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.
9000070804
Level:
A
Differentiate the following function.
\[
f(x) = 2x^{9} - x^{2} + 7
\]
\(f'(x) = 18x^{8} - 2x;\ x\in \mathbb{R}\)
\(f'(x) = 9x^{8} - 2x + 7;\ x\in \mathbb{R}\)
\(f'(x) = 18x^{8} - 2x + 7;\ x\in \mathbb{R}\)
\(f'(x) = 18x^{8} + 2x;\ x\in \mathbb{R}\)
9000070409
Level:
A
Given the function \(f(x) = \frac{x^{2}}
{x-2}\), find
the intervals where \(f\)
is a decreasing function.
\(\left (0;2\right )\)
\(\left (-\infty ;-1\right )\)
\(\left (5;\infty \right )\)
\(\left (2;5\right )\)
9000070805
Level:
A
Differentiate the following function.
\[
f(x) = -3x^{3} - x^{2} + 9x
\]
\(f'(x) = -9x^{2} - 2x + 9;\ x\in \mathbb{R}\)
\(f'(x) = 9x^{2} - 2x + 9;\ x\in \mathbb{R}\)
\(f'(x) = 27x^{2} - 2x;\ x\in \mathbb{R}\)
\(f'(x) = -9x^{2} - 2x;\ x\in \mathbb{R}\)
9000070410
Level:
A
Given the function \(f(x) = - \frac{x^{2}}
{x+3}\), find
the intervals where \(f\)
is an increasing function.
\(\left (-3;0\right )\)
\(\left (-\infty ;-6\right )\)
\(\left (0;\infty \right )\)
\(\left (-3;4\right )\)
9000070810
Level:
A
Differentiate the following function.
\[
f(x)=\log _{5}12
\]
\(f'(x) = 0;\ x\in \mathbb{R}\)
\(f'(x) = \frac{1}
{\ln 12};\ x\in \mathbb{R}\)
\(f'(x) = \frac{1}
{12\ln 5};\ x\in \mathbb{R}\)
\(f'(x) = 1;\ x\in \mathbb{R}\)
9000071204
Level:
A
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \left (2e^{x} -\frac{3}
{x}\right )\, \mathrm{d}x
\]
\(2e^{x} - 3\ln \left |x\right | + c,\ c\in \mathbb{R}\)
\(2\ln \left |x\right |- \frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
\(2e^{x} - 3 + c,\ c\in \mathbb{R}\)
9000071205
Level:
A
Evaluate the following integral on \(\mathbb{R}\).
\[
\int \left (x^{2} + 2^{x}\right )\, \mathrm{d}x
\]
\(\frac{x^{3}}
{3} + \frac{2^{x}}
{\ln 2} + c,\ c\in \mathbb{R}\)
\(\frac{x^{3}}
{3} + \frac{2^{x+1}}
{x+1} + c,\ c\in \mathbb{R}\)
\(2x + \frac{2^{x}}
{\ln \left |x\right |} + c,\ c\in \mathbb{R}\)
9000070806
Level:
A
Differentiate the following function.
\[
f(x) = \frac{\pi }
{x} +\ln 2
\]
\(f'(x) = - \frac{\pi }{x^{2}} ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 0;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) =\pi ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = \frac{\pi } {x^{2}} ;\ x\in \mathbb{R}\setminus \{0\}\)