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Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$ for all $x\in\mathbb{R}$. Find $f(2)$.
Let $f'(1)=a$, $f''(2)=b$, $f'''(3)=c$. Then $f(x)=x^3+ax^2+bx+c$.
$f'(x)=3x^2+2ax+b \Rightarrow f'(1)=3+2a+b=a \Rightarrow a+b=-3$ ...(i)
$f''(x)=6x+2a \Rightarrow f''(2)=12+2a=b$ ...(ii)
$f'''(x)=6 \Rightarrow f'''(3)=6=c$ ...(iii)
From (ii): $b=12+2a$. Sub in (i): $a+12+2a=-3\Rightarrow3a=-15\Rightarrow a=-5$, $b=2$, $c=6$.
$f(2)=8+4(-5)+2(2)+6=8-20+4+6=\mathbf{-2}$
Wait: $f(2)=8-20+4+6=-2$. So correct index is 2 (option C: $-2$).
Evaluate: $\displaystyle\lim_{x\to1^-}\dfrac{\sqrt{\pi}-\sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$
Let $\sin^{-1}x = \pi/2-\epsilon$ where $\epsilon\to0^+$ as $x\to1^-$. Then $x=\sin(\pi/2-\epsilon)=\cos\epsilon\approx1-\epsilon^2/2$, so $1-x\approx\epsilon^2/2$.
$\sqrt{\pi}-\sqrt{2(\pi/2-\epsilon)}=\sqrt{\pi}-\sqrt{\pi-2\epsilon}=\sqrt{\pi}\left(1-\sqrt{1-\frac{2\epsilon}{\pi}}\right)\approx\sqrt{\pi}\cdot\dfrac{\epsilon}{\pi}=\dfrac{\epsilon}{\sqrt{\pi}}$
$\sqrt{1-x}\approx\sqrt{\epsilon^2/2}=\epsilon/\sqrt{2}$
Limit $=\dfrac{\epsilon/\sqrt{\pi}}{\epsilon/\sqrt{2}}=\sqrt{\dfrac{2}{\pi}}$
Sensitivity and Specificity Definitions:
| Measure | Formula | Meaning |
|---|---|---|
| Sensitivity | \(\frac{TP}{TP + FN}\) | % of true positives correctly identified (rules OUT disease if negative β SnNOut) |
| Specificity | \(\frac{TN}{TN + FP}\) | % of true negatives correctly identified (rules IN disease if positive β SpPIn) |
- 95% sensitivity: The tool correctly identifies 95% of patients who have the condition
- 60% specificity: Only 60% of those without the condition are correctly identified as negative β meaning a high false positive rate (40%)
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