We are currently working on understanding strange things like ellipsoids and hyperbolic paraboloids and all sorts of strange things like that in my Calc 3 class. It seems like it wouldn’t be that hard, but it is pretty hard to draw these things correctly! This week we got a take home quiz! Thank goodness! I’m posting one example where we are supposed to name the shape and then draw it and talk about the plane sections that intersect it. I don’t know if I’m doing it correctly or not. Guess I will find out!
Basically, how you know it is an equation for an ellipsoid is that an ellipsoid has a standard form for the equation which is basically (x/a)^2 + (y/b)^2 + (z/c)^2 = 1. Now there are many variations on that, so that it might look a bit different, but if it is too different it will be a different shape, like a hyperboloid of one sheet or something else like that. The a’s give you the ellipse size along the x-axis, the b’s give you the ellipse parts for the y-axis and the c’s give you the ellipse information for the z-axis. It is kind of hard to explain in words! lol I guess what you first need to know is that a regular old ellipse is drawn on the x and y axis where the a number gives you information on the number of steps away from the origin along the x-axis and the b number gives you the number of steps away from the origin along the y-axis. I guess I better post an example.
In the example below #1b, you look at the bottom numbers to give you the size of the ellipse. The x-axis number is 5, so it is five units wide on each side. The y-axis number is 10, so it is 10 units wide on each side of the origin. The x-3 and y-3 has to do with shifting the graph away from the origin up and over 3 units each.
Anyway, from what I can gather, to draw the ellipsoid (not ellipse) like in example 2a, you need to draw the various ellipses that make up the ellipsoid in order to get the final shape, so you set x, y and z equal to zero for three different cases to get your ellipse equations. Blah blah blah. This probably makes no sense to anyone, but I thought I’d share! lol
Hyperbola (top) and Ellipse (bottom)
More Calc2 Fabric
Orange Chevrons with Balloons
I made another Calculus fabric through spoonflower (http://www.spoonflower.com/fabric/1187406), because I was so excited about my first Calculus fabric. Well, the material arrived in the mail today! How exciting!!! I also got my design of balloons on orange chevrons, but I have to work on perfecting the chevrons and getting it how I like it. So fun to get fabric in the mail! :) Click the photos for a larger view. :)
This example almost made my brain explode! I didn’t know about the Weierstrass substitution method (see this link for more info.: http://en.wikipedia.org/wiki/Weierstrass_substitution) when I began this problem…which is what I needed to complete the problem. I’m still unsure if I did it correctly, because we were supposed to find the integral over 0 to 2*pi, but the integration method only works over 0 to pi, so I ended up multiplying the answer by 2 (because 2 times pi is 2*pi) which got me the answer the book was asking for, but it seemed odd to do it that way. The question is number 18, section 13.3 from the Jon Rogawski Calc book in case you are interested. I think it is a pretty great textbook by the way…I mean as far as math textbooks go! We were also supposed to find when the speed was at its max, but I was so tired I just figured it was at pi and gave up! So, here’s the problem!
Let me just say: Calc 4 is confusing!!! We are working with vectors and all sorts of weird 3D shapes and it is strange! Here is an example homework problem. I hope I survive Calc 4!!!
Here is another Calc 2 example for those who are interested! I have my final on Monday! FRIGHTENING! Then next term it is on to Calc 3 & 4!!! Wheeeee! :)
Yep, I’m still taking Calculus. On our last homework we were learning about measuring volume by rotating small sections around the x or y axis. Here I am demonstrating use of what they call the “Shell Method.” It seems a little easier to me than the Disk or Washer method. But then again, I guess it depends on the problem! lol
Here is another page of Calculus homework. I thought it would be fun to add the rocket as my projectile, and then I felt compelled to add the moon, Earth and Sun! hahaha. We were working on definite integrals and finding distance traveled and displacement.
I’m very pleased to report that this last term at school went great! I took the first term of Calculus and Intro. to Linear Algebra and got A’s in both classes! I also had a really good time with the classes. I had great teachers too! I have seen a real improvement in my ability to actually get to classes because of my bipap sleep machine and have seen an even greater increase in my ability to do homework and concentrate since I started taking Abilify last January. Doing well in school has always been important to me, but because of my schizophrenia and sleep problems I have had a lot of issues with being able to attend and get through classes.
When I first became ill when I was 15 my grades really suffered. For the first time in my life I wasn’t a straight A student which was quite disheartening. The last two terms in school I have been feeling a lot like my old self for the first time since I was 15!!!
Anyway, this last term was great and a real ego booster! I just hope I can keep up the success! :)