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Let $S=\left\{t\in\mathbb{R}: f(x)=|x-\pi|\cdot(e^{|x|}-1)\sin|x|\right.$ is not differentiable at $t\}$. Find $S$.
Check $x=0$: $f(x)=|x-\pi|(e^{|x|}-1)\sin|x|$. Near $x=0$, both $(e^{|x|}-1)$ and $\sin|x|$ have factors of $|x|$, so $f(x)\approx |x-\pi|\cdot|x|^2$ near 0, which is differentiable at 0.
Check $x=\pi$: near $\pi$, $(e^{|x|}-1)\sin|x|$ is smooth and non-zero ($e^\pi\sin\pi=0$!). Since $\sin\pi=0$, the product $(e^{|x|}-1)\sin|x|$ vanishes at $\pi$, making $f(x)=0$ near $\pi$ โ wait, $\sin\pi=0$, so $f(\pi)=0$. Checking differentiability more carefully shows $f$ is differentiable at $\pi$ too.
Therefore $S=\boldsymbol{\emptyset}$.
First, calculate Least Count ($LC$): $LC = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}$.
Zero Error: The $45^{th}$ division coincides and the zero is barely visible, indicating a negative zero error. Error $= (45 - 50) \times 0.01 = -0.05 \text{ mm}$.
Observed Reading: $MSR + (CSR \times LC) = 0.5 + (25 \times 0.01) = 0.75 \text{ mm}$.
True Thickness: Observed Reading $-$ Zero Error $= 0.75 - (-0.05) = 0.80 \text{ mm}$.
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