1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

11 In a science experiment a substance is decaying exponentially. Its mass, \(M\) grams, at time \(t\) minutes is given by \(M = 300 e ^ { - 0.05 t }\).

1. Find the time taken for the mass to decrease to half of its original value. A second substance is also decaying exponentially. Initially its mass was 400 grams and, after 10 minutes, its mass was 320 grams.
2. Find the time at which both substances are decaying at the same rate.

7 The curve \(y = \left( x ^ { 2 } - 2 \right) \ln x\) has one stationary point which is close to \(x = 1\).

1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(2 x ^ { 2 } \ln x + x ^ { 2 } - 2 = 0\).
2. Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form \(x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } \ln x _ { n } + 3 x _ { n } ^ { 2 } + 2 } { 4 x _ { n } \left( \ln x _ { n } + 1 \right) }\).
3. Apply the Newton-Raphson formula with initial value \(x _ { 1 } = 1\) to find \(x _ { 2 }\) and \(x _ { 3 }\).
4. Find the coordinates of this stationary point, giving each coordinate correct to \(\mathbf { 3 }\) decimal places.

9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).

1. Explain why the path of the particle never crosses the \(x\)-axis.
2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.

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

1. Determine the \(x\)-coordinates of any stationary points on the curve.
2. Show that the curve is convex for all values of \(x\).

9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, \(B\), and the number of wildflower plants, \(F\), at time \(t\) weeks after the start of the study. In the model \(B = 20 + 2 t + \cos 3 t\) and \(F = 50 \mathrm { e } ^ { 0.1 t }\). The model assumes that \(B\) and \(F\) can be treated as continuous variables.

1. State the meaning of \(\frac { \mathrm { d } B } { \mathrm {~d} F }\).
2. Determine \(\frac { \mathrm { d } B } { \mathrm {~d} F }\) when \(t = 4\).
3. Suggest a reason why this model may not be valid for values of \(t\) greater than 12 .

1

1. Differentiate the following.
   1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
   2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in R\), \(x > 0\) $$\mathrm { f } ( x ) = ( 0.5 x - 8 ) \ln ( x + 1 ) \quad 0 \leq x \leq A$$ a. Find the value of \(A\).
b. Find \(\mathrm { f } ^ { \prime } ( x )\) The curve has a minimum turning point at \(B\).
c. Show that the \(x\)-coordinate of \(B\) is a solution of the equation $$x = \frac { 17 } { \ln ( x + 1 ) + 1 } - 1$$ d. Use the iteration formula $$x _ { n + 1 } = \frac { 17 } { \ln \left( x _ { n } + 1 \right) + 1 } - 1$$ with \(x _ { 0 } = 5\) to find the values of \(x _ { 1 }\) and the value of \(x _ { 6 }\) giving your answers to three decimal places.

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

14. A population of ants being studied on an island. The number of ants, \(P\), in the population, is modelled by the equation. $$P = \frac { 900 k e ^ { 0.2 t } } { 1 + k e ^ { 0.2 t } } , \text { where } k \text { is a constant. }$$ Given that there were 360 ants when the study started,
a. show that \(k = \frac { 2 } { 3 }\).
b. Show that \(P = \frac { 1800 } { 2 + 3 e ^ { - 0.2 t } }\). The model predicts an upper limit to the number of ants on the island.
c. State the value of this limit.
d. Find the value of \(t\) when \(P = 520\). Give your answer to one decimal place.
e. i. Show that the rate of growth, \(\frac { \mathrm { d } P } { d t } = \frac { P ( 900 - P ) } { 4500 }\) ii. Hence state the value of \(P\) at which the rate of growth is a maximum.

1. A curve has equation \(y = \mathrm { f } ( x )\), where

$$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$ where \(A\) and \(B\) are constants to be found.
2. Hence show that the \(x\) coordinates of the turning points of the curve are solutions of the equation $$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$ The equation \(x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) has two positive roots \(\alpha\) and \(\beta\) where \(\beta > \alpha\) A student uses the iteration formula $$x _ { n + 1 } = \frac { 2 \mathrm { e } ^ { 3 x _ { n } } - 4 } { \mathrm { e } ^ { 3 x _ { n } } + 4 }$$ in an attempt to find approximations for \(\alpha\) and \(\beta\) Diagram 1 shows a plot of part of the curve with equation \(y = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) and part of the line with equation \(y = x\) Using Diagram 1 on page 42
3. draw a staircase diagram to show that the iteration formula starting with \(x _ { 1 } = 1\) can be used to find an approximation for \(\beta\) Use the iteration formula with \(x _ { 1 } = 1\), to find, to 3 decimal places,
4. 1. the value of \(x _ { 2 }\)
   2. the value of \(\beta\) Using a suitable interval and a suitable function that should be stated
5. show that \(\alpha = 0.432\) to 3 decimal places. Only use the copy of Diagram 1 if you need to redraw your answer to part (c). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_736_812_372_143} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_738_815_370_1114} \captionsetup{labelformat=empty} \caption{copy of Diagram 1}
   \end{figure}

1. A large spherical balloon is deflating.

At time \(t\) seconds the balloon has radius \(r \mathrm {~cm}\) and volume \(V \mathrm {~cm} ^ { 3 }\) The volume of the balloon is modelled as decreasing at a constant rate.

