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

5. Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
3. state the range of f ,
4. find \(\mathrm { f } ^ { - 1 } ( x )\),
5. write down the domain of \(\mathrm { f } ^ { - 1 }\).

7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
2. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
3. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with the equation \(y = \left( 2 x ^ { 2 } - 5 x + 2 \right) \mathrm { e } ^ { - x }\).

1. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
2. Show that \(C\) crosses the \(x\)-axis at \(x = 2\) and find the \(x\)-coordinate of the other point where \(C\) crosses the \(x\)-axis.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Hence find the exact coordinates of the turning points of \(C\).

4. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , x \geqslant - 1$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Find the domain of \(\mathrm { f } ^ { - 1 }\). The function \(g\) is defined by $$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R }$$
3. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
4. Find the range of fg.

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \ln x , \quad x > 0 \end{aligned}$$

1. State the range of f.
2. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
4. Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .

1. Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
3. Show that the \(x\) coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
4. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\) coordinate of \(P\) is \(\alpha\).
5. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.

8. \begin{figure}[h] \end{figure} The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
The graph of \(P\) against \(t\) is shown in Figure 3. Use the given equation to

1. find the population at the start of the study,
2. find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,
3. calculate the value of \(k\) to 3 decimal places. Using this value for \(k\),
4. find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
5. Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study.

1. (a) On the same diagram, sketch and clearly label the graphs with equations

$$y = \mathrm { e } ^ { x } \quad \text { and } \quad y = 10 - x$$ Show on your sketch the coordinates of each point at which the graphs cut the axes.
(b) Explain why the equation \(\mathrm { e } ^ { x } - 10 + x = 0\) has only one solution.
(c) Show that the solution of the equation $$\mathrm { e } ^ { x } - 10 + x = 0$$ lies between \(x = 2\) and \(x = 3\) (d) Use the iterative formula $$x _ { n + 1 } = \ln \left( 10 - x _ { n } \right) , \quad x _ { 1 } = 2$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
Give your answers to 4 decimal places.

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left\{ \begin{array} { r r } 5 - 2 x , & x \leqslant 4 \\ \mathrm { e } ^ { 2 x - 8 } - 4 , & x > 4 \end{array} \right.$$

1. State the range of \(\mathrm { f } ( x )\).
2. Determine the exact value of ff(0).
3. Solve \(\mathrm { f } ( x ) = 21\) Give each answer as an exact answer.
4. Explain why the function f does not have an inverse.

2. A curve \(C\) has equation \(y = \mathrm { e } ^ { 4 x } + x ^ { 4 } + 8 x + 5\)

1. Show that the \(x\) coordinate of any turning point of \(C\) satisfies the equation $$x ^ { 3 } = - 2 - \mathrm { e } ^ { 4 x }$$
2. On the axes given on page 5, sketch, on a single diagram, the curves with equations
   1. \(y = x ^ { 3 }\),
   2. \(y = - 2 - e ^ { 4 x }\) On your diagram give the coordinates of the points where each curve crosses the \(y\)-axis and state the equation of any asymptotes.
3. Explain how your diagram illustrates that the equation \(x ^ { 3 } = - 2 - e ^ { 4 x }\) has only one root. The iteration formula $$x _ { n + 1 } = \left( - 2 - \mathrm { e } ^ { 4 x _ { n } } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = - 1$$ can be used to find an approximate value for this root.
4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 5 decimal places.
5. Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve \(C\). \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-04_1285_1294_308_331}

6. The function f is defined by $$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in \mathbb { R } , \quad k \text { is a positive constant. }$$

1. State the range of f .
2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain. The function g is defined by $$g : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
3. Solve the equation $$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$ giving your answer in its simplest form.
4. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
5. Find, in terms of the constant \(k\), the solution of the equation $$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$

8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$

1. Calculate the number of primroses at the start of the study.
2. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
3. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
4. Explain why the population of primroses can never be 270

2. Given that $$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$

1. sketch, on separate diagrams, the curve with equation
   1. \(y = \mathrm { f } ( x )\)
   2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
2. Deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\)
3. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 2\)

1. Water is being heated in an electric kettle. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water \(t\) seconds after the kettle is switched on, is modelled by the equation

$$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t } , \quad 0 \leqslant t \leqslant T$$

1. State the value of \(\theta\) when \(t = 0\) Given that the temperature of the water in the kettle is \(70 ^ { \circ } \mathrm { C }\) when \(t = 40\),
2. find the exact value of \(\lambda\), giving your answer in the form \(\frac { \ln a } { b }\), where \(a\) and \(b\) are integers. When \(t = T\), the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\) and the kettle switches off.
3. Calculate the value of \(T\) to the nearest whole number.

6. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing part of the curve with equation \(y = 2 ^ { x + 1 } - 3\) and part of the line with equation \(y = 17 - x\). The curve and the line intersect at the point \(A\).

