1.07o Increasing/decreasing: functions using sign of dy/dx

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}

10. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0\).
The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).

1. Find the exact area of \(R\).
2. Use calculus to show that \(y\) is increasing for \(x > 2\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).

1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
   The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
4. Hence calculate the exact area of \(R\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum.

9. \begin{figure}[h] \end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
3. Prove that this value of \(r\) gives a minimum value of \(S\).
4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).

3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).

8. \begin{figure}[h] \end{figure} A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, \(x \mathrm {~cm}\), as shown in Figure 2.
The volume of the cuboid is 81 cubic centimetres.

1. Show that the total length, \(L \mathrm {~cm}\), of the twelve edges of the cuboid is given by $$L = 12 x + \frac { 162 } { x ^ { 2 } }$$
2. Use calculus to find the minimum value of \(L\).
3. Justify, by further differentiation, that the value of \(L\) that you have found is a minimum.

8. \begin{figure}[h] \end{figure} A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius \(x \mathrm {~mm}\) and height \(h \mathrm {~mm}\), as shown in Figure 3. Given that the volume of each tablet has to be \(60 \mathrm {~mm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~mm} ^ { 2 }\), of a tablet is given by \(A = 2 \pi x ^ { 2 } + \frac { 120 } { x }\) The manufacturer needs to minimise the surface area \(A \mathrm {~mm} ^ { 2 }\), of a tablet.
3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.
5. Show that this value of \(A\) is a minimum.

9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).

1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
3. Find the maximum area of \(\triangle P Q R\).
4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { \sqrt { 4 x - 1 } } \quad x > \frac { 1 } { 4 }$$

1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the range of f.

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

9. Given that \(k\) is a negative constant and that the function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 2 - \frac { ( x - 5 k ) ( x - k ) } { x ^ { 2 } - 3 k x + 2 k ^ { 2 } } , \quad x \geqslant 0$$

1. show that \(\mathrm { f } ( x ) = \frac { x + k } { x - 2 k }\)
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer in its simplest form.
3. State, with a reason, whether \(\mathrm { f } ( x )\) is an increasing or a decreasing function. Justify your answer.

6

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } + 4\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. For what values of \(x\) does \(x ^ { 3 } - 3 x ^ { 2 } + 4\) increase as \(x\) increases?

8

1. Find the coordinates of the stationary points of the curve \(y = 27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine, in each case, whether the stationary point is a maximum or minimum point.
3. Hence state the set of values of \(x\) for which \(27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\) is an increasing function. \(9 \quad A\) is the point \(( 2,7 )\) and \(B\) is the point \(( - 1 , - 2 )\).

8

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. For what values of \(x\) does \(x ^ { 3 } + x ^ { 2 } - x + 3\) decrease as \(x\) increases?

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
3. Determine whether this stationary point is a minimum or maximum point.
4. Find the \(x\)-coordinate of the other stationary point.

8

1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.

7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
4. Determine whether each stationary point is a maximum or a minimum point.

8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$

1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
3. Determine whether the turning point is a maximum or minimum point.

3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

10. \includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),

1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
2. Find the value of \(r\) for which \(V\) is stationary.
3. Find the corresponding value of \(V\) in terms of \(\pi\).
4. Determine whether this is a maximum or a minimum value of \(V\).

2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).