• $P_{1} = 35 \text{ psi}$, $T_{1} = 27 + 273 = 300 \text{ K}$
• $P_{2} = 40 \text{ psi}$, $T_{2} = ?$
• $35/300 = 40/T_{2}$
• $T_{2} = (40 \times 300) / 35 \approx 342.8 \text{ K}$
• Converting to Celsius: $342.8 - 273 = 69.8^{\circ}\text{C} \approx 70^{\circ}\text{C}$

Vestigial structures are remnants of organs that had a function in ancestors but are reduced/non-functional in current species. Whale hind limb bones indicate evolution from four-legged land mammals.

• At C2: Priority order is $-OH > -COOH > -C(H)(Cl)CH_3 > -H$. The sequence is counter-clockwise, but since $H$ is on the horizontal line, it is 2S.
• At C3: Priority order is $-Cl > -CH(OH)COOH > -CH_3 > -H$. The sequence is clockwise, but with $H$ on the horizontal line, it is 3R.

Check $x=0$: Write $h(x)=\sin|x|-|x|$. For $x>0$: $h(x)=\sin x-x$, $h'(x)=\cos x-1$. For $x<0$: $h(x)=-\sin x+x$, $h'(x)=-\cos x+1$. At $x=0$: left limit $=0$, right limit $=0$ — differentiable.

The term $2(x-\pi)\cos|x|$ is clearly differentiable everywhere (product of smooth functions, since $\cos|x|$ is even and smooth).

At $x=\pi$: $\cos|\pi|=\cos\pi=-1$, and all parts are smooth for $x>0$.

Therefore $f$ is differentiable everywhere and $K=\boldsymbol{\emptyset}$.

In Shannon's information theory, redundancy describes those elements of a message that are predictable or repetitive — they add no new information, but they serve a vital communication function. Perfect redundancy is complete repetition (every message repeated identically). Zero redundancy means sheer unpredictability. The textbook notes: “No message can reach maximum efficiency unless it contains a balance between the unexpected and the predictable.” Redundancy makes communication more robust against noise — when part of a message is lost or distorted, the receiver can use the redundant elements to reconstruct the intended meaning. For example, the English language is estimated to be about 50% redundant (many letters in a word can be removed and the word is still recognisable). In mass communication, editorial repetition, summaries, and visual reinforcement are all forms of strategic redundancy.