Tuesday, June 30, 2009
An alternative perspective on Math.
"Math, my dear boy, is nothing more than the lesbian sister of biology." - Peter Griffin
Saturday, June 27, 2009
A Newton's Laws of Motion Problem!
Two adults and a child want to push a wheeled cart in the horizontal direction x. The two adults push with horizontal forces 
and 
as shown in the free-body diagram. (a) Find the magnitude and direction of the smallest force that the child should exert. You can ignore the effects of friction. (b) If the child exerts the minimum force found in part (a), the cart accelerates at 
in the +x-direction. What is the weight of the cart?
(a)If the box is pushed in the +x-direction by the total force of the two adults and the child...
]=-16.6=-17N=17N "F_{3y}=-[86.6+(-70)]=-16.6=-17N=17N")
Because the question asks for the magnitude, take the absolute value of the final answer -17N, which is 17N. The direction in which the little girl is pushing is 90 degrees with respect to the x-axis.
(b) Because we know the x-components of the forces, we can find the mass, which will allow us to calculate the weight of the cart by w=mg.
(9.8m/s^{2})=840N\therefore "w=mg=(85.6kg)(9.8m/s^{2})=840N\therefore")
(a)If the box is pushed in the +x-direction by the total force of the two adults and the child...
Because the question asks for the magnitude, take the absolute value of the final answer -17N, which is 17N. The direction in which the little girl is pushing is 90 degrees with respect to the x-axis.
(b) Because we know the x-components of the forces, we can find the mass, which will allow us to calculate the weight of the cart by w=mg.
Indefinite Integrals
Here is a problem on the first homework for my Calculus II class. I thought it was pretty nice so here you go!
Use
to help evaluate the integrals:
\;\int \sin^{2}(\frac{x}{2})dx\;\;\;\;\;\;and\;\;\;\;\;\;(b)\;\int\cos^{2}(\frac{x}{2})dx "(a)\;\int \sin^{2}(\frac{x}{2})dx\;\;\;\;\;\;and\;\;\;\;\;\;(b)\;\int\cos^{2}(\frac{x}{2})dx")
To be able to evaluate this integral using the stipulations above, we have to rewrite it in such a way that looks more familiar. One way this can be done is by rewriting the integrand using
as a substitution for x/2.
Because we have exchanged x/2 for theta we also have to change dx to
.
Substitute then we will have:
Now doesn't this look familiar?
If it does look familiar, you should notice that it is a part of the identity:
Do some rearranging and we get this:
Now we can rewrite the integral again and get:
Okay don't get too excited! We're not done yet! Remember that
Plug these back into the equation, do the cancellations and we get:
Integrate to get the final answer:
\;dx=\frac{1}{2}(x-\sin x)+C "\int \sin^{2}(\frac{x}{2})\;dx=\frac{1}{2}(x-\sin x)+C")
That was fun right?! I thought so! :]
I'm not going to go through the second integral because it's basically the same thing. Try it for yourself and you should get the answer:
\;dx=\frac{1}{2}(x+\sin x)+C "\int \cos^{2}(\frac{x}{2})\;dx=\frac{1}{2}(x+\sin x)+C")
Use
To be able to evaluate this integral using the stipulations above, we have to rewrite it in such a way that looks more familiar. One way this can be done is by rewriting the integrand using
Because we have exchanged x/2 for theta we also have to change dx to
Substitute then we will have:
Now doesn't this look familiar?
If it does look familiar, you should notice that it is a part of the identity:
Do some rearranging and we get this:
Now we can rewrite the integral again and get:
Okay don't get too excited! We're not done yet! Remember that
Plug these back into the equation, do the cancellations and we get:
Integrate to get the final answer:
That was fun right?! I thought so! :]
I'm not going to go through the second integral because it's basically the same thing. Try it for yourself and you should get the answer:
Wednesday, June 24, 2009
Willst Du Mit Mir Gehn?
So far this summer has been great. It is my first time taking summer classes and I really like the way they work. Most students would gawk at having to attend a class or two over the summer, but I thoroughly like it much better than the semesters. The thing I hate the most about going home and having no school during the summer is that I tend to lose track of things. I get lazy and its always difficult to get back into the swing of things at the beginning of the semester.
I just got done with the first summer session today. I took Calculus 1 and English 102. The final in Calculus was pretty easy. The only thing I was taken off for was because I wrote in the infinity symbol instead of DNE (does not exist) - which I think is dumb, because it's the same thing if x is approaching infinity from the left or the right side. But its cool though, I'm not worried about it. As long as I get an A in the class!
English 102 was actually to my surprise a very good class. Despite having to write a lot of papers and reading responses, the reading material was pretty interesting. Some of the articles were by people like Althusser, Pollan, Singer; just to name a few. The novel we read towards the end of the class is titled Nervous Conditions by Tsitsi Dangarembga. I'm not going to go into what the book is about because that would take another whole blog post! But basically the theme of the class was the Rhetoric of Food. For the final project we were put into groups based on our topic of interest to make a website. My group created a site to inform the general public about the dangers of fast food and how it is affecting our nation and the world. The website consists of a few papers we did in class such as a Food Narrative, Article Summary, and interviews of people with different views on fast food. You can view the website by going here:
http://fastfoodgroup.webs.com
Now that this first session is over, tomorrow will be starting anew with Calculus 2. I am very excited because I will taking the class with a couple of friends and will be able to have a study group again. In the future I will post entries on the progress of the class and possibly a problem here and there.
Also here is a song I really like!
I just got done with the first summer session today. I took Calculus 1 and English 102. The final in Calculus was pretty easy. The only thing I was taken off for was because I wrote in the infinity symbol instead of DNE (does not exist) - which I think is dumb, because it's the same thing if x is approaching infinity from the left or the right side. But its cool though, I'm not worried about it. As long as I get an A in the class!
English 102 was actually to my surprise a very good class. Despite having to write a lot of papers and reading responses, the reading material was pretty interesting. Some of the articles were by people like Althusser, Pollan, Singer; just to name a few. The novel we read towards the end of the class is titled Nervous Conditions by Tsitsi Dangarembga. I'm not going to go into what the book is about because that would take another whole blog post! But basically the theme of the class was the Rhetoric of Food. For the final project we were put into groups based on our topic of interest to make a website. My group created a site to inform the general public about the dangers of fast food and how it is affecting our nation and the world. The website consists of a few papers we did in class such as a Food Narrative, Article Summary, and interviews of people with different views on fast food. You can view the website by going here:
http://fastfoodgroup.webs.com
Now that this first session is over, tomorrow will be starting anew with Calculus 2. I am very excited because I will taking the class with a couple of friends and will be able to have a study group again. In the future I will post entries on the progress of the class and possibly a problem here and there.
Also here is a song I really like!
Tuesday, June 23, 2009
Looking for a good time?
Make the u substitution:
Differentiate:
Now distribute, solve for dy/dx and simplify! :D
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