2010011105
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to0}\frac{x^4+2x^3}{x-\sin x} \]
\(12\)
\( 0\)
\( -12\)
\( 6\)
\(-6\)
Applications of derivatives
2010011104
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\infty}\frac{2x+3}{\mathrm{e}^{2x}}\]
\(0\)
\( \frac12\)
\( \infty\)
\( 1\)
\(2\)
2010011103
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\infty}\frac{\mathrm{ln}\,(2x)+1}{5x+3}\]
\(0\)
\( \frac15\)
\( \frac25\)
\( \frac1{10}\)
\(\infty\)
2010011102
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\mathrm{tg}\,2x}{2x+\sin3x}\]
\( \frac25\)
\( \frac12\)
\( 10\)
\( 0\)
\(1\)
2010011101
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to2}\frac{x^3-4x^2+8}{2x^2-3x-2} \]
\( -\frac45\)
\( \frac45\)
\( -4\)
\( 0\)
\( \infty\)
2010010910
Level:
C
Find the minimum of the function \(g(x)=-{x^4}+2x^2+1\) when \( x \in [ -1;2]\).
\(x=2\)
\(x=-1\)
\(x=0\)
The function \(g\) does not have the minimum on the interval \([ -1;2]\).
2010010909
Level:
C
Find the maximum of the function \(g(x)=\frac{x^4}2-4x^2+2\) when \( x \in [ -2;3]\).
\(x=3\)
\(x=-2\)
\(x=0\)
The function \(g\) does not have the maximum on the interval \([ -2;3]\).
2010010908
Level:
C
How many of these functions do have a global minimum on the interval \( [ -1;\infty) \) at the point \(x=-1\)?
\[f(x)=x+\frac1{x+2}\]
\[g(x)=x^2+4x+4\]
\[h(x)=3x+1\]
\(3\)
\(1\)
\(2\)
\(0\)
2010010907
Level:
C
How many of these functions do have a global maximum on the interval \( (-\infty;-1]\) at the point \(x=-1\)?
\[f(x)=x+\frac1x+2\]
\[g(x)=-x^2+6x-9\]
\[h(x)=2x-3\]
\(3\)
\(1\)
\(2\)
\(0\)
2010010906
Level:
C
Find the maximum of the function \(g(x)=\frac{-1}{1-2x}+2x\) when \(x\in \left[ -2;\frac14\right]\).
\(x=0\)
\(x=-\frac{17}4\)
\( x=-\frac32\)
The function \(g\) does not have the maximum on the interval \(\left[ -2;\frac14\right]\).