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

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function \(gf(x)\), expressing your answer as simply as possible. [3]

The functions \(f(x)\) and \(g(x)\) are defined by \(f(x) = \ln x\) and \(g(x) = 2 + e^x\), for \(x > 0\). Find the exact value of \(x\), given that \(fg(x) = 2x\). [5]

The function f(x) is defined as \(f(x) = \frac{\ln x}{x}\). The graph of the function is shown in Fig. 6. \includegraphics{figure_6}

1. Give the coordinates of the point, P, where the curve crosses the \(x\)-axis. [1]
2. Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]

$$f(x) = 2x^2 + 3 \ln (2 - x), \quad x \in \mathbb{R}, \quad x < 2.$$

1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]

The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.

1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
2. Solve the equation \(f'(x) = 0\). [5]

The function f is defined by $$\text{f}(x) \equiv 2 + \ln (3x - 2), \quad x \in \mathbb{R}, \quad x > \frac{2}{3}.$$

1. Find the exact value of \(\text{f}(1)\). [2]
2. Find an equation for the tangent to the curve \(y = \text{f}(x)\) at the point where \(x = 1\). [4]
3. Find an expression for \(\text{f}^{-1}(x)\). [2]

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function gf(x), expressing your answer as simply as possible. [3]

Given that \(f(x) = \frac{1}{2}\ln(x - 1)\) and \(g(x) = 1 + e^{2x}\), show that g(x) is the inverse of f(x). [3]

Fig. 8 shows the line \(y = x\) and parts of the curves \(y = f(x)\) and \(y = g(x)\), where $$f(x) = e^{x-1}, \quad g(x) = 1 + \ln x.$$ The curves intersect the axes at the points A and B, as shown. The curves and the line \(y = x\) meet at the point C. \includegraphics{figure_8}

1. Find the exact coordinates of A and B. Verify that the coordinates of C are \((1, 1)\). [5]
2. Prove algebraically that \(g(x)\) is the inverse of \(f(x)\). [2]
3. Evaluate \(\int_0^1 f(x) \, dx\), giving your answer in terms of \(e\). [3]
4. Use integration by parts to find \(\int \ln x \, dx\). Hence show that \(\int_{e^{-1}}^1 g(x) \, dx = \frac{1}{e}\). [6]
5. Find the area of the region enclosed by the lines OA and OB, and the arcs AC and BC. [2]

A curve \(C\) is defined by the parametric equations $$x = \frac{4 - e^{-6t}}{4}, \quad y = \frac{e^{3t}}{3t}, \quad t \neq 0$$

1. Find the exact value of \(\frac{dy}{dx}\) at the point on \(C\) where \(t = \frac{2}{3}\). [5 marks]
2. Show that \(x = \frac{4 - e^{-6t}}{4}\) can be rearranged into the form \(e^{3t} = \frac{e}{2\sqrt{(1-x)}}\). [2 marks]
3. Hence find the Cartesian equation of \(C\), giving your answer in the form $$y = \frac{e}{f(x)[1 - \ln(f(x))]}$$ [2 marks]

Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \includegraphics{figure_7}

1. Find the lengths OA, OB and AC. [5]
2. Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
3. Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]

An object is formed by rotating the region OACB through \(360°\) about Ox.

1. Find the volume of the object. [5]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \frac{1}{x+1}\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} < \ln(n+1).$$ [5]
2. By considering the areas of another set of rectangles, show that $$1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > \ln(n+1).$$ [2]
3. Hence show that $$\ln(n+1) + \frac{1}{n+1} < \sum_{r=1}^{n+1} \frac{1}{r} < \ln(n+1) + 1.$$ [2]
4. State, with a reason, whether \(\sum_{r=1}^{\infty} \frac{1}{r}\) is convergent. [2]

The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by \begin{align} C &= 1 + \cos \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta + \cos 5\theta,
S &= \sin \theta + \sin 2\theta + \sin 3\theta + \sin 4\theta + \sin 5\theta. \end{align}

