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

The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant 4\).

1. The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = 4\). Find, in terms of e, the coordinates of the centroid of the region \(R\).
2. Show that \(\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\ & C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta } \end{aligned}$$ The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4 \sqrt { } 2\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the initial line, giving your answer correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The random variable \(T\) is the lifetime, in hours, of a randomly chosen decorative light bulb of a particular type. It is given that \(T\) has a negative exponential distribution with mean 1000 hours.

1. Write down the probability density function of \(T\).
2. Find the probability that a randomly chosen bulb of this type has a lifetime of more than 2000 hours. A display uses 10 randomly chosen bulbs of this type, and they are all switched on simultaneously. Find the greatest value of \(t\) such that the probability that they are all alight at time \(t\) hours is at least 0.9 .

9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).

1. Explain why the path of the particle never crosses the \(x\)-axis.
2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
   The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
   1. using model A,
   2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
3. State what is suggested about the long-term food supply by
   1. model A,
   2. model B.

1. The value of a car, \(\pounds V\), can be modelled by the equation

$$V = 15700 \mathrm { e } ^ { - 0.25 t } + 2300 \quad t \in \mathbb { R } , t \geqslant 0$$ where the age of the car is \(t\) years.
Using the model,

1. find the initial value of the car. Given the model predicts that the value of the car is decreasing at a rate of \(\pounds 500\) per year at the instant when \(t = T\),
2. 1. show that $$3925 \mathrm { e } ^ { - 0.25 T } = 500$$
   2. Hence find the age of the car at this instant, giving your answer in years and months to the nearest month.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) The model predicts that the value of the car approaches, but does not fall below, \(\pounds A\).
3. State the value of \(A\).
4. State a limitation of this model.

1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a cup of tea \(t\) minutes after it was placed on a table in a room, is modelled by the equation

$$\theta = 18 + 65 \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ Find, according to the model,

1. the temperature of the cup of tea when it was placed on the table,
2. the value of \(t\), to one decimal place, when the temperature of the cup of tea was \(35 ^ { \circ } \mathrm { C }\).
3. Explain why, according to this model, the temperature of the cup of tea could not fall to \(15 ^ { \circ } \mathrm { C }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-16_675_951_973_573} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The temperature, \(\mu ^ { \circ } \mathrm { C }\), of a second cup of tea \(t\) minutes after it was placed on a table in a different room, is modelled by the equation $$\mu = A + B \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ where \(A\) and \(B\) are constants.
   Figure 2 shows a sketch of \(\mu\) against \(t\) with two data points that lie on the curve.
   The line \(l\), also shown on Figure 2, is the asymptote to the curve.
   Using the equation of this model and the information given in Figure 2
4. find an equation for the asymptote \(l\).

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation $$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)

1. state the value of \(k\). From the instant when the tyre developed the puncture,
2. find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\) Give your answer in minutes to one decimal place.
3. Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
   Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.

1. The height, \(h\) metres, of a plant, \(t\) years after it was first measured, is modelled by the equation

$$h = 2.3 - 1.7 \mathrm { e } ^ { - 0.2 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ Using the model,

1. find the height of the plant when it was first measured,
2. show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. According to the model, there is a limit to the height to which this plant can grow.
3. Deduce the value of this limit.

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.

2. The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 } \\ & C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the \(x\)-coordinate of the point where \(C _ { 1 }\) and \(C _ { 2 }\) intersect is \(\frac { - 2 } { 5 }\).

10. The figure 4 shows the curves \(\mathrm { f } ( x ) = A - B e ^ { - 0.5 x }\) and \(\mathrm { g } ( x ) = 26 + e ^ { 0.5 x }\) \begin{figure}[h] \end{figure} Given that \(\mathrm { f } ( x )\) passes through \(( 0,8 )\) and has an horizontal asymptote \(y = 48\) a. Find the values of \(A\) and \(B\) for \(\mathrm { f } ( x )\) (3)
b. State the range of \(\mathrm { g } ( x )\) (1) The curves \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) meet at the points \(C\) and \(D\) c. Find the \(x\)-coordinates of the intersection points \(C\) and \(D\), in the form \(\ln k\), where \(k\) is an integer.

