1.06a Exponential function: a^x and e^x graphs and properties

1. Sketch, on the same set of axes, the graphs of $$y = 2 - e^{-x} \text{ and } y = \sqrt{x}.$$ [3] [It is not necessary to find the coordinates of any points of intersection with the axes.] Given that f(x) = \(e^{-x} + \sqrt{x} - 2\), \(x \geq 0\),
2. explain how your graphs show that the equation f(x) = 0 has only one solution, [1]
3. show that the solution of f(x) = 0 lies between \(x = 3\) and \(x = 4\). [2]

The iterative formula \(x_{n+1} = (2 - e^{-x_n})^2\) is used to solve the equation f(x) = 0.

1. Taking \(x_0 = 4\), write down the values of \(x_1\), \(x_2\), \(x_3\) and \(x_4\), and hence find an approximation to the solution of f(x) = 0, giving your answer to 3 decimal places. [4]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).

1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]

The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).

1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
2. Show that \(-1 < \alpha < 0\). [2]

The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).

1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = \text{f}^{-1}(x)\), [2]
2. \(y = 3\text{f}(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of f.
   [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function g is defined by $$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find fg(x), giving your answer in its simplest form. [3]

The temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).

Fig. 9 shows the line \(y = x\) and the curve \(y = f(x)\), where \(f(x) = \frac{1}{2}(e^x - 1)\). The line and the curve intersect at the origin and at the point P\((a, a)\). \includegraphics{figure_9}

1. Show that \(e^a = 1 + 2a\). [1]
2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac{1}{2}a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\). [6]
3. Show that the inverse function of f\((x)\) is g\((x)\), where g\((x) = \ln(1 + 2x)\). Add a sketch of \(y = g(x)\) to the copy of Fig. 9. [5]
4. Find the derivatives of f\((x)\) and g\((x)\). Hence verify that \(g'(a) = \frac{1}{f'(a)}\). Give a geometrical interpretation of this result. [7]

The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants.

1. Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of \(a\) and \(b\). [3]
2. Given also that the tree grows to a height of 6 metres in 8 years, find the value of \(k\), giving your answer correct to 2 decimal places. [3]

\includegraphics{figure_8} Fig. 8 shows the curve \(y = f(x)\), where \(f(x) = \frac{1}{e^x + e^{-x} + 2}\).

1. Show algebraically that \(f(x)\) is an even function, and state how this property relates to the curve \(y = f(x)\). [3]
2. Find \(f'(x)\). [3]
3. Show that \(f(x) = \frac{e^x}{(e^x + 1)^2}\). [2]
4. Hence, using the substitution \(u = e^x + 1\), or otherwise, find the exact area enclosed by the curve \(y = f(x)\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\). [5]
5. Show that there is only one point of intersection of the curves \(y = f(x)\) and \(y = \frac{1}{4}e^x\), and find its coordinates. [5]

The value \(£V\) of a car \(t\) years after it is new is modelled by the equation \(V = Ae^{-kt}\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.

1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 e^{-0.2t}.$$ Calculate how much value, to the nearest £100, this car has lost after 1 year. [2]
2. At the same time as Brian buys his car, Kate buys a new hatchback for £15000. Her car loses £2000 of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures. [3]
3. Find how long it is before Brian's and Kate's cars have the same value. [3]

Fig. 9 shows the curve \(y = xe^{-2x}\) together with the straight line \(y = mx\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P. The dashed line is the tangent at P. \includegraphics{figure_9}

1. Show that the \(x\)-coordinate of P is \(-\frac{1}{2}\ln m\). [3]
2. Find, in terms of \(m\), the gradient of the tangent to the curve at P. [4]

You are given that OP and this tangent are equally inclined to the \(x\)-axis.

1. Show that \(m = e^{-2}\), and find the exact coordinates of P. [4]
2. Find the exact area of the shaded region between the line OP and the curve. [7]

Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = e^{2x} + k e^{-2x}\) and \(k\) is a constant greater than 1. The curve crosses the \(y\)-axis at P and has a turning point Q. \includegraphics{figure_9}

1. Find the \(y\)-coordinate of P in terms of \(k\). [1]
2. Show that the \(x\)-coordinate of Q is \(\frac{1}{4}\ln k\), and find the \(y\)-coordinate in its simplest form. [5]
3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\ln k\). Give your answer in the form \(ak + b\). [4]

The function \(g(x)\) is defined by \(g(x) = f(x + \frac{1}{4}\ln k)\).

1. 1. Show that \(g(x) = \sqrt{k}(e^{2x} + e^{-2x})\). [3]
   2. Hence show that \(g(x)\) is an even function. [2]
   3. Deduce, with reasons, a geometrical property of the curve \(y = f(x)\). [3]

The function f is defined by $$f : x \to 3e^{x-1}, \quad x \in \mathbb{R}.$$

1. State the range of f. [1]
2. Find an expression for \(f^{-1}(x)\) and state its domain. [4]

The function g is defined by $$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$ Find, in terms of e,

1. the value of gf(ln 2), [3]
2. the solution of the equation $$f^{-1}g(x) = 4.$$ [4]

$$\text{f}(x) = e^{3x + 1} - 2, \quad x \in \mathbb{R}.$$

1. State the range of f. [1]

The curve \(y = \text{f}(x)\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).

1. Find the exact coordinates of \(P\) and \(Q\). [3]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 3ex + e - 2.$$ [4]
3. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\). [4]

Fig. 9 shows the curve \(y = f(x)\), where $$f(x) = (e^x - 2)^2 - 1, x \in \mathbb{R}.$$ The curve crosses the x-axis at O and P, and has a turning point at Q. \includegraphics{figure_9}

1. Find the exact x-coordinate of P. [2]
2. Show that the x-coordinate of Q is \(\ln 2\) and find its y-coordinate. [4]
3. Find the exact area of the region enclosed by the curve and the x-axis. [5]

The domain of f(x) is now restricted to \(x \geqslant \ln 2\).

1. Find the inverse function \(f^{-1}(x)\). Write down its domain and range, and sketch its graph on the copy of Fig. 9. [7]

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]

The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where \(m_0\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.

1. 1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004, correct to six decimal places. [1 mark]
   2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
2. The half-life of a radioactive substance is the time it takes for a mass of \(m_0\) to reduce to a mass of \(\frac{1}{2}m_0\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]

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]

\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.

1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).

1. Show that \(n = 14\). [1]

The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.

1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]

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]

A curve has the equation \(y = e^{-2x}\) At point \(P\) on the curve the tangent is parallel to the line \(x + 8y = 5\) Find the coordinates of \(P\) stating your answer in the form \((\ln p, q)\), where \(p\) and \(q\) are rational. [7 marks]

Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.

1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
4. Comment on the long-term accuracy of the model. [1 mark]

Which one of these functions is decreasing for all real values of \(x\)? Circle your answer. \(f(x) = e^x\) \quad \(f(x) = -e^{1-x}\) \quad \(f(x) = -e^{x-1}\) \quad \(f(x) = -e^{-x}\) [1 mark]