Exponential Functions

4 The graph of \(y = a b ^ { x }\) passes through the points \(( 1,6 )\) and \(( 2,3.6 )\). Find the values of \(a\) and \(b\).

1. The graph \(G\) shows the relationship between the variables \(y\) and \(x\), where \(y \propto x\). Sketch the graph \(G\). [1]
2. Mary and Jeff work for a company which pays its employees by hourly rates. Mary's hourly rate is twice Jeff's hourly rate. On a certain day, Jeff worked three times as long as Mary and was paid £120. Calculate Mary's earnings on that day. [3]
3. Atmospheric pressure, \(P\) units, decreases as the height, \(H\) metres, above sea level increases. The rate of decrease is 12% for every 1000m. At sea level, the pressure \(P\) is 1013 units. Write down the model for \(P\) in terms of \(H\) and find the pressure at the top of Mount Everest, which is 8848m above sea level. [3]

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1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$
   1. Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   2. Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
   3. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.

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

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

1. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, \(T ^ { \circ } \mathrm { C }\), \(t\) minutes after it enters the liquid, is given by

$$T = 400 \mathrm { e } ^ { - 0.05 t } + 25 , \quad t \geqslant 0$$

1. Find the temperature of the ball as it enters the liquid.
2. Find the value of \(t\) for which \(T = 300\), giving your answer to 3 significant figures.
3. Find the rate at which the temperature of the ball is decreasing at the instant when \(t = 50\). Give your answer in \({ } ^ { \circ } \mathrm { C }\) per minute to 3 significant figures.
4. From the equation for temperature \(T\) in terms of \(t\), given above, explain why the temperature of the ball can never fall to \(20 ^ { \circ } \mathrm { C }\).

8. \begin{figure}[h] \end{figure} The value of Lin's car is modelled by the formula $$V = 18000 \mathrm { e } ^ { - 0.2 t } + 4000 \mathrm { e } ^ { - 0.1 t } + 1000 , \quad t \geqslant 0$$ where the value of the car is \(V\) pounds when the age of the car is \(t\) years.
A sketch of \(t\) against \(V\) is shown in Figure 1.

1. State the range of \(V\). According to this model,
2. find the rate at which the value of the car is decreasing when \(t = 10\) Give your answer in pounds per year.
3. Calculate the exact value of \(t\) when \(V = 15000\)

Two quantities are related by the equation \(Q = 1.25P^3\). Explain why the graph of \(\log_{10} Q\) against \(\log_{10} P\) is a straight line. State the gradient of the straight line and the intercept on the \(\log_{10} Q\) axis of the graph. [4]

1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation

$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$

1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
2. With reference to the model, interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find, according to the model,
   1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
   2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
4. State one reason why this may not be a realistic model in the long term.

3 \includegraphics[max width=\textwidth, alt={}, center]{733c3711-0429-415d-a8f3-8de86097635a-2_550_843_769_651} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { - x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0,1.3\) ) and ( \(1.6,0.9\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.

1. Sketch the curve \(y = 3^{-x}\) [2]
2. Solve the inequality \(3^{-x} < 27\) [2]

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]

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1. 1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
      1. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
   2. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
   3. 1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\) 14
   4. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\) Find the exact value of the total area of the eight rectangles.
      Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
      [0pt] [3 marks]
      14
   5. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)

1. On a single set of axes, sketch the curves \(y = e^x - 1\) and \(y = 2e^{-x}\). [3]
2. Find the exact coordinates of the point of intersection of these curves. [5]

8 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).

1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
   (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
   (c) State a possible value for \(a\) and a possible value for \(b\).
2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$

9

1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).

1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).

1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

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] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)

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] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)

4. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by \(P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }\), where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of \(a\),
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
3. With reference to this model, give a reason why the population of deer cannot exceed 2000.

2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t } .$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).

2. The current, \(I\) amps, in an electric circuit at time \(t\) seconds is given by $$I = 16 - 16 ( 0.5 ) ^ { t } , \quad t \geqslant 0$$ Use differentiation to find the value of \(\frac { \mathrm { d } I } { \mathrm {~d} t }\) when \(t = 3\).
Give your answer in the form \(\ln a\), where \(a\) is a constant.

5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where \(k\) and \(p\) are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

1. Write down the value of \(p\).
2. Show that \(k = \frac { 1 } { 4 } \ln 3\).
3. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).

5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$

1. Find the time taken for the substance to halve its mass.
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.

Robert wants to deposit \(£P\) into a savings account. He has a choice of two accounts. • Account \(A\) offers an annual compound interest rate of \(1\%\). • Account \(B\) offers an interest rate of \(5\%\) for the first year and an annual compound interest rate of \(0.6\%\) for each subsequent year. After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\). [5]

11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).

1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
2. State the rate of increase in calls according to model 1.
3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .

9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.

1. Find a model to represent the annual sales, defining any variables used.
2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.

1 Solve the equation \(\frac { 3 ^ { x } + 2 } { 3 ^ { x } - 2 } = 8\), giving your answer correct to 3 decimal places.

1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).

4 The curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether the stationary point is a maximum or a minimum point.

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.

12. The table shows the average weekly pay of a footballer at a certain club on 1 August 1990 and 1 August 2010. The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where \(\pounds P\) is the average weekly pay \(t\) years after 1 August 1990, and \(A\) and \(k\) are constants.
a. i. Write down the value of \(A\).
ii. Show that the value of \(k\) is 1.16159 , correct to five decimal places.
b. With reference to the model, interpret
i. the value of the constant \(A\),
ii. the value of the constant \(k\), Using the model,
c. find the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed \(\pounds 100000\).

1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.