1. Using this model, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = - \frac { k } { r ^ { 2 } }$$ where \(k\) is a positive constant. Given that
   • the initial radius of the balloon is 40 cm
   • after 5 seconds the radius of the balloon is 20 cm
   • the volume of the balloon continues to decrease at a constant rate until the balloon is empty
   • solve the differential equation to find a complete equation linking \(r\) and \(t\).
   • Find the limitation on the values of \(t\) for which the equation in part (b) is valid.

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 4 \sin 2 x } { \mathrm { e } ^ { \sqrt { 2 } x - 1 } } , \quad 0 \leqslant x \leqslant \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2 x = \sqrt { 2 }$$
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation
   1. \(y = \mathrm { f } ( 2 x )\).
   2. \(y = 3 - 2 \mathrm { f } ( x )\).

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 \(P\).

1. The function f is defined by

$$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.

1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point, (b) find the range of possible values of \(k\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) where $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 4 x ^ { 2 } - 1 } & x > 0 \\ \mathrm {~g} ( x ) = 8 \ln x & x > 0 \end{array}$$

1. Find
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { g } ^ { \prime } ( x )\) Given that \(\mathrm { f } ^ { \prime } ( x ) = \mathrm { g } ^ { \prime } ( x )\) at \(x = \alpha\)
2. show that \(\alpha\) satisfies the equation $$4 x ^ { 2 } + 2 \ln x - 1 = 0$$ The iterative formula $$x _ { n + 1 } = \sqrt { \frac { 1 - 2 \ln x _ { n } } { 4 } }$$ is used with \(x _ { 1 } = 0.6\) to find an approximate value for \(\alpha\)
3. Calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(\alpha\)

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 4 x ^ { 2 } + x } { 2 \sqrt { x } } - 4 \ln x \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 x ^ { 2 } + x - 16 \sqrt { x } } { 4 x \sqrt { x } }$$ The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).
2. Show that the \(x\) coordinate of \(P\) is a solution of $$x = \left( \frac { 4 } { 3 } - \frac { \sqrt { x } } { 12 } \right) ^ { \frac { 2 } { 3 } }$$
3. Use the iteration formula $$x _ { n + 1 } = \left( \frac { 4 } { 3 } - \frac { \sqrt { x _ { n } } } { 12 } \right) ^ { \frac { 2 } { 3 } } \quad \text { with } x _ { 1 } = 2$$ to find (i) the value of \(x _ { 2 }\) to 5 decimal places,
   (ii) the \(x\) coordinate of \(P\) to 5 decimal places.

1. A company decides to manufacture a soft drinks can with a capacity of 500 ml .

The company models the can in the shape of a right circular cylinder with radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). In the model they assume that the can is made from a metal of negligible thickness.

1. Prove that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the can is given by $$S = 2 \pi r ^ { 2 } + \frac { 1000 } { r }$$ Given that \(r\) can vary,
2. find the dimensions of a can that has minimum surface area.
3. With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.

8. \begin{figure}[h] \end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl. When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)

1. Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\) Given that the flow of water into the bowl is increased to a constant rate of \(300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(12 < h \leqslant 24\)
2. find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 20\)

10 A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins \(B\) and \(C\). \(A B\) is a 30 cm rod and \(A C\) is a 15 cm rod. The angle \(B A C\) is \(\theta\) radians as shown in the diagram. The angle \(\theta\) increases at a rate of 0.1 radians per second.
Determine the rate of change of the length BC when \(\theta = \frac { 1 } { 3 } \pi\).

4 Find the second derivative of \(\left( x ^ { 2 } + 5 \right) ^ { 4 }\), giving your answer in factorised form.

4 Differentiate the following.

1. \(\sqrt { 1 - 3 x ^ { 2 } }\)
2. \(\frac { x ^ { 2 } } { 3 x + 2 }\)

2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).

15 Functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows. \(\mathrm { f } ( x ) = \sqrt { x }\) for \(x > 0\) and \(\mathrm { g } ( x ) = x ^ { 3 } - x - 6\) for \(x > 2\). The function \(\mathrm { h } ( x )\) is defined as \(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).

1. Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
2. Find \(\mathrm { h } ( 3 )\). Fig. 15 shows \(\mathrm { h } ( x )\) and \(\mathrm { h } ^ { - 1 } ( x )\), together with the straight line \(y = x\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242} \captionsetup{labelformat=empty} \caption{Fig. 15}
   \end{figure}
3. Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).

8

1. The curve \(y = \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }\) is shown in Fig. 8. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-08_495_1058_1105_315} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Show that \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 20 x ^ { 2 } - 4 } { \left( 1 + x ^ { 2 } \right) ^ { 4 } }\).
   2. In this question you must show detailed reasoning. Find the set of values of \(x\) for which the curve is concave downwards.
2. Use the substitution \(x = \tan \theta\) to find the exact value of \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x\). Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).