Differentiation

Increase, decrease and concavity

1. The function has the equation \(f(x)=x^3-6x^2+9x+2\).
   

   (a)   Find and classify the stationary points.

   (b)   Find the intervals of increase and decrease.

   Solutions:    (a)  Maximum: \(P_1(1,6)\), minimum: \(P_2(3,2)\);     (b)  the function increases on \((-\infty,1)\) and on \((3,\infty)\), it decreases on \((1,3)\)
2. The function has the equation \(f(x)=\frac{\textstyle 2}{\textstyle x^2-2x+2}\).
   

   (a)   Find and classify the stationary points.

   (b)   Find the increasing and decreasing intervals.

   Solutions:    (a)  Maximum: \(P_1(1,2)\);     (b)  the function increases on \((-\infty,1)\), it decreases on \((1,\infty)\)
3. The function has the equation \(f(x)=\frac{1}{3}x^3-2x^2+5x-\frac{5}{3}\).
   

   (a)   Show that this function is increasing for all values of \(x\).

   (b)   Find the inflexion point.

   (c)   Write the equation of the tangent at the inflexion point.

   (d)   Write the intervals where the function is concave down and concave up.

   Solutions:    (a)  \(f'(x)=(x-2)^2+1\gt0\);     (b)  inflexion point: \(P_1(2,3)\);     (c)  tangent: \(y=x+1\);     (d)  concave down on \((-\infty,2)\), concave up on \((2,\infty)\)
4. The function has the equation \(f(x)=\frac{\textstyle 2x}{\textstyle x-2}\).
   

   (a)   Find the intervals of increase and decrease.

   (b)   Find the intervals of concavity and convexity.

   Solutions:    (a)  The function is decreasing on \((-\infty,2)\) and on \((2,\infty)\);     (b)  it's concave (= concave down) on \((-\infty,2)\), its convex (= concave up) on \((2,\infty)\)
5. The function has the equation \(f(x)=\frac{\textstyle x^2-9}{\textstyle x^2+3}\).
   

   (a)   Find the zeros and asymptotes.

   (b)   Find the stationary points.

   (c)   Find the intervals of increase and decrease.

   (d)   Find the inflexion points.

   (e)   Find the intervals of concavity and convexity.

   (f)   Draw the graph.

   Solutions:    (a)  Zeros: \(x_1=-3,~ x_2=3\), no vertical asymptotes, horizontal asymptote: \(y=1\);     (b)  minimum: \(P_1(0,-3)\);     (c)  it increases on \((0,\infty)\), it decreases on \((-\infty,0)\);     (d)  inflexion points: \(P_2(-1,-2),~ P_2(1,-2)\);     (e)  it's concave on \((-\infty,-1)\) and on \((1,\infty)\), it's convex on \((-1,1)\)

Applications of the derivative

6. A particle is moving along \(y\)-axis. It's displacement in metres is:   \(s=\frac{1}{3}t^2+2t+4\),   where \(t\) represents time in seconds.
   

   (a)   Write down the coordinates of this particle at \(t=0\) and at \(t=6\).

   (b)   Find the velocity of this particle at \(t=0\) and at \(t=6\).

   (c)   Find the acceleration of this particle.

   Solutions:    (a)  \(P_0(0,4),~ P_6(0,28)\);     (b)  \(v_0=2~\mathrm{m/s}\) and \(v_6=6~\mathrm{m/s}\);     (c)  \(a=\frac{2}{3}~\mathrm{m/s}^2\)
7. A particle is moving along \(x\)-axis. It's displacement (in \(\mathrm{cm}\)) is:   \(s=2t^2+5t+m\),   where \(t\) represents time in seconds. We started observing the particle at \(t=0\) and in that moment the particle was passing through the origin of the coordinate system.
   

   (a)   Find the value of \(m\).

   (b)   Find the velocity of this particle at \(t=0\) and at \(t=10\).

   (c)   Find the acceleration of this particle.

   Solutions:    (a)  \(m=0\);     (b)  \(v_0=5~\mathrm{cm/s}\) and \(v_{10}=45~\mathrm{cm/s}\);     (c)  \(a=4~\mathrm{cm/s}^2\)
8. The velocity of a car (in \(\mathrm{ms}^{-1}\)) changes as   \(v=30-6t\),   where \(t\) is time in seconds.
   

   (a)   Find the initial velocity (velocity at \(t=0\) seconds).

   (b)   Find the velocity at \(t=2~\mathrm{s}\).

   (c)   Find the acceleration of this car.