1. Show that \(C + iS = \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta} - e^{-i\theta}} \cdot e^{i\theta}\). [4]
2. Deduce that \(C = \sin 3\theta \cos \frac{5}{2}\theta \operatorname{cosec} \frac{1}{2}\theta\) and write down the corresponding expression for \(S\). [4]
3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\). [4]

1. Given that \(y = \ln [t + \sqrt{(1 + t^2)}]\), show that \(\frac{dy}{dt} = \frac{1}{\sqrt{(1+t^2)}}\). [3]

The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{(1+t^2)}}, \quad y = \ln [t + \sqrt{(1 + t^2)}], \quad t \in \mathbb{R}.$$ A student was asked to prove that, for \(t > 0\), the gradient of the tangent to \(C\) is negative. The attempted proof was as follows: $$y = \ln \left(t + \frac{1}{x}\right)$$ $$= \ln \left(\frac{tx + 1}{x}\right)$$ $$= \ln (tx + 1) - \ln x$$ $$\therefore \frac{dy}{dx} = \frac{t}{tx + 1} - \frac{1}{x}$$ $$= \frac{\frac{t}{x}}{t + \frac{1}{x}} - \frac{1}{x}$$ $$= \frac{t\sqrt{(1+t^2)}}{t + \sqrt{(1+t^2)}} - \sqrt{(1 + t^2)}$$ $$= -\frac{(1+t^2)}{t + \sqrt{(1+t^2)}}$$ As \((1 + t^2) > 0\), and \(t + \sqrt{(1 + t^2)} > 0\) for \(t > 0\), \(\frac{dy}{dx} < 0\) for \(t > 0\).

1. 1. Identify the error in this attempt.
   2. Give a correct version of the proof. [6]
2. Prove that \(\ln [-t + \sqrt{(1 + t^2)}] = -\ln [t + \sqrt{(1 + t^2)}]\). [3]
3. Deduce that \(C\) is symmetric about the \(x\)-axis and sketch the graph of \(C\). [3]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]

The function f is defined by \(f(x) = e^x + 1\) for \(x \in \mathbb{R}\) Find an expression for \(f^{-1}(x)\) Tick \((\checkmark)\) one box. [1 mark] \(f^{-1}(x) = \ln(x - 1)\) \(\square\) \(f^{-1}(x) = \ln(x) - 1\) \(\square\) \(f^{-1}(x) = \frac{1}{e^x + 1}\) \(\square\) \(f^{-1}(x) = \frac{x - 1}{e}\) \(\square\)

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]

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

1. Determine the value of \(x\) when \(y = 10\). [2]
2. Determine the gradient of the tangent to the curve at the point where \(x = 2\). [2]

Write down the inverse function of \(y = e^x\). On the same set of axes, sketch the graphs of \(y = e^x\) and its inverse function, clearly labelling the coordinates of the points where the graphs cross the \(x\) and \(y\) axes. [3]

Solve the following equations for values of \(x\).

1. \(\ln(2x + 5) = 3\) [2]
2. \(5^{2x+1} = 14\) [3]
3. \(3\log_7(2x) - \log_7(8x^2) + \log_7 x = \log_3 81\) [6]

A function \(f\) is defined by \(f(x) = \log_{10}(2 - x)\). Another function \(g\) is defined by \(g(x) = \log_{10}(5 - x)\). The diagram below shows a sketch of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_17}

1. The point \((c, 1)\) lies on \(y = f(x)\). Find the value of \(c\). [2]
2. A point P lies on \(y = f(x)\) and has \(x\)-coordinate \(\alpha\). Another point Q lies on \(y = g(x)\) and also has \(x\)-coordinate \(\alpha\). The distance between P and Q is 1.2 units. Find the value of \(\alpha\), giving your answer correct to three decimal places. [5]

In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]

The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$

1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
2. Find the range of the function fg. [2]
3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]