1.06d Natural logarithm: ln(x) function and properties

10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h] \end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.

11 The height \(h \mathrm {~cm}\) of a sunflower plant \(t\) days after planting the seed is modelled by \(\mathrm { h } = \mathrm { a } + \mathrm { b }\) Int for \(t \geqslant 9\), where \(a\) and \(b\) are constants. The sunflower is 10 cm tall 10 days after planting and 200 cm tall 85 days after planting.

1. 1. Show that the value of \(b\) which best models these values is 88.8 correct to \(\mathbf { 3 }\) significant figures.
   2. Find the corresponding value of \(a\).
2. 1. Explain why the model is not suitable for small positive values of \(t\).
   2. Explain why the model is not suitable for very large positive values of \(t\).
3. Show that the model indicates that the sunflower grows to 1 m in height in less than half the time it takes to grow to 2 m .
4. Find the value of \(t\) for which the rate of growth is 3 cm per day.

10 In this question you must show detailed reasoning.
Show that \(\int _ { \mathrm { e } } ^ { \pi } \frac { 1 } { x } \mathrm {~d} x = \ln \pi - 1\) as given in line 37.

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 \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).

8

1. Given that \(\log _ { a } b = c\), express \(b\) in terms of \(a\) and \(c\).
2. By forming a quadratic equation, show that there is only one value of \(x\) which satisfies the equation \(2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0\).

1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.

   \(x\)	2	3	4	5	6
   \(y\)	2.89	6.36

   The shaded area is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
2. Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
3. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.

5

1. The diagram shows part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the point \(( a , 0 )\) and the \(y\)-axis at the point \(( 0 , - b )\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-4_569_853_1206_589} On separate diagrams, sketch the curves with the following equations. On each diagram, indicate, in terms of \(a\) or \(b\), the coordinates of the points where the curve crosses the coordinate axes.
   1. \(y = | \mathrm { f } ( x ) |\).
   2. \(\quad y = 2 \mathrm { f } ( x )\).
2. 1. Describe a sequence of geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x + 1 ) - 2\).
   2. Find the exact values of the coordinates of the points where the graph of \(y = 4 \ln ( x + 1 ) - 2\) crosses the coordinate axes.

2 For \(0 < x \leqslant 2\), the curves with equations \(y = 4 \ln x\) and \(y = \sqrt { x }\) intersect at a single point where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1.5.
2. Show that the equation \(4 \ln x = \sqrt { x }\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \sqrt { x _ { n } } } { 4 } \right) }$$ with \(x _ { 1 } = 0.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
4. Figure 1, on the page 3, shows a sketch of parts of the graphs of \(y = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-3_1285_1543_356_296}
   \end{figure}

8 The curve with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}

1. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { f } ^ { - 1 }\).
   3. Sketch, on the axes given on page 9 , the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
      (2 marks)
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
   1. Find \(\mathrm { gf } ( x )\), giving your answer in the form \(( a x - b ) ^ { 2 } - c\), where \(a , b\) and \(c\) are integers.
   2. Write down an expression for \(\mathrm { fg } ( x )\), and hence find the exact solution of the equation \(\operatorname { fg } ( x ) = \ln 5\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}

3. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$

4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).

1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
3. Find the \(x\)-coordinate of \(R\) in exact form.

6. The function f is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0 .$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find an expression for the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
4. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.

1. The function f is defined by

$$\mathrm { f } ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 } .$$

1. Find the exact value of \(\mathrm { ff } ( 1 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

8. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).

1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
2. Show that \(1 < q < 2\).
3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
4. Use the iterative formula $$x _ { n + 1 } = \frac { 12 } { 7 } \ln \left( 2 x _ { n } + 5 \right) - 2 ,$$ with \(x _ { 0 } = 1.5\), to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.

7. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$

1. State the range of f .
2. Evaluate fg(-2).
3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
5. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures. \end{document}

7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, \(N\), in the population \(t\) weeks after the start of the investigation. Use this model to answer the following questions.

