2010011102
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\mathrm{tg}\,2x}{2x+\sin3x}\]
\( \frac25\)
\( \frac12\)
\( 10\)
\( 0\)
\(1\)
Applications of derivatives
2010011103
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\infty}\frac{\mathrm{ln}\,(2x)+1}{5x+3}\]
\(0\)
\( \frac15\)
\( \frac25\)
\( \frac1{10}\)
\(\infty\)
2010011104
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\infty}\frac{2x+3}{\mathrm{e}^{2x}}\]
\(0\)
\( \frac12\)
\( \infty\)
\( 1\)
\(2\)
2010011105
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to0}\frac{x^4+2x^3}{x-\sin x} \]
\(12\)
\( 0\)
\( -12\)
\( 6\)
\(-6\)
2010017801
Level:
B
Let \(p\) be the tangent to the graph of the function \(f(x) = x^{2} -6x +1\) perpendicular to the line \(x - 2y + 3 = 0\). Find the point \(A\), where \(p\) touches the graph of the function \(f\).
\(A = \left [2;-7\right ]\)
\(A = \left [4;-7\right ]\)
\(A = \left [1;-4\right ]\)
\(A = \left [0;1\right ]\)
2010017802
Level:
B
Find the tangent to the graph of the function
\(f(x) = x^{2} - 6x + 1\) perpendicular
to the line \(x -2y + 4 = 0\).
\(2x + y +3 = 0\)
\(-2x - y - 1= 0\)
\(4x +2y - 1= 0\)
\(-4x - 2y +3 = 0\)
9000062407
Level:
B
Find the tangent to the function \(f(x) =\ln x\)
at the point \(T = [1;y_{0}]\).
\(y = x - 1\)
\(y = x\)
\(y = x + 1\)
\(y = -x\)
9000062408
Level:
B
Find the points in which the tangent to the curve
\(y = x^{3}\) has a
slope \(m = 3\)?
\(T_{1} = [1;1],\ T_{2} = [-1;-1]\)
\(T_{1} = [1;-1],\ T_{2} = [-1;1]\)
\(T_{1} = [-1;1],\ T_{2} = [-1;-1]\)
\(T_{1} = [1;-1],\ T_{2} = [-1;-1]\)
9000062410
Level:
B
Find the line perpendicular to the graph of the function
\(f(x) = x^{3}\) at the
point \(T = [-1;y_{0}]\).
\(x + 3y + 4 = 0\)
\(3x - y + 2 = 0\)
\(3x + y - 4 = 0\)
\(x - 3y - 2 = 0\)
9000064101
Level:
B
Find the slope of the tangent to the graph of
\(f(x) = x^{2} + 3x - 2\) at the
point \([1;2]\).
\(-\frac{1}
{5}\)
\(5\)
\(- 5\)
\(\frac{1}
{5}\)