9000070306
Level:
B
Given function \(f(x) = x^{4} + 6x^{3} - 24x^{2} + x + 3\), find
the intervals where \(f\)
is a strictly concave down function.
\((-4,1)\)
\((-6,2)\)
\((2,4)\)
\((-5,4)\)
B
9000070706
Level:
B
Differentiate the following function.
\[
f(x) = \sqrt{x^{2 } + 3x}
\]
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{\sqrt{x^{2 } +3x}}
{2x+3} ,\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)
9000070707
Level:
B
Differentiate the following function.
\[
f(x) = \root{5}\of{x^{2} - 7x}
\]Remark: The function \(f\colon y = \root{5}\of{x}\)
is defined for \(x\in \left < 0,\infty \right )\).
\(f^{\prime}(x) = \frac{2x-7}
{5(x^{2}-7x)^{\frac{4}
{5} }} ,\ x\in \left (-\infty ,0\right )\cup \left (7,\infty \right )\)
\(f^{\prime}(x) = \frac{2x-7}
{5(x^{2}-7x)^{\frac{4}
{5} }} ,\ x\in \left (-\infty ,0\right ] \cup \left [ 7,\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x},\ x\in \left (-\infty ,0\right )\cup \left (7,\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x},\ x\in \left (-\infty ,0\right ] \cup \left [ 7,\infty \right )\)
9000070708
Level:
B
Differentiate the following function.
\[
f(x) =\ln \left (\frac{1 + x}
{1 - x}\right )
\]
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ,\ x\in \left (-1,1\right )\)
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ,\ x\in \mathbb{R}\setminus \left \{-1,1\right \}\)
\(f^{\prime}(x) = \frac{1-x}
{1+x},\ x\in \left (-1,1\right )\)
\(f^{\prime}(x) = \frac{1-x}
{1+x},\ x\in \mathbb{R}\setminus \left \{-1,1\right \}\)
9000070801
Level:
B
Differentiate the following function.
\[
f(x) = 3\sin x\cos x
\]
\(f'(x) = 3\cos (2x),\ x\in \mathbb{R}\)
\(f'(x) = 3,\ x\in \mathbb{R}\)
\(f'(x) = -3\cos x\sin x,\ x\in \mathbb{R}\)
\(f'(x) = 3(\cos x)^{2},\ x\in \mathbb{R}\)
9000070807
Level:
B
Differentiate the following function.
\[
f(x) = \frac{x^{4} + 3}
{x^{2}} + x^{3}
\]
\(f'(x) = 3x^{2} + 2x - \frac{6}
{x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x - \frac{6}
{x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 3x^{2} + 2x + \frac{6}
{x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x + \frac{6}
{x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
9000070808
Level:
B
Differentiate the following function.
\[
f(x)= \frac{x}
{x + 1}
\]
\(f'(x) = \frac{1}
{(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{1}
{(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = \frac{x}
{(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{x}
{(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
9000070809
Level:
B
Differentiate the following function.
\[
f(x)= 3x^{2}\sin x
\]
\(f'(x) = 6x\sin x + 3x^{2}\cos x,\ x\in \mathbb{R}\)
\(f'(x) = 6x\cos x,\ x\in \mathbb{R}\)
\(f'(x) = 3x^{2}\sin x\cos x,\ x\in \mathbb{R}\)
\(f'(x) = -3x^{2}\sin x\cos x,\ x\in \mathbb{R}\)
9000071202
Level:
B
Evaluate the following integral on the interval \((0,+\infty)\).
\[
\int \frac{11\sqrt{x^{3}} - 2}
{\root{3}\of{x^{2}}} \, \mathrm{d}x
\]
\(6(x\root{6}\of{x^{5}} -\root{3}\of{x}) + c,\ c\in \mathbb{R}\)
\(\frac{\frac{22}
{5} \sqrt{x^{5}}-2x}
{\frac{3}
{5} \root{3}\of{x^{5}}} + c,\ c\in \mathbb{R}\)
\(\frac{121}
{6} \root{6}\of{x^{11}} -\frac{2}
{3}\root{3}\of{x} + c,\ c\in \mathbb{R}\)
9000070502
Level:
B
Consider the geometric sequence with the first term
\(a_{1} = 243\) and quotient
\(q = \frac{1}
{3}\). Find
\(n\) such that the
sum of the first \(n\)
terms equals \(363\).
\(n = 5\)
\(n = 2\)
\(n = 3\)
\(n = 4\)
\(n = 6\)