1. 1. Find the size of the population at the start of the investigation.
   2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
   3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form \(t = r \ln s\), where \(r\) and \(s\) are integers.
2. 1. Show that the rate of growth, \(\frac { \mathrm { d } N } { \mathrm {~d} t }\), is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
   2. The maximum rate of growth occurs after \(T\) weeks. Find the value of \(T\).

2 The average weekly pay of a footballer at a certain club was \(\pounds 80\) on 1 August 1960. By 1 August 1985, this had risen to \(\pounds 2000\). The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where \(\pounds P\) is the average weekly pay \(t\) years after 1 August 1960, and \(A\) and \(k\) are constants.

1. 1. Write down the value of \(A\).
   2. Show that the value of \(k\) is 1.137411 , correct to six decimal places.
2. Use this model to predict the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed \(\pounds 100000\).

4 The value, \(\pounds V\), of an initial investment, \(\pounds P\), at the end of \(n\) years is given by the formula $$V = P \left( 1 + \frac { r } { 100 } \right) ^ { n }$$ where \(r \%\) per year is the fixed interest rate.
Mr Brown invests \(\pounds 1000\) in Barcelona Bank at a fixed interest rate of \(3 \%\) per year.

1. 1. Find the value of Mr Brown's investment at the end of 5 years. Give your value to the nearest \(\pounds 10\).
   2. The value of Mr Brown's investment will first exceed \(\pounds 2000\) after \(N\) complete years. Find the value of \(N\).
2. Mrs White invests \(\pounds 1500\) in Bilbao Bank at a fixed interest rate of \(1.5 \%\) per year. Mr Brown and Mrs White invest their money at the same time. The value of Mr Brown's investment will first exceed the value of Mrs White's investment after \(T\) complete years. Find the value of \(T\).

2. A particle \(P\) of mass 0.25 kg is moving on a horizontal plane. At time \(t\) seconds the velocity, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), of \(P\) relative to a fixed origin \(O\) is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane.

1. Find the acceleration of \(P\) in terms of \(t\).
2. Find, correct to 3 significant figures, the magnitude of the resultant force acting on \(P\) when \(t = 1\).
   (4 marks)

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross section \(A B C D\) of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain. Using these measurements, the curve \(B D\) is modelled by the equation $$y = \ln ( 3.6 x - k ) \quad 1 \leqslant x \leqslant 1.18$$ as shown in Figure 2.

1. Find the value of \(k\).
2. Find the depth of the paddling pool according to this model. The pool is being filled with water from a tap.
3. Find, in terms of \(h\), the volume of water in the pool when the pool is filled to a depth of \(h \mathrm {~m}\). Given that the pool is being filled at a constant rate of 15 litres every minute,
4. find, in \(\mathrm { cmh } ^ { - 1 }\), the rate at which the water level is rising in the pool when the depth of the water is 0.2 m .

1. (a) Given that

$$y = \ln \left( 3 + x ^ { 2 } \right)$$ complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures. In part (b) you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} (b) Use Simpson's rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of $$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$ (c) Using your answer to part (b) and making your method clear, estimate the value of $$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$

10 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-7_579_764_255_651} The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.

1. (a) Determine the value of \(\alpha\).
   (b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value.
2. Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as $$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
3. Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
4. Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.

4 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is \(\mathrm { f } ( x ) \geqslant - 2\).

1. Show that \(a = \frac { 3 } { 2 }\).
2. Find the value of \(b\).
3. Find the exact value of \(\mathrm { ff } ( - 1 )\).
4. Explain why the function f does not have an inverse.

5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at ( \(0 , a\) ) and ( \(b , 0\) ). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}

1. State the value of \(a\), and show that \(b = \ln 3\).
2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).

6 An on-line science website states:
'To find a dog's equivalent human age in years, multiply the natural logarithm of the dog's age in years by 16 then add 31.' 6

1. Calculate the equivalent age to the nearest human year of a dog aged 5 years. 6
2. A dog's equivalent age in human years is 40 years. Find the dog's actual age to the nearest month.
   6
3. Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs.