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If $y = \sec(\tan^{-1}x)$, find $\dfrac{dy}{dx}$ at $x=1$.
Let $\theta = \tan^{-1}x$, so $\tan\theta = x$ and $y = \sec\theta$.
$\frac{dy}{dx} = \sec\theta\tan\theta\cdot\frac{d\theta}{dx} = \sec\theta\tan\theta\cdot\frac{1}{1+x^2}$
Since $\tan\theta=x$ and $\sec\theta=\sqrt{1+x^2}$:
$\frac{dy}{dx} = \sqrt{1+x^2}\cdot x\cdot\frac{1}{1+x^2} = \frac{x}{\sqrt{1+x^2}}$
At $x=1$: $\frac{dy}{dx} = \frac{1}{\sqrt{2}}$
CTG classification is based on NICE (2022) / FIGO guidelines. Assess 4 features:
| Feature | Reassuring | Non-reassuring | Abnormal | This Strip |
|---|---|---|---|---|
| Baseline FHR | 110–160 bpm | 100–109 or 161–180 | <100 or >180 | 155 bpm ✓ Reassuring |
| Variability | 5–25 bpm | 3–4 bpm (>30 min) or >25 bpm | <3 bpm (>60 min) or sinusoidal | 2 bpm = Abnormal |
| Decelerations | None or early | Variable (typical) | Late decelerations | Late = Abnormal |
| Accelerations | Present | Absent | — | Not mentioned |
Classification (NICE):
- Normal: All 4 features reassuring
- Suspicious: 1 non-reassuring feature
- Pathological: ≥2 non-reassuring OR ≥1 abnormal feature
This CTG has: variability 2 bpm (abnormal) + late decelerations (abnormal) = PATHOLOGICAL
Action: Inform senior doctor immediately, consider fetal blood sampling (FBS), prepare for expedited delivery. Offer resuscitative measures: maternal lateral position, IV fluids, stop oxytocin.
A satellite orbits at height $h$ from Earth's surface where $h \ll R$ ($R$ = Earth's radius). The minimum increase in speed for the satellite to escape Earth's gravitational field is:
For $h \ll R$, orbital radius $\approx R$:
Orbital speed: $v_o = \sqrt{gR}$
Escape speed from surface (approximately same radius): $v_e = \sqrt{2gR}$
Minimum increase: $\Delta v = v_e - v_o = \sqrt{2gR} - \sqrt{gR} = \sqrt{gR}(\sqrt{2}-1)$
Initial charge (without dielectric):
$Q_0 = C_0 V = 90 \times 10^{-12} \times 20 = 1800\ \text{pC} = 1.8\ \text{nC}$
New capacitance with dielectric:
$C = KC_0 = \frac{5}{3} \times 90 = 150\ \text{pF}$
New free charge:
$Q = CV = 150 \times 10^{-12} \times 20 = 3000\ \text{pC} = 3\ \text{nC}$
Induced charge on dielectric:
$Q_{induced} = Q\left(1 - \frac{1}{K}\right) = 3 \times \left(1 - \frac{3}{5}\right) = 3 \times \frac{2}{5} = 1.2\ \text{nC}$
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