8. \begin{figure}[h] \end{figure} The curves with equation \(y = 21 - 2 ^ { x }\) meet the curve with equation \(y = 2 ^ { 2 x + 1 }\) at the point \(A\) as shown in Figure 2. Find the exact coordinates of point \(A\).

9. A cup of tea is cooling down in a room. The temperature of tea, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) minutes after the tea is made, is modelled by the equation $$\theta = A + 70 e ^ { - 0.025 t }$$ where \(A\) is a positive constant.
Given that the initial temperature of the tea is \(85 ^ { \circ } \mathrm { C }\) a. find the value of \(A\).
b. Find the temperature of the tea 20 minutes after it is made.
c. Find how long it will take the tea to cool down to \(43 ^ { \circ } \mathrm { C }\).
(4)

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

1. The height above ground, \(H\) metres, of a passenger on a roller coaster can be modelled by the differential equation

$$\frac { \mathrm { d } H } { \mathrm {~d} t } = \frac { H \cos ( 0.25 t ) } { 40 }$$ where \(t\) is the time, in seconds, from the start of the ride. Given that the passenger is 5 m above the ground at the start of the ride,

1. show that \(H = 5 \mathrm { e } ^ { 0.1 \sin ( 0.25 t ) }\)
2. State the maximum height of the passenger above the ground. The passenger reaches the maximum height, for the second time, \(T\) seconds after the start of the ride.
3. Find the value of \(T\).

1. The value, \(\pounds V\), of a vintage car \(t\) years after it was first valued on 1 st January 2001, is modelled by the equation

$$V = A p ^ { t } \quad \text { where } A \text { and } p \text { are constants }$$ Given that the value of the car was \(\pounds 32000\) on 1st January 2005 and \(\pounds 50000\) on 1st January 2012

1. 1. find \(p\) to 4 decimal places,
   2. show that \(A\) is approximately 24800
2. With reference to the model, interpret
   1. the value of the constant \(A\),
   2. the value of the constant \(p\). Using the model,
3. find the year during which the value of the car first exceeds \(\pounds 100000\)

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
      Given that \(l\) is a tangent to \(C\),
2. find the possible values of \(k\), giving your answers as simplified surds.

5. A curve \(C\) has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve \(C\) can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where \(a\) and \(b\) are integers to be found.

1. A scientist is studying a population of mice on an island.

The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation $$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$

1. Find the number of mice in the population at the start of the study.
2. Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\) The rate of growth is a maximum after \(T\) months.
3. Find, according to the model, the value of \(T\). According to the model, the maximum number of mice on the island is \(P\).
4. State the value of \(P\).

1. (a) Sketch the curve with equation

$$y = 4 ^ { x }$$ stating any points of intersection with the coordinate axes.
(b) Solve $$4 ^ { x } = 100$$ giving your answer to 2 decimal places.

1. Coffee is poured into a cup.

The temperature of the coffee, \(H ^ { \circ } \mathrm { C } , t\) minutes after being poured into the cup is modelled by the equation $$H = A \mathrm { e } ^ { - B t } + 30$$ where \(A\) and \(B\) are constants.
Initially, the temperature of the coffee was \(85 ^ { \circ } \mathrm { C }\).

1. State the value of \(A\). Initially, the coffee was cooling at a rate of \(7.5 ^ { \circ } \mathrm { C }\) per minute.
2. Find a complete equation linking \(H\) and \(t\), giving the value of \(B\) to 3 decimal places.

1. A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that

• the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
• after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
  1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.

Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)

Use this information to evaluate the model.

6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula. \(P = 100 \mathrm { e } ^ { t }\)

1. Use this model to answer the following.
   1. Find the value of \(P\) when \(t = 4\).
   2. Find the value of \(t\) when the population is 9000 .
2. It is suspected that a more appropriate model would be the following formula. \(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
   1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line. Some observations of \(t\) and \(P\) gave the following results.

      \(t\)	1	2	3	4	5
      \(P\)	100	500	1800	7000	19000
      \(\log _ { 10 } P\)	2.00	2.70	3.26	3.85	4.28

   2. On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
   3. Hence estimate the values of \(k\) and \(a\).

6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.

1. Write down the value of \(a\).
2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
3. Calculate the value of \(k\).
4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.