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What does 'subtle' most likely mean in the sentence: 'The detective noticed a subtle change in the suspect's tone...'
'Subtle' means so slight or delicate as to be difficult to notice. In this context, the detective noticed a small, not obvious change. 'Slight' is the best synonym among the options.
The functional group present in ketones is called:
Ketones contain a carbonyl group (C=O) bonded to two carbon atoms. Aldehydes have -CHO (formyl group). Carboxyl = -COOH, amino = -NH₂.
Number of orbitals = \(2\ell + 1 = 2(3) + 1 = 7\)
Max electrons = \(7 \times 2 = 14\)
This is the 4f subshell. Note that \(n = 4\) is valid since \(\ell\) can be 0, 1, 2, or 3 for \(n = 4\). The 4f subshell holds exactly 14 electrons, which accounts for the 14 lanthanoids.
Spot size $b \approx \text{geometric spread} + \text{diffraction spread} = a + \frac{L\lambda}{a}$.
To minimize $b$, differentiate with respect to $a$ and set to zero:
$\frac{db}{da} = 1 - \frac{L\lambda}{a^2} = 0 \implies a = \sqrt{\lambda L}$.
Substituting back: $b_{min} = \sqrt{\lambda L} + \frac{L\lambda}{\sqrt{\lambda L}} = 2\sqrt{\lambda L} = \sqrt{4\lambda L}$.
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 $x=1$: $\frac{dy}{dx} = \frac{1}{\sqrt{2}}$
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