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.
9000070803
Level:
A
Differentiate the following function.
\[
f(x) = 3x^{3} + 2x +\mathrm{e} ^{x}
\]
\(f'(x) = 9x^{2} + 2 +\mathrm{e} ^{x},\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x,\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x +\mathrm{e} ^{x},\ x\in \mathbb{R}\)
\(f'(x) = 9x^{2} + 2,\ x\in \mathbb{R}\)
9000070410
Level:
A
Given the function \(f(x) = - \frac{x^{2}}
{x+3}\), find
the intervals where \(f\)
is an increasing function.
\(\left (-3,0\right )\)
\(\left (-\infty ,-6\right )\)
\(\left (0,\infty \right )\)
\(\left (-3,4\right )\)
9000070804
Level:
A
Differentiate the following function.
\[
f(x) = 2x^{9} - x^{2} + 7
\]
\(f'(x) = 18x^{8} - 2x,\ x\in \mathbb{R}\)
\(f'(x) = 9x^{8} - 2x + 7,\ x\in \mathbb{R}\)
\(f'(x) = 18x^{8} - 2x + 7,\ x\in \mathbb{R}\)
\(f'(x) = 18x^{8} + 2x,\ x\in \mathbb{R}\)
9000070805
Level:
A
Differentiate the following function.
\[
f(x) = -3x^{3} - x^{2} + 9x
\]
\(f'(x) = -9x^{2} - 2x + 9,\ x\in \mathbb{R}\)
\(f'(x) = 9x^{2} - 2x + 9,\ x\in \mathbb{R}\)
\(f'(x) = 27x^{2} - 2x,\ x\in \mathbb{R}\)
\(f'(x) = -9x^{2} - 2x,\ x\in \mathbb{R}\)
9000070806
Level:
A
Differentiate the following function.
\[
f(x) = \frac{\pi }
{x} +\ln 2
\]
\(f'(x) = - \frac{\pi }{x^{2}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 0,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) =\pi ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = \frac{\pi } {x^{2}} ,\ x\in \mathbb{R}\setminus \{0\}\)
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 \((0,+\infty)\).
\[
\int \left (2e^{x} -\frac{3}
{x}\right )\, \mathrm{d}x
\]
\(2e^{x} - 3\ln \left |x\right | + c,\ c\in \mathbb{R}\)
\(2\ln \left |x\right |- \frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
\(2e^{x} - 3 + c,\ c\in \mathbb{R}\)
9000071206
Level:
A
Given the function \(f(x) =\sin x +\cos x\), find its
primitive function \(F\) so that the graph of \(F\) passes through the point \(A = \left [ \frac{\pi }{2},3\right ]\).
\(F(x) =\sin x -\cos x + 2\)
\(F(x) =\cos x -\sin x + 4\)
\(F(x) = -\cos x +\sin x + 4\)