Study questions platform-wide or filter by specific tests with correct answers revealed.
Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable with $|f(x)-f(y)|\le 2|x-y|^{3/2}$ for all $x,y\in\mathbb{R}$. If $f(0)=1$, evaluate $\displaystyle\int_0^1 f^2(x)\,dx$.
From the condition $|f(x)-f(y)|\le 2|x-y|^{3/2}$, divide both sides by $|x-y|$:
$\left|\frac{f(x)-f(y)}{x-y}\right|\le 2|x-y|^{1/2}$
Taking $y\to x$: $|f'(x)|\le 2\cdot0=0$, so $f'(x)=0$ for all $x$.
Hence $f$ is constant. Since $f(0)=1$, we have $f(x)=1$ for all $x$.
$\int_0^1 f^2(x)\,dx = \int_0^1 1\,dx = \mathbf{1}$
Which philosopher wrote the 'Two Treatises of Government'?
John Locke's work argued against absolute monarchy and for consent of the governed.
Therefore: IE₁(Al) lies between Na and Mg → between 496 and 737 kJ/mol (approximately 577 kJ/mol). This is the well-known anomaly where Al < Mg in ionisation energy despite being to the right of Mg.
ICU Severity Scoring Systems:
| Score | Full Name | Use |
|---|---|---|
| APACHE II | Acute Physiology and Chronic Health Evaluation II | ICU mortality prediction (first 24h data) |
| SOFA | Sequential Organ Failure Assessment | Organ failure tracking, sepsis diagnosis |
| SAPS II | Simplified Acute Physiology Score II | ICU outcome prediction |
| CURB-65 | — | Pneumonia severity (community setting) |
| GCS | Glasgow Coma Scale | Neurological status, not general ICU severity |
Volume $V = \frac{\pi d^2 h}{4} = \frac{\pi \times 12.6^2 \times 34.2}{4} \approx 4262.2 \text{ cm}^3$.
Since inputs have 3 significant figures, $V \approx 4260 \text{ cm}^3$.
Error calculation: $\frac{\Delta V}{V} = 2\frac{\Delta d}{d} + \frac{\Delta h}{h}$
$\Delta V = 4260 \left( 2 \frac{0.1}{12.6} + \frac{0.1}{34.2} \right) \approx 4260 (0.01587 + 0.00292) \approx 80 \text{ cm}^3$.
Final value: $4260 \pm 80 \text{ cm}^3$.
CVP interpretation:
| CVP | Interpretation |
|---|---|
| \(0 - 5 \text{ mmHg}\) | Low — hypovolemia, vasodilation |
| \(6 - 12 \text{ mmHg}\) | Normal (mechanically ventilated: \(8-12\)) |
| \(> 12 \text{ mmHg}\) | High — fluid overload, right heart failure, tension pneumothorax |
- Hypovolemic: Low CVP, low CO, high SVR — fluid loss (hemorrhage, burns, diarrhea)
- Cardiogenic: High CVP, low CO, high SVR — pump failure
- Distributive: Low/normal CVP, high CO, low SVR — sepsis, anaphylaxis, neurogenic
- Obstructive: High CVP, low CO — PE, cardiac tamponade, tension pneumothorax
Sign in to join the conversation and share your thoughts.
Log In to Comment