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

5 It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r }\), where \(u _ { r } = x ^ { \mathrm { f } ( r ) } - x ^ { \mathrm { f } ( r + 1 ) }\) and \(x > 0\).

1. Find \(S _ { n }\) in terms of \(n , x\) and the function f .
2. Given that \(\mathrm { f } ( r ) = \ln r\), find the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-10_2716_31_106_2016} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-11_2723_35_101_20}
3. Given instead that \(\mathrm { f } ( r ) = 2 \log _ { x } r\) where \(x \neq 1\), use standard results from the List of formulae (MF19) to find \(\sum _ { n = 1 } ^ { N } S _ { n }\) in terms of \(N\). Fully factorise your answer.

3

1. Solve the inequality \(| 2 x - 5 | < | x + 3 |\).
2. Hence find the largest integer \(y\) satisfying the inequality \(| 2 \ln y - 5 | < | \ln y + 3 |\).

5

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation $$\frac { 1 } { x } = \ln x$$
2. Verify by calculation that this root lies between 1 and 2 .
3. Show that this root also satisfies the equation $$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
4. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$ with initial value \(x _ { 1 } = 1.8\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 1 } \frac { 1 } { 6 + 2 \mathrm { e } ^ { x } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
2. Find \(\int \frac { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).

2 The curve with equation \(y = \frac { \sin 2 x } { \mathrm { e } ^ { 2 x } }\) has one stationary point in the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the exact \(x\)-coordinate of this point.

6

1. Find \(\int 4 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
2. Show that \(\int _ { 1 } ^ { 3 } \frac { 6 } { 3 x - 1 } \mathrm {~d} x = \ln 16\).

6

1. Find
   1. \(\int \frac { \mathrm { e } ^ { 2 x } + 6 } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\),
   2. \(\int 3 \cos ^ { 2 } x \mathrm {~d} x\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { 6 } { \ln ( x + 2 ) } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
   1. Express \(3 \cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
   2. Hence solve the equation $$3 \cos 2 x + \sin 2 x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1

1. Find \(\int \frac { 2 } { 4 x - 1 } \mathrm {~d} x\).
2. Hence find \(\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x\), expressing your answer in the form \(\ln a\), where \(a\) is an integer.

2 The curve \(y = \frac { \mathrm { e } ^ { 3 x - 1 } } { 2 x }\) has one stationary point. Find the coordinates of this stationary point.

5 The parametric equations of a curve are $$x = 1 + \sqrt { } t , \quad y = 3 \ln t$$

1. Find the exact value of the gradient of the curve at the point \(P\) where \(y = 6\).
2. Show that the tangent to the curve at \(P\) passes through the point \(( 1,0 )\).

3

1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).

3 A curve has equation $$3 \ln x + 6 x y + y ^ { 2 } = 16$$ Find the equation of the normal to the curve at the point \(( 1,2 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
2. Verify by calculation that the root lies between 2 and 2.8.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 3 } { 2 } } \left( 4 + \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\), expressing your answer in logarithmic form.
2. By considering \(\frac { d } { d x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } n } \right)\), or otherwise, show that $$4 n l _ { n + 2 } = \frac { 3 } { 2 } \left( \frac { 2 } { 5 } \right) ^ { n } + ( n - 1 ) l _ { n } .$$
3. Find the value of \(I _ { 5 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{23b06b1c-997f-425d-ae3d-bd4cc1295605-10_771_1146_260_497} The diagram shows the curve with equation \(\mathrm { y } = \ln ( 1 + \mathrm { x } )\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles each of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) d x < U _ { n }\), where $$U _ { n } = \frac { 1 } { n } \ln \frac { ( 2 n ) ! } { n ! } - \ln n$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(\mathrm { L } _ { \mathrm { n } }\) for \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) \mathrm { d } x\).
3. By simplifying \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } }\), show that \(\lim _ { \mathrm { n } \rightarrow \infty } \left( \mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \log _ { 10 } x\) The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 14\) Using the trapezium rule with four strips of equal width,

1. show that the area of \(R\) is approximately 10.10
2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(R\).
3. Using the answer to part (a) and making your method clear, estimate the value of
   1. \(\quad \int _ { 2 } ^ { 14 } \log _ { 10 } \sqrt { x } \mathrm {~d} x\)
   2. \(\int _ { 2 } ^ { 14 } \log _ { 10 } 100 x ^ { 3 } \mathrm {~d} x\)

3. (a) \(\quad y = 5 ^ { x } + \log _ { 2 } ( x + 1 ) , \quad 0 \leqslant x \leqslant 2\) Complete the table below, by giving the value of \(y\) when \(x = 1\) (b) Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) \mathrm { d } x$$ giving your answer to 2 decimal places.
(c) Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 + 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) d x$$ giving your answer to 2 decimal places.

2. The function \(f\) and the function \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 12 } { x + 1 } & x > 0 , x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { 5 } { 2 } \ln x & x > 0 , x \in \mathbb { R } \end{array}$$

1. Find, in simplest form, the value of \(\mathrm { fg } \left( \mathrm { e } ^ { 2 } \right)\)
2. Find f-1
3. Hence, or otherwise, find all real solutions of the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$$

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

1. The function f is defined by

$$f ( x ) = \frac { 2 x ^ { 2 } - 32 } { 3 x ^ { 2 } + 7 x - 20 } + \frac { 8 } { 3 x - 5 } \quad x \in \mathbb { R } \quad x > 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 5 }\)
2. Show, using calculus, that f is a decreasing function. You must make your reasoning clear. The function g is defined by $$g ( x ) = 3 + 2 \ln x \quad x \geqslant 1$$
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 5$$

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0 \\ & \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
2. Use differentiation to prove that g is an increasing function.
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the range of fg

7 The mass, \(M \mathrm {~kg}\), of a species of tree can be modelled by the equation $$\log _ { 10 } M = 1.93 \log _ { 10 } r + 0.684$$ where \(r \mathrm {~cm}\) is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm .
According to the model,

1. find the mass of this tree, giving your answer to 2 significant figures.
2. Show that the equation of the model can be written in the form $$M = p r ^ { q }$$ giving the values of the constants \(p\) and \(q\) to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(p\). Q

4. $$y = \log _ { 10 } ( 2 x + 1 )$$

1. Express \(x\) in terms of \(y\).
2. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 4.
   1. Express \(x\) in terms of \(y\).

      	\(\begin{array} { c } \text { Leave }
      \text { blank } \end{array}\)

      (2)
   2. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-10_2662_111_107_1950}

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = | 2 - 4 \ln ( x + 1 ) | \quad x > k$$ where \(k\) is a constant.
Given that the curve

• has an asymptote at \(x = k\)
• cuts the \(y\)-axis at point \(A\)
• meets the \(x\)-axis at point \(B\) as shown in Figure 2,
  1. state the value of \(k\)
  2. 1. find the \(y\) coordinate of \(A\)
     2. find the exact \(x\) coordinate of \(B\)
  3. Using algebra and showing your working, find the set of values of \(x\) such that

$$| 2 - 4 \ln ( x + 1 ) | > 3$$