Study questions platform-wide or filter by specific tests with correct answers revealed.
Find the set $K$ of all real $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable.
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 $=0$ โ 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 $K=\boldsymbol{\emptyset}$.
A firm maintains an allowance for doubtful debts at 5% of trade receivables. This year, trade receivables were $8,000, which represented a 20% decrease from the previous year. How will this change affect the current year's profit?
1. Current Receivables = $8,000. Current Allowance = $8,000 $\times$ 5% = $400.
2. Last Year Receivables = $8,000 / 0.80 = $10,000. Last Year Allowance = $10,000 $\times$ 5% = $500.
3. Change = $500 - $400 = $100 reduction in allowance.
A reduction in allowance is credited to the income statement, thus increasing profit by $100.
Which Islamic thinker developed the 'Rise and Fall of Sovereign Power' cyclical theory?
Ibn-e-Khaldun's 'Muqaddimah' introduces the theory of Asabiyyah and the cyclical nature of dynasties.
For $h \ll R$, orbital radius $\approx R$.
Orbital velocity: $v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR}$
Escape velocity from orbit at $R$: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$
Minimum increase in speed:
$\Delta v = v_e - v_o = \sqrt{2gR} - \sqrt{gR} = \sqrt{gR}(\sqrt{2}-1)$
Sign in to join the conversation and share your thoughts.
Log In to Comment