User:ZaheerChothia

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${x}^{2}-2\,x-4\leq 4\,y-{y}^{2}$

Derivation of cooling equation:

$\frac{d \theta}{d t} \propto - (\theta - \theta_{amb})$

$\frac{d \theta}{d t} = - k (\theta - \theta_{amb})$

Using: $θ(t = 0) = θ 0$

$\int_{\theta_0}^{\theta} \frac{1}{\theta - \theta_{amb}} \, . \, d\theta = - k \int_0^t 1 \, . \, dt$

$\Big [ln{(\theta - \theta_{amb})} \Big]_{\theta_0}^{\theta} = -k (t - 0)$

$ln{(\theta - \theta_{amb})} - ln{(\theta_0 - \theta_{amb})} = -kt \,$

$ln{\left (\frac{\theta - \theta_{amb}}{\theta_0 - \theta_{amb}} \right )} = -kt$

${\theta} = \theta_{\mathrm{amb}} + (\theta_{\mathrm{0}} - \theta_{\mathrm{amb}}) \ e^{-k t}$

Extra:

$ln{(\theta - \theta_{amb})} = -kt + ln{(\theta_0 - \theta_{amb})} \,$

$y = m x + c$

Reference: $Δ E = V c Δθ$

$\Delta E \propto V_{water}$

$\int_a^b x.dx$

$\lim_{t \rightarrow \infty} \theta(t) = \theta_{amb}$

Multiline Equations:

$f (n + 1)$	$= (n + 1) 2$
	$= n 2 + 2 n + 1$