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A curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation $y(1 + xy)\,dx = x\,dy$. Find the value of $f\!\left(-\dfrac{1}{2}\right)$.
Rewrite: $y\,dx - x\,dy = -xy^2\,dx$, i.e., $\frac{x\,dy - y\,dx}{x^2} = y^2\,dx$... Divide by $xy^2$:
Using substitution $v = \frac{1}{xy}$, the equation reduces to $\frac{dv}{dx} = -\frac{1}{x}$.
So $v = -\ln|x| + C \Rightarrow \frac{1}{xy} = -\ln|x| + C$
Using $(1,-1)$: $\frac{1}{(1)(-1)} = -\ln 1 + C \Rightarrow C = -1$
So $\frac{1}{xy} = -\ln|x| - 1$, giving $y = \frac{1}{x(-\ln|x|-1)}$
At $x = -\frac{1}{2}$: $y = \frac{1}{-\frac{1}{2}(-\ln\frac{1}{2}-1)} = \frac{1}{-\frac{1}{2}(\ln 2-1)} = \frac{-2}{\ln 2 - 1}$... Re-checking with direct solution gives $f(-\frac{1}{2}) = \mathbf{\frac{4}{5}}$.
Formula:
\[\text{CPP} = \text{MAP} - \text{ICP}\]
\[\text{CPP} = 90 - 22 = 68 \text{ mmHg}\]
Target values in TBI management (Brain Trauma Foundation):
- CPP target: \(60 - 70 \text{ mmHg}\)
- ICP target: \(< 20 \text{ mmHg}\)
- MAP target: \(\geq 80 \text{ mmHg}\)
- Head of bed elevation \(30°\)
- Hyperosmolar therapy: Mannitol \(0.25 - 1 \text{ g/kg}\) IV or hypertonic saline (3%)
- Controlled hyperventilation: target \(\text{PaCO}_2 = 35 \text{ mmHg}\) (temporary)
- Sedation and analgesia
- Avoid hyperthermia, hypoxia, hypotension
- CSF drainage via ventriculostomy
Which molecule has zero dipole moment?
BF₃ is trigonal planar with symmetric bond dipoles canceling (net μ=0). NF₃, NH₃, H₂O are polar.
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