2010010908
Level:
C
How many of these functions do have a global minimum on the interval \( [ -1;\infty) \) at the point \(x=-1\)?
\[f(x)=x+\frac1{x+2}\]
\[g(x)=x^2+4x+4\]
\[h(x)=3x+1\]
\(3\)
\(1\)
\(2\)
\(0\)
C
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 \(\left[ -2;\frac14\right]\).
2010010905
Level:
C
Find the maximum of the function \(g(x)=3x-\frac{1}{1-3x}\) when \(x\in \left[ -1;\frac19\right]\).
\(x=0\)
\(x=-\frac{13}4\)
\( x=-\frac76\)
The function \(g\) does not have the maximum on the interval \(\left[ -1;\frac19\right]\).
2010010904
Level:
C
Find the minimum value of the function \(f(x)=-3x+6\) on the interval \( (0;4)\).
The function \(f\) does not have the minimum on the interval \((0; 4)\).
\(6\)
\(-6\)
\(-9\)
2010010903
Level:
C
Find the minimum value of the function \(f(x)=2x-4\) on the interval \( (0;4)\).
The function \(f\) does not have the minimum on the interval \((0; 4)\).
\(-4\)
\(4\)
\(-6\)
2010010902
Level:
C
Find the minimum value of the function \(f(x)=-x^2-x+2\) on the interval \( [ -2;0 ]\).
\( 0\)
\(2\)
\( \frac94\)
\(-4\)
2010010901
Level:
C
Find the minimum value of the function \(f(x)=-x^2+x-1\) on the interval \( [ 0;2 ]\).
\( -3\)
\(-1\)
\( -\frac34\)
\(-7\)
2000010806
Level:
C
Let’s have a coil of \(0.06\,\mathrm{H}\) inductance. The current flowing through the coil is given by
\[
i=0.2\sin(100\pi t),\]
where time \(t\) is measured in seconds and current \(i\) is measured in amperes. Determine the voltage induced in the coil at time \(t=2\) seconds. (Hint: Instantaneous voltage can be expressed as the derivative of current function with respect to time: \(u(t)=-L\frac{\mathrm{d}i}{\mathrm{d}t}\). The negative sign indicates only that voltage induced opposes the change in current through the coil per unit time. It does not affect the magnitude of the voltage.)
\( 1.2\pi \,\mathrm{V}\)
\( 20\pi \,\mathrm{V}\)
\( 0 \,\mathrm{V}\)
\( 12 \,\mathrm{V}\)
2000010805
Level:
C
A flywheel rotates such that it sweeps out an angle at the rate of
\[
\varphi = 4t^2,
\]
where an angle \(\varphi\) is measured in radians and time \(t\) is measured in seconds. At what time is instantaneous angular velocity of the flywheel equal to \(36\,\frac{\mathrm{rad}}{s}\)? (Hint: Instantaneous angular velocity can be expressed as the derivative of the function \(\varphi(t)\) with respect to time: \(\omega(t)=\frac{\mathrm{d}\varphi}{\mathrm{d}t}\).)
\( 4.5 \,\mathrm{s}\)
\( 3\,\mathrm{s}\)
\( 288 \,\mathrm{s}\)
\( 9 \,\mathrm{s}\)