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

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at A. [3]
2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).

1. Find the exact value of the gradient of the curve at \(P\). [4]
2. Find the exact coordinates of \(M\). [4]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]

\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\). [4]
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]

A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k\lambda^t$$ where

• \(N\) grams is the amount of carbon-14 currently present in the item
• \(k\) grams was the initial amount of carbon-14 present in the item
• \(t\) is the number of years since the item was made
• \(\lambda\) is a constant, with \(0 < \lambda < 1\)

1. Sketch the graph of \(N\) against \(t\) for \(k = 1\) [2]

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

1. show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. [2]

Given that Item A

• is known to have had 15 grams of carbon-14 present initially
• is thought to be 3250 years old

1. calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item A. [2]

Item B is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

1. Use algebra to calculate the age of Item B to the nearest 100 years. [3]

The speed, \(v\) m s⁻¹, of a train at time \(t\) seconds is given by \(v = \sqrt{(1.2^t - 1)}, \quad 0 \leq t \leq 30.\) The following table shows the speed of the train at 5 second intervals.

1. Complete the table, giving the values of \(v\) to 2 decimal places. [3]

The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int_0^{30} \sqrt{(1.2^t - 1)} \, dt.$$

1. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\). [3]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P(\ln 2, p + 2q)\) on \(C\), the gradient is \(5\).

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately \(53.8\). [5]

\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 graphs 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 \(y = f(x), x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = f^{-1}(x)\), [2]
2. \(y = 3f(2x)\). [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 $$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

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 $$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

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

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 \(f(x) = e^{-x} + \sqrt{x} - 2, x \geq 0\),

1. explain how your graphs show that the equation \(f(x) = 0\) has only one solution, [1]
2. 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]

28a.

1. Given that \(\cos(x + 30)° = 3 \cos(x - 30)°\), prove that \(\tan x° = -\frac{\sqrt{3}}{2}\). [5]
2. 1. Prove that \(\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta\). [3]
   2. Verify that \(\theta = 180°\) is a solution of the equation \(\sin 2\theta = 2 - 2 \cos 2\theta\). [1]
   3. Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360°\), of the equation using \(\sin 2\theta = 2 - 2 \cos 2\theta\). [4]

The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).

1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]