   (d)   When will the car stop?

   Solutions:    (a)  \(v_0=30~\mathrm{m/s}\);     (b)  \(v_2=18~\mathrm{m/s}\);     (c)  \(a=-6~\mathrm{m/s}^2\);     (d)  it'll stop at \(t=5~\mathrm{s}\)
9. A car has a certain initial velocity \(v_0\) and then it starts to accelerate. It's velocity \(t\) seconds after it starts to accelerate is:  \(v=10+4t\)   (in \(\mathrm{m/s}\)).
   

   (a)   Find the initial velocity \(v_0\). Write it in \(\mathrm{m/s}\) and in \(\mathrm{km/h}\).

   (b)   Find the velocity at \(t=5~\mathrm{s}\). Write it in \(\mathrm{m/s}\) and in \(\mathrm{km/h}\).

   (c)   When will the car reach the velocity of \(144~\mathrm{km/h}\)?

   (d)   Find the acceleration of this car.

   Solutions:    (a)  \(v_0=10~\mathrm{m/s}=36~\mathrm{km/h}\);     (b)  \(v_2=30~\mathrm{m/s}=108~\mathrm{km/h}\);     (c)  at \(t=7.5~\mathrm{s}\);     (d)  \(a=4~\mathrm{m/s}^2=51\,840~\mathrm{km/h}^2\)
10. An oscillating particle is moving along \(x\)-axis. It's displacement in centimetres is:   \(s=5\cos \frac{\textstyle \pi\, t}{\textstyle 2}\),   where \(t\) represents time in seconds.
    

    (a)   Write down the displacement of this particle at \(t=0,~ 1,~ 2,~ 3\) and \(4\).

    (b)   Find the velocity of this particle at \(t=0,~ 1,~ 2,~ 3\) and \(4\).

    (c)   Find the acceleration of this particle at \(t=0,~ 1,~ 2,~ 3\) and \(4\).

    Solutions:    (a)  \(s_0=5,~ s_1=0,~ s_2=-5,~ s_3=0,~ s_4=5\);     (b)  \(v_0=0,~ v_1=-\frac{5\pi}{2}\approx-7.85,~ v_2=0,~ v_3=\frac{5\pi}{2}\approx7.85,~ v_4=0\);     (c)  \(a_0=-\frac{5\pi^2}{4}\approx-12.34,~ a_1=0, a_2=\frac{5\pi^2}{4}\approx12.34,~ a_3=0,~ a_4=-\frac{5\pi^2}{4}\approx-12.34 \)
11. We would like to build a swimming pool. It should have the form of a cuboid with concrete square bottom and concrete side walls and with the volume \(256~\mathrm{m}^3\). Determine the lengths of the sides so that the building costs will be minimal.
    Solutions:    The area of the bottom and side walls is minimal when \(a=b=8~\mathrm{m},~ h=4~\mathrm{m}\)
12. A math teacher wants to make wire models of a square and of an oblong (a non-square rectangle) with the sides \(a:b=3:1\). He has a wire \(140~\mathrm{cm}\) long and he's going to split it in two pieces to make both models at the same time. Find the lengths of the sides so that the total area of both figures will be extreme. Is this extreme a maximum or a minimum?
    Solutions:    Square: \(x=15~\mathrm{cm}\), oblong: \(a=30~\mathrm{cm},~ b=10~\mathrm{cm}\), it's a minimum
13. A craftsman is going to make a cardboard box, open at the top. He has a square piece of cardboard with the dimensions \(60\times60~\mathrm{cm}\). He will cut off a small square with the side \(x\) at each corner and then fold the sides upwards.
    

    (a)   Find the value of \(x\) that gives the maximum volume of the box.

    (b)   Calculate the sides and the volume of the box in this case.

    Solutions:    (a)  Small square(s): \(x=10~\mathrm{cm}\);     (b)  box: \(a=b=40~\mathrm{cm},~ h=10~\mathrm{cm},~ V=16\,000~\mathrm{cm}^3\)
14. A sphere has the radius \(R\). The cylinder with the largest volume possible is inscribed in this sphere.
    

    (a)   Find the radius \(r\) and height \(h\) of this cylinder.

    (b)   How many times larger in volume is the sphere compared with the cylinder?

    Solutions:    (a)  Cylinder: \(r=\frac{\sqrt{6}}{3}R,~ h=\frac{2\sqrt{3}}{3}R\);     (b)  volumes: \(V_S=V_C\cdot\sqrt{3}\)

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