Applications of derivatives

2010013708

Level: 
C
Suppose we throw an object vertically upwards at the initial speed \(v_0=80\,\mathrm{m}/\mathrm{s}\). Determine the time needed for the object to reach the maximum height and determine the corresponding maximum height as well. \[\] Hint: The vertical upwards motion of a body is the movement composed of uniformly rectilinear motion (vertically upwards) and free fall. The dependence of the instantaneous height of a body on the time is given by the relation \(h=v_0t-\frac12gt^2\), where \(v_0\) is the magnitude of the initial velocity and \(g\) is the gravitational acceleration. In this problem, calculate with the rounded value of \(g=10\,\frac{\mathrm{m}}{\mathrm{s}^2}\). We measure time \(t\) in second and height \(h\) in meters.
\(8\,\mathrm{s}\), \(320\,\mathrm{m}\)
\(8\,\mathrm{s}\), \(600\,\mathrm{m}\)
\(16\,\mathrm{s}\), \(1190\,\mathrm{m}\)
\(4\,\mathrm{s}\), \(230\,\mathrm{m}\)

2010013707

Level: 
C
Suppose we throw an object vertically upwards at the initial speed \(v_0=60\,\mathrm{m}/\mathrm{s}\). Determine the time needed for the object to reach the maximum height and determine the corresponding maximum height as well. \[\] Hint: The vertical upwards motion of a body is the movement composed of uniformly rectilinear motion (vertically upwards) and free fall. The dependence of the instantaneous height of a body on the time is given by the relation \(h=v_0t-\frac12gt^2\), where \(v_0\) is the magnitude of the initial velocity and \(g\) is the gravitational acceleration. In this problem, calculate with the rounded value of \(g=10\,\frac{\mathrm{m}}{\mathrm{s}^2}\). We measure time \(t\) in second and height \(h\) in meters.
\(6\,\mathrm{s}\), \(180\,\mathrm{m}\)
\(6\,\mathrm{s}\), \(330\,\mathrm{m}\)
\(12\,\mathrm{s}\), \(660\,\mathrm{m}\)
\(3\,\mathrm{s}\), \(135\,\mathrm{m}\)

2010013706

Level: 
C
An electrical source is characterized by the electromotive force \(U_e=40\,\mathrm{V}\) and the internal resistance \(R_i=2\,\Omega\). Determine the value of the electric current for which the appliance power will be at its maximum and determine the value of this maximum power as well. \[\] Hint: The dependence of the power of an appliance (\(P\), unit Watt (\(\mathrm{W}\))) on the magnitude of the folowing current (\(I\), unit Ampere (\(\mathrm{A}\))) is given by the relation \(P=U_eI-R_iI^2\). The source properties have a role of parameters: \(U_e\) is the electromotive force and \(R_i\) is the internal resistance of the source.
\(10\,\mathrm{A},\ 200\,\mathrm{W}\)
\(10\,\mathrm{A},\ 380\,\mathrm{W}\)
\(20\,\mathrm{A},\ 760\,\mathrm{W}\)
\(4\,\mathrm{A},\ 128\,\mathrm{W}\)

2010013705

Level: 
C
An electrical source is characterized by the electromotive force \(U_e=60\,\mathrm{V}\) and the internal resistance \(R_i=2\,\Omega\). Determine the value of the electric current for which the appliance power will be at its maximum and determine the value of this maximum power as well. \[\] Hint: The dependence of the power of an appliance (\(P\), unit Watt (\(\mathrm{W}\))) on the magnitude of the folowing current (\(I\), unit Ampere (\(\mathrm{A}\))) is given by the relation \(P=U_eI-R_iI^2\). The source properties have a role of parameters: \(U_e\) is the electromotive force and \(R_i\) is the internal resistance of the source.
\(15\,\mathrm{A},\ 450\,\mathrm{W}\)
\(15\,\mathrm{A},\ 870\,\mathrm{W}\)
\(30\,\mathrm{A},\ 1740\,\mathrm{W}\)
\(10\,\mathrm{A},\ 400\,\mathrm{W}\)

2010017806

Level: 
C
We want to lift an edge of a large square plate with a side of \(4\,\mathrm{m}\) so that it creates a shelter (see the picture). To what height \(h\) do we have to lift the edge of the plate, if the created shelter shall be of the largest possible volume?
$h=2\sqrt2\,\mathrm{m}$
$h=4\cdot \sqrt{\frac23}\,\mathrm{m}$
$h=\frac43\sqrt3\,\mathrm{m}$
$h=\left( -\frac12 + \sqrt{65}\right)\,\mathrm{m}$

2010017805

Level: 
C
What dimensions (in centimeters) must a glass aquarium in the shape of a cuboid with a square bottom have, so that its volume is \(20\) liters and the surface of the aquarium is as small as possible. (We consider the cuboid without the lid.)
$a\doteq 34.2\,\mathrm{cm}$, $v\doteq 17.1\,\mathrm{cm}$
$a\doteq 27.1\,\mathrm{cm}$, $v\doteq 27.1\,\mathrm{cm}$
$a\doteq 63.2\,\mathrm{cm}$, $v\doteq 5\,\mathrm{cm}$
$a\doteq 13.6\,\mathrm{cm}$, $v\doteq 108.6\,\mathrm{cm}$

2010017804

Level: 
C
Using a \(60\,\mathrm{m}\) long wire mash we shall fence a rectangular garden with two inner walls (see the picture). What will the dimensions \(a\) and \(b\) of the garden be, if there is \(2\,\mathrm{m}\) wide opening in one outside wall and the area of the garden shall be as large as possible? (The wire mesh is used to make inner walls as well.)
$a=7.75\,\mathrm{m}$, $b=15.5\,\mathrm{m}$
$a=7.25\,\mathrm{m}$, $b=16.5\,\mathrm{m}$
$a=7.5\,\mathrm{m}$, $b=16\,\mathrm{m}$
$a=10\,\mathrm{m}$, $b=11\,\mathrm{m}$

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 \(f\).
\(A = \left [2;-7\right ]\)
\(A = \left [4;-7\right ]\)
\(A = \left [1;-4\right ]\)
\(A = \left [0;1\right ]\)

2010012502

Level: 
C
Identify a true statement about the function \(f(x) = x^{3} +6x^{2} + 12x -1\).
There is neither local minimum nor maximum of \(f\).
The function \(f\) has a local maximum at the point \(x = -2\).
The function \(f\) has a local minimum at the point \(x = -2\).
The global minimum of \(f\) on \(\mathbb{R}\) is at \(x = -2\).