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.
A
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 \(A = \left [ \frac{\pi }{2},3\right ]\).
\(F(x) =\sin x -\cos x + 2\)
\(F(x) =\cos x -\sin x + 4\)
\(F(x) = -\cos x +\sin x + 4\)
9000070401
Level:
A
Given the function \(f(x) = x^{2} + x - 2\), find
the intervals where \(f\)
is an increasing function.
\(\left (-\frac{1}
{2},\infty \right )\)
\(\left (-3,\infty \right )\)
\(\left (-2,\infty \right )\)
\(\left (-1,\infty \right )\)
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.
\(\left (-\infty ,-5\right )\)
\(\left (-\infty ,-3\right )\)
\(\left (-1,\infty \right )\)
\(\left (-3,\infty \right )\)