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

6 \includegraphics[max width=\textwidth, alt={}, center]{388d7076-636c-417d-84cb-e6e2a3e9a6a0-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).

1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
2. Find the exact \(x\)-coordinate of \(M\).

6 \includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-08_451_1086_260_525} The diagram shows the curve with equation $$y = ( \ln x ) ^ { 2 } - 2 \ln x$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a minimum point \(M\).

1. Find the exact value of the gradient of the curve at each of the points \(A\) and \(B\).
2. Find the exact \(x\)-coordinate of \(M\).

4

1. Use the trapezium rule with three intervals to show that the value of \(\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x\) is approximately \(\ln 12\).
2. Use a graph of \(y = \ln x\) to show that \(\ln 12\) is an under-estimate of the true value of \(\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x\).

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

2 A curve has equation \(y = 7 + 4 \ln ( 2 x + 5 )\).
Find the equation of the tangent to the curve at the point ( \(- 2,7\) ), giving your answer in the form \(y = m x + c\).

6

1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\ln x = 2 \mathrm { e } ^ { - x }$$ has exactly one root.
2. Verify by calculation that the root lies between 1.5 and 1.6.
3. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$ converges, then it converges to the root of the equation in part (a).
4. Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{1cd04df5-3fe3-4573-b880-d49262afd16a-10_456_598_269_762} The diagram shows the curve with equation \(y = \frac { 2 \ln x } { 3 x + 1 }\). The curve crosses the \(x\)-axis at the point \(A\) and has a maximum point \(B\). The shaded region is bounded by the curve and the lines \(x = 3\) and \(y = 0\).

1. Find the gradient of the curve at \(A\).
2. Show by calculation that the \(x\)-coordinate of \(B\) lies between 3.0 and 3.1.
3. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has exactly one root.
2. Verify by calculation that the root lies between 1.0 and 1.4 .
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ to determine the root correct to 2 decimal places, showing the result of each iteration.

6 \includegraphics[max width=\textwidth, alt={}, center]{08210e25-0f0e-405b-b72d-1bf989689b0a-3_641_865_264_641} The diagram shows the part of the curve \(y = \frac { \ln x } { x }\) for \(0 < x \leqslant 4\). The curve cuts the \(x\)-axis at \(A\) and its maximum point is \(M\).

1. Write down the coordinates of \(A\).
2. Show that the \(x\)-coordinate of \(M\) is e, and write down the \(y\)-coordinate of \(M\) in terms of e.
3. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x } \mathrm {~d} x$$ correct to 2 decimal places.
4. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (iii).

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation \(x = 9 \mathrm { e } ^ { - 2 x }\).
2. Verify, by calculation, that this root lies between 1 and 2 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( \ln 9 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation given in part (i).
4. Use the iterative formula, with \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

1 Given that \(13 ^ { x } = ( 2.8 ) ^ { y }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

6

1. By sketching a suitable pair of graphs, show that the equation $$3 \mathrm { e } ^ { x } = 8 - 2 x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
3. Show that this root also satisfies the equation $$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
4. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4

1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
2. Show by calculation that the root lies between 2.0 and 2.5.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6

1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. \includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

6

1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. \includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

4 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-06_355_839_260_653} The diagram shows the curve with equation \(y = \frac { 5 \ln x } { 2 x + 1 }\). The curve crosses the \(x\)-axis at the point \(P\) and has a maximum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \frac { x + 0.5 } { \ln x }\).
3. Use an iterative formula based on the equation in part (ii) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Show the result of each iteration to 6 significant figures.

2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).

1. Find the \(x\)-coordinate of the stationary point.
2. Determine whether the stationary point is a maximum or minimum point.

10 \includegraphics[max width=\textwidth, alt={}, center]{0f081749-4fe0-46e3-96c2-466e69cf49d3-4_620_894_338_687} The function f is defined by \(\mathrm { f } ( x ) = ( \ln x ) ^ { 2 }\) for \(x > 0\). The diagram shows a sketch of the graph of \(y = \mathrm { f } ( x )\). The minimum point of the graph is \(A\). The point \(B\) has \(x\)-coordinate e .

1. State the \(x\)-coordinate of \(A\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(B\).
3. Use the substitution \(x = \mathrm { e } ^ { u }\) to show that the area of the region bounded by the \(x\)-axis, the line \(x = \mathrm { e }\), and the part of the curve between \(A\) and \(B\) is given by $$\int _ { 0 } ^ { 1 } u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u .$$
4. Hence, or otherwise, find the exact value of this area.

2 Showing all necessary working, solve the equation \(\ln ( 2 x - 3 ) = 2 \ln x - \ln ( x - 1 )\). Give your answer correct to 2 decimal places.

2 Solve the equation $$\ln ( 1 + x ) = 1 + \ln x$$ giving your answer correct to 2 significant figures.

2 Solve the equation $$\ln \left( 1 + x ^ { 2 } \right) = 1 + 2 \ln x$$ giving your answer correct to 3 significant figures.

1 Using the substitution \(u = \mathrm { e } ^ { x }\), or otherwise, solve the equation $$\mathrm { e } ^ { x } = 1 + 6 \mathrm { e } ^ { - x }$$ giving your answer correct to 3 significant figures.

8 Let \(f ( x ) = \frac { 12 + 8 x - x ^ { 2 } } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 4 + x ^ { 2 } }\).
2. Show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 2 } \right)\).

1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).

1 Solve the equation \(5 \ln \left( 4 - 3 ^ { x } \right) = 6\). Show all necessary working and give the answer correct to 3 decimal places.