.
Then $\hat{v}=\sin(x)$
Then $\int x\cos(x)dx = x \sin(x)- \int\sin(x)dx = x\sin (x)+\cos(x) +\hat{C}$
We only concern ourself with a particular $v$, so we assume $\hat{C}=0$.
Then $f(x) = e^x \left[ x\sin(x) + \cos (x) \right] -\int \left[x\sin(x)+\cos(x) \right] e^x dx$
What remarks can we make? For this specific problem, regardless of the above three choices for $u$ and $dv$, we will have to do integration by parts again. DETAIL recommended choice A, but choice B was just as short. Though as a personal opinion, I find it easier to make sign errors when determining antiderivatives to trigonometric functions than when determining antiderivatives to exponential functions.
Recall that $\frac{d}{dt} \sin (t) = \cos (t)$, while $\frac {d}{dt} \cos (t)=-\sin(t)$.
Useful Mnemonic: DETAIL
[20120202]
A student had an interesting random thought.
What to do when the integrand is a product of three functions?
In particular he asked about $\int x e^x \cos (x)dx$.
What do you think?
[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.
A student had an interesting random thought.
What to do when the integrand is a product of three functions?
In particular he asked about $\int x e^x \cos (x)dx$.
What do you think?
[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.
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