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

A zoologist is investigating the growth of a population of red squirrels in a forest. She uses the equation \(N = \frac{200}{1 + 9e^{-\frac{t}{5}}}\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation? Circle your answer. [1 mark] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)

At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally. The velocity, \(\mathbf{v}\) m s\(^{-1}\), of the parachutist at time \(t\) seconds is given by: $$\mathbf{v} = (40e^{-0.2t})\mathbf{i} + 50(e^{-0.2t} - 1)\mathbf{j}$$ The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that the parachutist is at the origin when \(t = 0\) Model the parachutist as a particle.

1. Find an expression for the position vector of the parachutist at time \(t\). [4 marks]
2. The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. [4 marks]
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model. [3 marks]

A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75°C\) cools so that the temperature, \(\theta °C\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5(4 + \lambda e^{-kt})$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68°C\).

1. Find the temperature of the liquid after 15 minutes. Give your answer to three significant figures. [7 marks]
2. 1. Find the room temperature of the laboratory, giving a reason for your answer. [2 marks]
   2. Find the time taken in minutes for the liquid to cool to \(1°C\) above the room temperature of the laboratory. [2 marks]
3. Explain why the model might need to be changed if the experiment was conducted in a different place. [1 mark]

A curve has equation \(y = 3e^{2x}\) Find the gradient of the curve at the point where \(y = 10\) [3 marks]

One of the graphs shown below **cannot** have an equation of the form $$y = a^x \quad \text{where } a > 0$$ Identify this graph. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}

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]

In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).

1. Write down the values of \(A\) and \(k\). [2]
2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
3. What does the model suggest about the number of fruit flies in the long run? [1]

Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$

1. Identify three ways in which this second model is consistent with the known data. [3]
2. 1. Identify one feature which is not accounted for by the second model. [1]
   2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]

Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] \(\lim_{x \to \infty} (x^2e^{-x})\) \quad \(\lim_{x \to 0} (x^5 \ln x)\) \quad \(\lim_{x \to \infty} \left(\frac{e^x}{x^5}\right)\) \quad \(\lim_{x \to 0^+} (x^3e^x)\)

\includegraphics{figure_6} **In this question you must show all stages of your working.** **Solutions relying on calculator technology are not acceptable.** Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2^{2x}$$ The point \(P\left(a, 96\sqrt{2}\right)\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3 \times 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is $$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]

Sketch the curve \(y = 2^{2x+3}\), stating the coordinates of any points of intersection with the axes. [2] The point \(P\) on the curve \(y = 3^{3x+2}\) has \(y\)-coordinate equal to 180. Use logarithms to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. [2] The curves \(y = 2^{2x+3}\) and \(y = 3^{3x+2}\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as $$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$ [3]

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

1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]

\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, \(N\), in the first population is modelled by the equation $$N = Ae^{kt} \quad t \geq 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that • there were 1000 bacteria in this population at the start of the study • it took exactly 5 hours from the start of the study for this population to double

1. find a complete equation for the model. [4]
2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. [2]

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500e^{1.4t} \quad t \geq 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study. Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

1. find the value of \(T\). [3]

A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N\) ppm (parts per million), is modelled by the equation $$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$ where \(t\) hours is the time after the treatment was applied. Use the equation of the model to answer parts (a) and (b).

1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3] For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. [3]

$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$ $$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$

1. Find and simplify an expression for the composite function \(gf(x)\). [2]
2. State the domain and range of \(gf(x)\). [2]
3. Solve the equation $$gf(x) = 27.$$ [3]

The equation \(gf(x) = k\), where \(k\) is a constant, has solutions.

1. State the range of the possible values of \(k\). [1]

In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
2. Find the time when the mass is 30 grams. [3]
3. Sketch the graph of \(m\) against \(t\). [2]

The number of members of a social networking site is modelled by \(m = 150e^{2t}\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.

1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\). [2]
2. What is the significance of the integer 150 in the model? [1]
3. Find the week in which the model predicts that the number of members first exceeds 60 000. [3]
4. The social networking site only expects to attract 60 000 members. Suggest how the model could be refined to take account of this. [1]

Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3e^{-0.02t}\) units and the concentration of Coldcure is \(5e^{-0.07t}\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

1. Show that Coldcure becomes ineffective before Antiflu. [3]
2. Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]