B

9000063109

Level: 
B
Differentiate the following function. \[ f(x) = 3^{x}\cdot x^{3} \]
\(f'(x) = 3^{x}x^{2}(x\ln 3 + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x+1}x^{2}\ln 3,\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x\ln x + 3),\ x\in \mathbb{R}^{+}\)

9000046609

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left ( \frac{\pi }{4}; \frac{3\pi } {4}\right )\).
\(\sin x\geq \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{tg}}\nolimits x > 1\)
\(\cos x > 0\)
\(\mathop{\mathrm{cotg}}\nolimits x\geq \frac{\sqrt{3}} {3} \)

9000046610

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left (\frac{5\pi } {6}; \frac{3\pi } {2}\right )\).
\(\cos x < \frac{1} {2}\)
\(\mathop{\mathrm{tg}}\nolimits x < 0\)
\(\sin x\geq -\frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits x < 1\)

9000062905

Level: 
B
A semicircle is a half of the full circle. An infinite spiral is built from semicircles with an increasing radius. The radius of the first semicircle is \(2\, \mathrm{cm}\). The radius of each semicircle in the spiral is a double of the radius of the previous semicircle. Find the total length of the spiral.
\(\infty \)
\(4\pi \)
\(\frac{4} {3}\pi \)
\(- 4\pi \)