9000064803
Level:
C
Suppose three numbers form three consecutive terms of an arithmetic sequence. The sum of these
numbers is \(33\) and
the product is \(1\: 155\).
Find the smallest of these three numbers.
\(7\)
\(9\)
\(11\)
\(13\)
\(15\)
C
9000064807
Level:
C
Find the sum of the integers which satisfy the following inequality.
\[
x^{2} - 8x - 153\leq 0
\]
\(108\)
\(162\)
\(91\)
\(78\)
\(56\)
9000064010
Level:
C
Find the limit of the following sequence.
\[
{\left({\Bigl (\frac{2n + 1}
{n} \Bigr )}^{n}\right)}_{
n=1}^{\infty }
\]
Hint: The limit of the sequence \({\bigl ({\bigl (1 + \frac{1}
{n}\bigr )}^{n}\bigr )}_{n=1}^{\infty }\)
is the Euler number \(\mathrm{e}\).
\(\infty \)
\(2\mathrm{e}\)
\(\mathrm{e}^{2}\)
\(\mathrm{e} + 2\)
9000063610
Level:
C
Find the following limit.
\[
\lim _{n\to \infty } \frac{3 + 6 + 9 +\cdots +3n}
{6 + 12 + 18 +\cdots +6n}
\]
\(\frac{1}
{2}\)
\(0\)
\(1\)
\(\infty \)
9000063303
Level:
C
Differentiate the following function.
\[
f(x) = \sqrt{\sin x}
\]
\(f'(x) = \frac{\cos x}
{2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ;\pi + 2k\pi \right )\)
\(f'(x) = \frac{\sin x}
{2\sqrt{\cos x}},\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ; \frac{\pi } {2} + 2k\pi \right )\)
\(f'(x) = \frac{1}
{2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (2k\pi ;\pi + 2k\pi \right )\)
\(f'(x) = \frac{\cos x}
{2\sqrt{\sin x}},\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left [ 2k\pi ; \frac{\pi } {2} + 2k\pi \right ] \)
9000063304
Level:
C
Differentiate the following function.
\[
f(x) =\ln \sqrt{x}
\]
\(f'(x) = \frac{1}
{2x},\ x > 0\)
\(f'(x) = \frac{1}
{2x},\ x\neq 0\)
\(f'(x) = \frac{1}
{x},\ x > 0\)
\(f'(x) = \frac{1}
{x},\ x\neq 0\)
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.
\[
f(x) =\ln\left(\cos 2x\right)
\]
\(f'(x) = -2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = -2,\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 1 -\ln\left(\sin 2x\right),\ x\in \mathop{\mathop{\bigcup
}}\nolimits _{k\in \mathbb{Z}}\left (k\pi ; \frac{\pi } {2} + k\pi \right )\)
9000063308
Level:
C
Differentiate the following function.
\[
f(x) = \frac{1}
{\ln x}
\]
\(f'(x) = \frac{-1}
{x\ln ^{2}x},\ x > 0,\ x\neq 1\)
\(f'(x) =\ln x,\ x > 0\)
\(f'(x) = \frac{\ln x}
{x},\ x\neq 0\)
\(f'(x) = \frac{1}
{x\ln ^{2}x},\ x\neq 0\)