User:WraithM/math

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1 Test
2 Proof of Volume and Surface Area of a Sphere
3 Partial Derivatives Test
4 Chemistry
5 Lindsay
6 Physics

[edit] Test

$\int {d \over dx} f(x) \, dx = f(x)$

[edit] Proof of Volume and Surface Area of a Sphere

$sin \, \theta = {x \over r}$
$x = r sin \, \theta$
$2 \int_0^{\pi \over 2} {2 \pi r \, sin \theta \, r} d\theta$
$4 \pi r^2 \int_0^{\pi \over 2} {sin \theta} \, d\theta = 4 \pi r^2$
$\int {4 \pi r^2} \, dr = 4 \pi \int {r^2} \, dr = {4 \over 3} \pi r^3$

[edit] Partial Derivatives Test

$f (x, y) = x 2 + y 2$
$\frac{\partial}{\partial x} f(x,y) = 2x$

Goofing around with second partial derivatives and double integrals.
$f (x, y) = x 2 y + y 2 x$
$\frac{\partial}{\partial x} f(x,y) = 2xy + y^2$
$\frac{\partial^2}{\partial x \partial y} f(x,y) = 2x + 2y$
$\iint {2x + 2y} \, dxdy = x^2y + y^2x = f(x,y)$
$\iint {\frac{\partial^2}{\partial x \partial y} f(x,y)} \, dxdy = f(x,y)$

[edit] Chemistry

Percent error $Percent Error = \frac{(.507-.512)}{.512} \times 100% = -97%$

Titration $\frac{11.5 \, mL + 14.5 \, mL}{2} = 13 \, mL$
$\frac{12 \, mL + 14 \, mL}{2} = 13 \, mL$
$[HCH_3CO_2] \times 13 \, mL = .1 \, N \times 13 \, mL$
$[HCH_3CO_2] = \frac{.1 \, \times 13 \, mL}{13 \, mL} = .1 \, N$

$Q = C p m Δ t$
$Cp_{Al} \, m_{Al} \, \Delta t_{Al} = Cp_{H_2O} \, m_{H_2O} \, \Delta t_{H_2O}$
$Cp_{Al} = \frac{Cp_{H_2O} \, m_{H_2O} \, \Delta t_{H_2O}}{m_{Al} \, \Delta t_{Al}}$
$Cp_{Al} = \frac{4.18 j/gC \, 39.73 g \, 12 C}{35.67 g \, 68 C} = .822 j/gC$
$Percent Error = \frac{(.822 - .898)}{.898} \times 100% = -8.4%$

[edit] Lindsay

$The\,Amount\,of\,Love\,Matthew\,Wraith\,Has\,for\,Lindsay\,Preseau = (\displaystyle\lim_{x\to\infty}{\displaystyle\prod_{n=1}^\infty{\displaystyle\sum_{j=1}^{\infty}{x^{n\infty}+\mathbb{R}^{\mathbb{C}+\infty j x n})!}}}$
$f(x) = \infty$