9000073405
Level:
A
Find the sum of the following infinite series.
\[
\sqrt{2} - 1 + \frac{\sqrt{2}}
{2} -\frac{1}
{2} + \frac{\sqrt{2}}
{4} -\frac{1}
{4}+\cdots
\]
\(2\sqrt{2} - 2\)
\(\sqrt{2} - 1\)
\(2\sqrt{2} + 2\)
\(\infty \)
A
9000073406
Level:
A
Find the sum of the following infinite series.
\[
\sum _{n=1}^{\infty }\left (\frac{\sqrt{2} - 1}
{\sqrt{2}} \right )^{n-1}
\]
\(\sqrt{2}\)
\(\frac{\sqrt{2}+1}
{\sqrt{2}} \)
\(\frac{\sqrt{2}}
{2} \)
Series diverges.
9000070402
Level:
A
Given the function \(f(x)= x^{2} - 2x - 8\), find
the intervals where \(f\)
is a decreasing function.
\(\left (-\infty ;1\right )\)
\(\left (-\infty ;8\right )\)
\(\left (-\infty ;2\right )\)
\(\left (-\infty ;4\right )\)
9000070403
Level:
A
Given function \(f(x) = -x^{2} + 2x + 3\), find
the intervals where \(f\)
is an increasing function.
\(\left (-\infty ;1\right )\)
\(\left (-\infty ;2\right )\)
\(\left (-\infty ;3\right )\)
\(\left (-\infty ;6\right )\)
9000070404
Level:
A
Given function \(f(x)= -x^{2} + 4x + 12\), find
the intervals where \(f\)
is a decreasing function.
\(\left (2;\infty \right )\)
\(\left (1;\infty \right )\)
\(\left (-4;\infty \right )\)
\(\left (-6;\infty \right )\)
9000070405
Level:
A
Given function \(f(x)= x^{3} + 3x^{2} - 24x + 5\), find
the intervals where \(f\)
is a decreasing function.
\(\left (-4;2\right )\)
\(\left (-3;5\right )\)
\(\left (-3;3\right )\)
\(\left (-5;1\right )\)
9000070406
Level:
A
Given function \(f(x)= x^{3} + 6x^{2} - 15x + 7\), find
the intervals where \(f\)
is an increasing function.
\(\left (-\infty ;-5\right )\)
\(\left (-\infty ;-3\right )\)
\(\left (-1;\infty \right )\)
\(\left (-3;\infty \right )\)
9000070407
Level:
A
Given the function \(f(x) = -x^{3} + 3x^{2} + 9x - 1\), find
the intervals where \(f\)
is a decreasing function.
\(\left (3;\infty \right )\)
\(\left (-\infty ;1\right )\)
\(\left (-1;3\right )\)
\(\left (1;\infty \right )\)
9000070803
Level:
A
Differentiate the following function.
\[
f(x) = 3x^{3} + 2x +\mathrm{e} ^{x}
\]
\(f'(x) = 9x^{2} + 2 +\mathrm{e} ^{x};\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x;\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x +\mathrm{e} ^{x};\ x\in \mathbb{R}\)
\(f'(x) = 9x^{2} + 2;\ x\in \mathbb{R}\)
9000070408
Level:
A
Given the function \(f(x) = -x^{3} + 3x^{2} + 45x - 12\), find
the intervals where \(f\)
is an increasing function.
\(\left (-3;5\right )\)
\(\left (-\infty ;-3\right )\)
\(\left (5;\infty \right )\)
\(\left (-12;45\right )\)