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

8

1. Differentiate \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) with respect to \(x\).
2. Hence show that \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) is increasing for all values of \(x\).

5

1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
   1. \(\quad y = \mathrm { f } ( x ) + 3\)
   2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
   3. \(y = \mathrm { f } ^ { - 1 } ( x )\)
2. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
   1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
   2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
   3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius \(r\), height \(h\) and slant height \(l\), the following formulae are given.
Curved surface area, \(S = \pi r l\) Volume, \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) A container is to be designed in the shape of an inverted cone with no lid. The base radius is \(r \mathrm {~m}\) and the volume is \(V \mathrm {~m} ^ { 3 }\). The area of the material to be used for the cone is \(4 \pi \mathrm {~m} ^ { 2 }\).

1. Show that \(V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }\).
2. In this question you must show detailed reasoning. It is given that \(V\) has a maximum value for a certain value of \(r\).
   Find the maximum value of \(V\), giving your answer correct to 3 significant figures.

2 A curve has equation \(y = x ^ { 5 } - 5 x ^ { 4 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Verify that the curve has a stationary point when \(x = 4\).
3. Determine the nature of this stationary point.

1. A curve has equation

$$y = 3 x ^ { 2 } + \frac { 24 } { x } + 2 \quad x > 0$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the exact range of values of \(x\) for which the curve is increasing.

1. A curve has equation \(y = \mathrm { g } ( x )\).

Given that

• \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
• the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
• the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
  1. find \(\mathrm { g } ( x )\),
  2. prove that the stationary point at \(( 2,9 )\) is a maximum.

1. (a) Factorise completely \(9 x - x ^ { 3 }\)

The curve \(C\) has equation $$y = 9 x - x ^ { 3 }$$ (b) Sketch \(C\) showing the coordinates of the points at which the curve cuts the \(x\)-axis. The line \(l\) has equation \(y = k\) where \(k\) is a constant.
Given that \(C\) and \(l\) intersect at 3 distinct points,
(c) find the range of values for \(k\), writing your answer in set notation. Solutions relying on calculator technology are not acceptable.

1. A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\) and the capacity of each container is \(355 \mathrm {~cm} ^ { 3 }\) The metal used

• for the circular base and the curved side costs 0.04 pence/ \(\mathrm { cm } ^ { 2 }\)
• for the circular top costs 0.09 pence/ \(\mathrm { cm } ^ { 2 }\)

Both metals used are of negligible thickness.

1. Show that the total cost, \(C\) pence, of the metal for one container is given by $$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
2. Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
3. Using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} r ^ { 2 } }\) prove that the cost is minimised for the value of \(r\) found in part (b).
4. Hence find the minimum value of \(C\), giving your answer to the nearest integer.

1. A curve has equation

$$y = \frac { 2 } { 3 } x ^ { 3 } - \frac { 7 } { 2 } x ^ { 2 } - 4 x + 5$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form.
2. Hence find the range of values of \(x\) for which \(y\) is decreasing.

1. A curve has equation

$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.

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3. The curve \(C\) has equation $$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$ a. Find
i) \(\frac { d y } { d x }\) ii) \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) b. Use calculus to find the coordinates of the stationary point of \(C\).
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.

6. \begin{figure}[h] \end{figure} The figure 1 shows sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). $$f ( x ) = a x ( x - b ) ^ { 2 } , x \in R$$ where \(a\) and \(b\) are constants.
The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\).
There is a minimum point at \(( 1 , - 4 )\) and a maximum point at \(( 3,0 )\).
a. Find the equation of \(C\).
b. Deduce the values of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) > 0$$ Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
c. State the possible values for \(k\).

3. $$y = \frac { 5 x ^ { 2 } + 10 x } { ( x + 1 ) ^ { 2 } } \quad x \neq - 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A } { ( x + 1 ) ^ { n } }\) where \(A\) and \(n\) are constants to be found.
2. Hence deduce the range of values for \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\)

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a cubic expression in \(X\). The curve

• passes through the origin
• has a maximum turning point at \(( 2,8 )\)
• has a minimum turning point at \(( 6,0 )\)
  1. Write down the set of values of \(x\) for which

$$\mathrm { f } ^ { \prime } ( x ) < 0$$ The line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at only one point.

Find the set of values of \(k\), giving your answer in set notation.

Find the equation of \(C\). You may leave your answer in factorised form.

1. The function f is defined by

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

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be found.
2. Hence, using algebra, find the values of \(x\) for which f is decreasing. You must show each step in your working.

11. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$

1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \quad x > 3$$
2. Prove that \(\mathrm { f } ( x )\) is a decreasing function.

1. A scientist is studying a population of mice on an island.

The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation $$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$

1. Find the number of mice in the population at the start of the study.
2. Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\) The rate of growth is a maximum after \(T\) months.
3. Find, according to the model, the value of \(T\). According to the model, the maximum number of mice on the island is \(P\).
4. State the value of \(P\).

13. \begin{figure}[h] \end{figure} [A sphere of radius \(r\) has volume \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(4 \pi r ^ { 2 }\) ]
A manufacturer produces a storage tank.
The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness.
The cylinder has radius \(r\) metres and height \(h\) metres and the hemisphere has radius \(r\) metres.
The volume of the tank is \(6 \mathrm {~m} ^ { 3 }\).

1. Show that, according to the model, the surface area of the tank, in \(\mathrm { m } ^ { 2 }\), is given by $$\frac { 12 } { r } + \frac { 5 } { 3 } \pi r ^ { 2 }$$ The manufacturer needs to minimise the surface area of the tank.
2. Use calculus to find the radius of the tank for which the surface area is a minimum.
   (4)
3. Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

10 In this question you must show detailed reasoning.

1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.

13 Determine the range of values of \(x\) for which \(y = 4 x ^ { 3 } + 7 x ^ { 2 } - 6 x + 8\) is a decreasing function.

4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Determine the coordinates of the stationary points on the curve.
2. Determine the nature of each stationary point.
3. Write down the equation of the vertical asymptote.
4. Deduce the set of values of \(x\) for which the curve is concave upwards.

11 Show that \(\mathrm { e } ^ { x }\) is an increasing function for all values of \(x\), as stated in line 39 .

12

1. Show that the only stationary point on the curve \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } }\) occurs where \(x = \mathrm { e }\), as given in line 45.
2. Show that the stationary point is a maximum.
3. It follows from part (b) that, for any positive number \(a\) with \(a \neq \mathrm { e }\), \(\frac { \ln \mathrm { e } } { \mathrm { e } } > \frac { \ln a } { a }\).
   Use this fact to show that \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).

5 A model car moves so that its distance, \(x\) centimetres, from a fixed point \(O\) after time \(t\) seconds is given by $$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$

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

3 The depth of water, \(y\) metres, in a tank after time \(t\) hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether it is a maximum value or a minimum value.
3. 1. Find the rate of change of the depth of water, in metres per hour, when \(t = 1\).
   2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when \(t = 1\).