1. Show that the \(x\) coordinate of \(A\) satisfies the equation $$x = \frac { \ln ( 20 - x ) } { \ln 2 } - 1$$
2. Use the iterative formula $$x _ { n + 1 } = \frac { \ln \left( 20 - x _ { n } \right) } { \ln 2 } - 1 , \quad x _ { 0 } = 3$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
3. Use your answer to part (b) to deduce the coordinates of the point \(A\), giving your answers to one decimal place.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = g ( x )\), where $$\mathrm { g } ( x ) = \left| 4 \mathrm { e } ^ { 2 x } - 25 \right| , \quad x \in \mathbb { R }$$ The curve cuts the \(y\)-axis at the point \(A\) and meets the \(x\)-axis at the point \(B\). The curve has an asymptote \(y = k\), where \(k\) is a constant, as shown in Figure 1

1. Find, giving each answer in its simplest form,
   1. the \(y\) coordinate of the point \(A\),
   2. the exact \(x\) coordinate of the point \(B\),
   3. the value of the constant \(k\). The equation \(\mathrm { g } ( x ) = 2 x + 43\) has a positive root at \(x = \alpha\)
2. Show that \(\alpha\) is a solution of \(x = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x + 17 \right)\) The iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 1 } { 2 } x _ { n } + 17 \right)$$ can be used to find an approximation for \(\alpha\)
3. Taking \(x _ { 0 } = 1.4\) find the values of \(x _ { 1 }\) and \(x _ { 2 }\) Give each answer to 4 decimal places.
4. By choosing a suitable interval, show that \(\alpha = 1.437\) to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-07_2258_47_315_37}

9. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.

1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
   

   VIIIV SIHI NITIIIM I I N O C

   VI4V SIHI NI IHIHM ION OC

   VI4V SIHI NI JIIIM ION OO

   \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-16_2258_47_315_37}

8. \begin{figure}[h] \end{figure} The number of rabbits on an island is modelled by the equation $$P = \frac { 100 \mathrm { e } ^ { - 0.1 t } } { 1 + 3 \mathrm { e } ^ { - 0.9 t } } + 40 , \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of rabbits, \(t\) years after they were introduced onto the island.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 3.

1. Calculate the number of rabbits that were introduced onto the island.
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The number of rabbits initially increases, reaching a maximum value \(P _ { T }\) when \(t = T\)
3. Using your answer from part (b), calculate
   1. the value of \(T\) to 2 decimal places,
   2. the value of \(P _ { T }\) to the nearest integer.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) For \(t > T\), the number of rabbits decreases, as shown in Figure 3, but never falls below \(k\), where \(k\) is a positive constant.
4. Use the model to state the maximum value of \(k\).

1. The number of bacteria, \(N\), present in a liquid culture at time \(t\) hours after the start of a scientific study is modelled by the equation

$$N = 5000 ( 1.04 ) ^ { t } , \quad t \geqslant 0$$ where \(N\) is a continuous function of \(t\).

1. Find the number of bacteria present at the start of the scientific study.
2. Find the percentage increase in the number of bacteria present from \(t = 0\) to \(t = 2\) Given that \(N = 15000\) when \(t = T\),
3. find the value of \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) when \(t = T\), giving your answer to 3 significant figures.

2. The curve \(C\) has equation $$3 ^ { x - 1 } + x y - y ^ { 2 } + 5 = 0$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(( 1,3 )\) on the curve \(C\) can be written in the form \(\frac { 1 } { \lambda } \ln \left( \mu \mathrm { e } ^ { 3 } \right)\), where \(\lambda\) and \(\mu\) are integers to be found.

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation $$y = 10 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad 0 \leqslant x \leqslant 10$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the volume, \(V\), of this solid is given by $$V = k \int _ { 0 } ^ { 10 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ where \(k\) is a constant to be found.
2. Find \(\int x ^ { 2 } e ^ { - x } d x\) Figure 3 represents an exercise weight formed by joining two of these solids together.
   The exercise weight has mass 5 kg and is 20 cm long.
   Given that $$\text { density } = \frac { \text { mass } } { \text { volume } }$$ and using your answers to part (a) and part (b),
3. find the density of this exercise weight. Give your answer in grams per \(\mathrm { cm } ^ { 3 }\) to 3 significant figures.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$2 ^ { x } - 4 x y + y ^ { 2 } = 13 \quad y \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 2

1. Find the \(y\) coordinate of \(P\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(\frac { a \ln 2 + b } { c \ln 2 + d }\) where \(a , b , c\) and \(d\) are integers to be found.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a car engine, \(t\) minutes after the engine is turned off, is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 15 ) ^ { 2 }$$ where \(k\) is a constant.
Given that the temperature of the car engine

• is \(85 ^ { \circ } \mathrm { C }\) at the instant the engine is turned off
• is \(40 ^ { \circ } \mathrm { C }\) exactly 10 minutes after the engine is turned off
  1. solve the differential equation to show that, according to the model

$$\theta = \frac { a t + b } { c t + d }$$ where \(a , b , c\) and \(d\) are integers to be found.

Hence find, according to the model, the time taken for the temperature of the car engine to reach \(20 ^ { \circ } \mathrm { C }\). Give your answer to the nearest minute.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { 0.5 x } - 2\) The region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
Show that the volume of this solid can be written in the form \(a \ln 2 + b\), where \(a\) and \(b\) are constants to be found.

3 The profit \(\pounds P\) made by a company in its \(n\)th year is modelled by the exponential function $$P = A \mathrm { e } ^ { b n }$$ In the first year (when \(n = 1\) ), the profit was \(\pounds 10000\). In the second year, the profit was \(\pounds 16000\).

1. Show that \(\mathrm { e } ^ { b } = 1.6\), and find \(b\) and \(A\).
2. What does this model predict the profit to be in the 20th year?