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

1 The curve with equation \(y = 13 + 18 x + 3 x ^ { 2 } - 4 x ^ { 3 }\) passes through the point \(P\) where \(x = - 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P\) is a stationary point of the curve and find the other value of \(x\) where the curve has a stationary point.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
      (l mark)

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

2 A bird flies from a tree. At time \(t\) seconds, the bird's height, \(y\) metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
2. 1. Find the rate of change of height of the bird in metres per second when \(t = 1\).
   2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when \(t = 1\).
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) when \(t = 2\).
   2. Given that \(y\) has a stationary value when \(t = 2\), state whether this is a maximum value or a minimum value.

3 A curve has equation \(y = 7 - 2 x ^ { 5 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation for the tangent to the curve at the point where \(x = 1\).
3. Determine whether \(y\) is increasing or decreasing when \(x = - 2\).

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.

9. \begin{figure}[h] \end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),

1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
2. Find the value of \(x\) for which \(P\) is a minimum.
3. Show that \(P\) is a minimum for this value of \(x\).
4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).

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

7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$

1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
2. Determine whether each stationary point is a maximum or minimum point.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions.

9. \begin{figure}[h] \end{figure} Figure 2 shows a tray made from sheet metal.
The horizontal base is a rectangle measuring \(8 x \mathrm {~cm}\) by \(y \mathrm {~cm}\) and the two vertical sides are trapezia of height \(x \mathrm {~cm}\) with parallel edges of length \(8 x \mathrm {~cm}\) and \(10 x \mathrm {~cm}\). The remaining two sides are rectangles inclined at \(45 ^ { \circ }\) to the horizontal. Given that the capacity of the tray is \(900 \mathrm {~cm} ^ { 3 }\),

1. find an expression for \(y\) in terms of \(x\),
2. show that the area of metal used to make the tray, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 18 x ^ { 2 } + \frac { 200 ( 4 + \sqrt { 2 } ) } { x } ,$$
3. find to 3 significant figures, the value of \(x\) for which \(A\) is stationary,
4. find the minimum value of \(A\) and show that it is a minimum.

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

1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.

6. A curve has the equation \(y = \mathrm { e } ^ { 3 x } \cos 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
4. Determine whether the stationary point is a maximum or minimum point and justify your answer.

3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$

1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

3

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d\), where \(d\) is a constant. When \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ), the remainder is - 2 . Use the Remainder Theorem to find the value of \(d\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3\).
   1. Given that \(x = - \frac { 1 } { 2 }\) is a solution of the equation \(\mathrm { g } ( x ) = 0\), write \(\mathrm { g } ( x )\) as a product of three linear factors.
   2. The function h is defined by \(\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }\) for \(x > 2\). Simplify \(\mathrm { h } ( x )\), and hence show that h is a decreasing function.
      [0pt] [4 marks]

7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).

1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
2. determine the nature of each of the turning points.
3. Sketch the curve \(C\).

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).

• Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
• Show that this stationary point is a maximum turning point.

5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width \(x\) metres and length \(2 x\) metres, and the height of the tank is \(h\) metres. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is \(54 \mathrm {~m} ^ { 2 }\).

1. 1. Show that \(x ^ { 2 } + 3 x h = 27\).
   2. Hence express \(h\) in terms of \(x\).
   3. Hence show that the volume of water, \(V \mathrm {~m} ^ { 3 }\), that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Verify that \(V\) has a stationary value when \(x = 3\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 3\).
   (2 marks)

2 The curve with equation \(y = x ^ { 4 } - 32 x + 5\) has a single stationary point, \(M\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(x\)-coordinate of \(M\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence, or otherwise, determine whether \(M\) is a maximum or a minimum point.
4. Determine whether the curve is increasing or decreasing at the point on the curve where \(x = 0\).

4 A model helicopter takes off from a point \(O\) at time \(t = 0\) and moves vertically so that its height, \(y \mathrm {~cm}\), above \(O\) after time \(t\) seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether this stationary value is a maximum value or a minimum value.
   (4 marks)
3. Find the rate of change of \(y\) with respect to \(t\) when \(t = 1\).
4. Determine whether the height of the helicopter above \(O\) is increasing or decreasing at the instant when \(t = 3\).

6. (a) Use the substitution \(u = \sqrt { t }\) to show that $$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$ (b) The function g is such that $$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$

1. Use differentiation to find the function g .
2. Evaluate \(\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t\) and simplify your answer.
   (c) Find the value of \(x\) (where \(x > 1\) ) that gives the maximum value of $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$

10 A square sheet of metal has edges 30 cm long. Four squares each with edge \(x \mathrm {~cm}\), where \(x < 15\), are removed from the corners of the sheet. The four rectangular sections are bent upwards to form an open-topped box, as shown in the diagrams. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_392_460_630_347} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_387_437_635_872} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_282_380_703_1318} 10

1. Show that the capacity, \(C \mathrm {~cm} ^ { 3 }\), of the box is given by $$C = 900 x - 120 x ^ { 2 } + 4 x ^ { 3 }$$ 10
2. Find the maximum capacity of the box. Fully justify your answer.