Exponential functions

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A new type of car is released for sale. The total number of this type of car sold, N, in a particular region, t months after the cars were released for sale, is modelled by the equation

Use the equation of the model to answer parts (a), (b), (c) and (d).

Find the total number of cars sold in the first 3 months. Given that N = 3000 when t = T

find the value of T giving the answer to 2 decimal places.

Find the rate of increase in the total number of cars sold when t = 3, giving the answer to 3 significant figures. After a marketing campaign, the total number of cars sold is expected to rise and have an upper limit of 6500

Using this information, suggest one refinement to the model.

Sketch the curve with equation

stating any points of intersection with the coordinate axes.

Solve

giving your answer to 2 decimal places.

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

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

find a complete equation for the model.

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. The number of bacteria, M, in the second population is modelled by the equation

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,

find the value of T.

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

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

find a complete equation for the model.

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.

The number of bacteria, M, in the second population is modelled by the equation

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,

find the value of T.

Given that

sketch, on separate diagrams, the curve with equation

y = f(x)

y = |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.

Deduce the set of values of x for which f(x) = |f(x)|

Find the exact solutions of the equation |f(x)| = 2

Water is being heated in an electric kettle. The temperature, θ °C, of the water t seconds after the kettle is switched on, is modelled by the equation

State the value of θ when t = 0

Given that the temperature of the water in the kettle is 70°C when t = 40,

find the exact value of , giving your answer in the form , where a and b are integers.

When t = T, the temperature of the water reaches 100°C and the kettle switches off.

Calculate the value of T to the nearest whole number.

A curve C has equation y = e^4x + x⁴ + 8x + 5

Show that the x coordinate of any turning point of C satisfies the equation

On the axes given on page 5, sketch, on a single diagram, the curves with equations

y = –2 – e^4x On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes.

Explain how your diagram illustrates that the equation x³ = –2 – e^4x has only one root.

The iteration formula

can be used to find an approximate value for this root.

Calculate the values of x₁ and x₂, giving your answers to 5 decimal places.

Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

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

Calculate the number of primroses at the start of the study.

Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers.

Find the exact value of when t =10. Give your answer in its simplest form.

Explain why the population of primroses can never be 270

Find algebraically the exact solutions to the equations

ln(4 – 2 x) + ln(9 – 3 x) = 2ln(x + 1), –1 < x < 2

2^x e^3x+1 = 10 Give your answer to (b) in the form where a, b, c and d are integers.

The value of Bob’s car can be calculated from the formula

where V is the value of the car in pounds (£) and t is the age in years.

Find the value of the car when t 0

Calculate the exact value of t when V = 9500

Find the rate at which the value of the car is decreasing at the instant when t 8. Give your answer in pounds per year to the nearest pound.

The functions f and g are defined by

State the range of f.

Find fg(x), giving your answer in its simplest form.

Find the exact value of x for which f (2x 3) 6

Find f ¹, the inverse function of f, stating its domain.

On the same axes sketch the curves with equation y = f (x) and y = f ¹(x), giving the coordinates of all the points where the curves cross the axes.

The mass, m grams, of a leaf t days after it has been picked from a tree is given by

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.

Write down the value of p.

Find the value of t when 0 6. ln 3.

Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation

Write down the number of rabbits that were introduced to the island.

Find the number of years it would take for the number of rabbits to first exceed 1000.

The curve meets the coordinate axes at the points A(0,1–k) and B (–¹₂ k ln,0), where k is a constant and k 1, as shown in Figure 2. On separate diagrams, sketch the curve with equation

y = 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 f (x) = e^2x − k,

state the range of f,

write down the domain of f ¹.

y = 5^x + log₂(x + 1), 0 x ≤ 2 Complete the table below, by giving the value of y when x = 1

Use the trapezium rule, with all the values of y from the completed table, to find an approximate value for

giving your answer to 2 decimal places.

Use your answer to part (b) to find an approximate value for

giving your answer to 2 decimal places.

Sketch the graph of

showing the coordinates of any points at which the graph crosses the axes.

Use algebra to solve the equation

giving your answers to 2 decimal places where appropriate.