1103143702
Level:
A
Choose the graph of a function \( f \) for which \( f(0)=0 \), \( f'(1)=-1 \), \( f'(2)=0 \), \( f'(3)=3 \) (\( f' \) is the derivative of the function \( f \)).
Derivative
1103143701
Level:
A
Choose the graph of a function \( f \) for which \( f(0)=0 \), \( f'(0)=3 \), \( f'(1)=0 \), \( f'(2)=-1 \) (\( f' \) is the derivative of the function \( f \)).
1003112011
Level:
B
Differentiate the following function:
\[ f(x)=\frac{x-5\sqrt[3]{x^2}}{\sqrt[3]x} \]
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3] x}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3]x }{3}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]x-5\sqrt[3]{x^2}}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{\sqrt[3]x}{3x}-5\text{, }x\in\mathbb{R}\setminus\{0\} \)
1003112010
Level:
B
Differentiate the following function:
\[ f(x)=\frac{\sqrt{x\sqrt x}}x \]
\( f'(x)=-\frac{\sqrt[4]{x^3}}{4x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]{x^3}}{x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)
1003112009
Level:
A
Which of the statements A, B, C, D given bellow are incorrect?
\[
\begin{array}{l}
\text{A: }\ \left(\sqrt{x}-\cos x+\mathrm{ln}\, x\right)'=\frac1{2\sqrt x}+\sin x+\frac1x\text{, }x\in\mathbb{R}^+ \\
\text{B: }\ \left(3\sin x-3^x+x^3 \right)'=3\cos x-3^x+3x^2 \text{, }x\in\mathbb{R} \\
\text{C: }\ \left(\mathrm{cotg}\, x-\mathrm{log}_3 5\right)'=-\frac1{\sin^2 x}\text{, }x\in\mathbb{R}\setminus\bigcup\limits_{k\in\mathbb{Z}}\{k\pi\} \\
\text{D: }\ \left(3\sqrt[3]{x^2}-2\mathrm{e}^x \right)'=\frac2{\sqrt[3]x}-2\mathrm{e}^x\text{, }x\in\mathbb{R}\setminus\{0\}
\end{array}
\]
The only incorrect statements are:
B
B, C
A, B, C
C, D
B, C, D
C
1003112008
Level:
B
Which of the statements A, B, C, D given bellow are correct?
\[
\begin{array}{l}
\text{A: }\ \left(3x^{-3}-\frac5{x^2} +7\right)'=-\frac9{x^4}-\frac{10}{x^3}\text{, }x\in\mathbb{R}\setminus\{0\} \\
\text{B: }\ \left(\frac{x^3-4}{3x}\right)'=2x+\frac4{3x^2}\text{, }x\in\mathbb{R}\setminus\{0\} \\
\text{C: }\ \left(\frac{x^4-x+1}{x}\right)'=3x^2-\frac{1}{x^2}\text{, }x\in\mathbb{R}\setminus\{0\} \\
\text{D: }\ \left(\frac2{x^2}-\frac3{x^3} \right)'=-\frac4{x^3}+\frac9{x^4}\text{, }x\in\mathbb{R}\setminus\{0\}
\end{array}
\]
The only correct statements are:
C, D
B, C
A, B
A, C, D
B, C, D
A, D
1003112007
Level:
A
Which of the statements A, B, C, D given bellow are correct?
\[
\begin{array}{l}
\text{A: }\ \left(3x^3-5x^2+7x\right)'=9x^2-10x+7\text{, }x\in\mathbb{R} \\
\text{B: }\ \left(x^2+2x^{-2}-5\right)'=2x-4x^{-3}\text{, }x\in\mathbb{R}\setminus\{0\} \\
\text{C: }\ \left(5x^4-7x^3+2\pi\right)'=20x^3-21x^2+2\text{, }x\in\mathbb{R} \\
\text{D: }\ \left(x^4-\frac4{x^4}\right)'=4x^3-\frac{16}{x^3}\text{, }x\in\mathbb{R}\setminus\{0\}
\end{array}
\]
The only correct statements are:
A, B
A, C
A, B, C
A, D
A, C, D
1003112006
Level:
A
Given the function \( f(x)=x^2+7 \), determine all the values of \( x \) (\( x\in\mathbb{R} \)) for which \(|f' (x)|=1 \).
\( x_1=\frac12 \), \( x_2 =-\frac12 \)
\( x=\frac12 \)
\( x_1=1 \), \( x_2 =-1 \)
\( x=1 \)
\( x=-\frac12 \)
1003112005
Level:
A
Given the function \( f(x)=-x^2-4x \), determine all the values of \( x \) (\( x\in\mathbb{R} \)) for which \( f'(x)=0 \).
\( x=-2 \)
\( x=2 \)
\( x=-4 \)
\( x=4 \)
\( x_1=0 \), \( x_2=4 \)
1003112004
Level:
A
Given the function \( f(x)=x^2-6x+5 \), determine all the values of \( x \) (\( x\in\mathbb{R} \)) for which \( f'(x)=0 \).
\( x=3 \)
\( x=1 \)
\( x=-3 \)
\( x=5 \)
\( x_1=1 \), \(x_2=5 \)