C

9000064801

Level: 
C
The sides of a right triangle form three consecutive terms of an arithmetic sequence. The perimeter of the triangle is \(60\, \mathrm{cm}\). Find the hypotenuse of the triangle (the longest side).
\(25\, \mathrm{cm}\)
\(12\, \mathrm{cm}\)
\(15\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(30\, \mathrm{cm}\)

9000063303

Level: 
C
Differentiate the following function. \[ f(x) = \sqrt{\sin x} \]
\(f'(x) = \frac{\cos x} {2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ;\pi + 2k\pi \right )\)
\(f'(x) = \frac{\sin x} {2\sqrt{\cos x}},\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ; \frac{\pi } {2} + 2k\pi \right )\)
\(f'(x) = \frac{1} {2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ;\pi + 2k\pi \right )\)
\(f'(x) = \frac{\cos x} {2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left [ 2k\pi ; \frac{\pi } {2} + 2k\pi \right ] \)

9000063305

Level: 
C
Differentiate the following function. \[ f(x) = \sqrt{\frac{x - 1} {x + 1}} \]
\(f'(x) = \frac{1} {(x+1)^{2}} \sqrt{\frac{x+1} {x-1}},\ x\in (-\infty ;-1)\cup (1;\infty )\)
\(f'(x) = \frac{\sqrt{x-1}} {(x-1)^{2}\sqrt{x+1}},\ x\in (-\infty ;-1)\cup [ 1;\infty )\)
\(f'(x) = \frac{x-1} {2\sqrt{(x+1)^{3}}} ,\ x\neq - 1\)
\(f'(x) = \frac{x-1} {\sqrt{(x+1)^{3}}} ,\ x\in (-\infty ;-1)\cup [ 1;\infty )\)

9000063306

Level: 
C
Differentiate the following function. \[ f(x) =\mathrm{e} ^{\sin 2x} \]
\(f'(x) = 2\mathrm{e}^{\sin 2x}\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) = x\mathrm{e}^{\sin 2x}\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) =\mathrm{e} ^{\sin 2x}\sin 2x,\ x\in \mathbb{R}\)
\(f'(x) =\mathrm{e} ^{\cos 2x},\ x\in \mathbb{R}\)

9000063307

Level: 
C
Differentiate the following function. \[ f(x) =\ln\left(\cos 2x\right) \]
\(f'(x) = -2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = -2,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 1 -\ln\left(\sin 2x\right),\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (k\pi ; \frac{\pi } {2} + k\pi \right )\)

9000064003

Level: 
C
Consider the convergent sequence \[ (a_{n})_{n=1}^{\infty } = \left (\frac{4n^{2} + 3n - 250} {2n^{2}} \right )_{n=1}^{\infty } \] and its limit \(L\). Find the maximal difference between \(L\) and the subsequence \((a_{n})_{n=250}^{\infty }\). (In other words, find the maximal difference between \(L\) and the terms of the sequence starting at \(a_{250}\).)
\(0.004\)
\(0.04\)
\(0.504\)
\(0.54\)