So here we have a little grapher that solves quadratic equations visually. Just enter the coefficients in the equation.
Quadratic Equation: y = a x2 + b x + c
Enter:
y =
x2 +
x +
Solution:
Analytical solution by factoring (with a little help from the quadratic equation if necessary).
Hopefully, this can help students learn about factoring and quadratics in a more graphical way.
Notes:
This is my first interactive post. I use Javascript in combination with HTML5. Now I need to figure out how to interact with the image itself, instead of the textboxes.
The Centers for Disease Control (CDC) has been thinking about the potential for a zombie apocalypse. You can find a page on Zombie Preparedness on their website, as well as a graphic novel (9Mb pdf).
If you are generally well equipped to deal with a zombie apocalypse you will be prepared for a hurricane, pandemic, earthquake, or terrorist attack.
-- Ali Khan (2011) (Head of the Office of Public Health Preparedness and Response at the Centers for Disease Control and Prevention): Quoted in Zombie Preparedness on the CDC website.
NOTE: The CDC recommends you quarantine zombies rather than kill them; Kyle Munkittrick, of the Pop Bioethics blog disagrees.
Say you have the equation that gives you the slope of a curve (let ) be the slope):
When you use integration to solve the equation, there are quite the number of possible solutions (infinite actually), because when you integrate:
you get:
where c is a constant. Unfortunately, you don't know what c is without more information; it could be anything.
However, even without integrating, we can get a feel for what the curve will look like by plotting what the slope will look like at a bunch of different points in space. This comes in really handy when you end up with a equation for slope that is really hard -- or even impossible -- to solve.
The graph below show a curve of possible solutions to the slope equation. You should be able to see, as the graph slowly moves up and down, how the slope of the graph corresponds to the slope field.
This is an attempt to illustrate numerical integration by animating an HTML5's canvas.
We're trying to find the area between x = 1 and x = 5, beneath the parabola:
By integrating, the area under the curve can be calculated as being 17 ⅔ (see below for the analytical solution). For numerical integration, however, the area under the curve is filled with trapezoids and the total area is calculated from the sum of all the areas. As you increase the number of trapezoids, the approximation becomes more accurate. The reduction in the error can be seen on the graph: with 1 trapezoid there is a large gap between the shaded area and the curve; more trapezoids fill in the gap better and better.
The table below show how the error (defined as the difference between the calculation using trapezoids and the analytic solution) gets smaller with increasing numbers of trapezoids (n).
Number of trapezoids
Area (units2)
Error (difference from 17.66)
1
15.00
2.66
2
17.00
0.66
3
17.37
0.29
4
17.50
0.16
5
17.56
0.10
6
17.59
0.07
7
17.61
0.05
8
17.63
0.03
9
17.63
0.03
10
17.64
0.02
Analytic solution
The area under the curve, between x = 1 and x = 5 can be figured out analytically by integrating between these limits.
(See also WolframAlpha's solution).
So I'm experimenting with creating drawings using HTML5. And it works (mostly)!! I have to hardwire in paragraph breaks, but otherwise it works so far.
The key references:
w3schools.com: HTML5 Canvas: for the code for the drawing above.
"Soviet means excellent." Soviet progaganda poster via How to be a Retronaut.
An excellent set of Soviet propaganda posters from How to be a Retronaut. The collection contains a fascinating blend of of triumphalism sprinkled with some attempts at modesty.
"Glory to the conquerors of the universe."
The posters make wonderful subjects for the study of propaganda and the space race.
My favorite:
) be the slope):
By integrating, the area under the curve can be calculated as being 17 ⅔ (see below for the analytical solution). For numerical integration, however, the area under the curve is filled with trapezoids and the total area is calculated from the sum of all the areas. As you increase the number of trapezoids, the approximation becomes more accurate. The reduction in the error can be seen on the graph: with 1 trapezoid there is a large gap between the shaded area and the curve; more trapezoids fill in the gap better and better.
The table below show how the error (defined as the difference between the calculation using trapezoids and the analytic solution) gets smaller with increasing numbers of trapezoids (n).
![Area = \left[-\frac{x^3}{12} + \frac{x^2}{2} + 4x \right]_1^5](http://l.wordpress.com/latex.php?latex=Area%20%3D%20%5Cleft%5B-%5Cfrac%7Bx%5E3%7D%7B12%7D%20%20%2B%20%5Cfrac%7Bx%5E2%7D%7B2%7D%20%2B%204x%20%5Cright%5D_1%5E5%20&bg=FFFFAA&fg=000000&s=3 "Area = \left[-\frac{x^3}{12} + \frac{x^2}{2} + 4x \right]_1^5")
![Area = \left[-\frac{(5)^3}{12} + \frac{(5)^2}{2} + 4(5) \right] - \left[-\frac{(1)^3}{12} + \frac{(1)^2}{2} + 4(1) \right]](http://l.wordpress.com/latex.php?latex=Area%20%3D%20%5Cleft%5B-%5Cfrac%7B%285%29%5E3%7D%7B12%7D%20%20%2B%20%5Cfrac%7B%285%29%5E2%7D%7B2%7D%20%2B%204%285%29%20%5Cright%5D%20-%20%5Cleft%5B-%5Cfrac%7B%281%29%5E3%7D%7B12%7D%20%20%2B%20%5Cfrac%7B%281%29%5E2%7D%7B2%7D%20%2B%204%281%29%20%5Cright%5D%20%20&bg=FFFFAA&fg=000000&s=3 "Area = \left[-\frac{(5)^3}{12} + \frac{(5)^2}{2} + 4(5) \right] - \left[-\frac{(1)^3}{12} + \frac{(1)^2}{2} + 4(1) \right]")
![Area = \left[-\frac{125}{12} + \frac{25}{2} + 16 \right] - \left[-\frac{1}{12} + \frac{1}{2} + 4 \right]](http://l.wordpress.com/latex.php?latex=Area%20%3D%20%5Cleft%5B-%5Cfrac%7B125%7D%7B12%7D%20%20%2B%20%5Cfrac%7B25%7D%7B2%7D%20%2B%2016%20%5Cright%5D%20-%20%5Cleft%5B-%5Cfrac%7B1%7D%7B12%7D%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%2B%204%20%5Cright%5D%20%20&bg=FFFFAA&fg=000000&s=3 "Area = \left[-\frac{125}{12} + \frac{25}{2} + 16 \right] - \left[-\frac{1}{12} + \frac{1}{2} + 4 \right]")
![Area = \left[ 22 \frac{1}{12} \right] - \left[4 \frac{5}{12} \right]](http://l.wordpress.com/latex.php?latex=Area%20%3D%20%5Cleft%5B%2022%20%5Cfrac%7B1%7D%7B12%7D%20%5Cright%5D%20%20-%20%5Cleft%5B4%20%5Cfrac%7B5%7D%7B12%7D%20%5Cright%5D%20&bg=FFFFAA&fg=000000&s=3 "Area = \left[ 22 \frac{1}{12} \right] - \left[4 \frac{5}{12} \right]")
(See also WolframAlpha's 
