Chain rule

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The value of a car, £V, is modelled by the equation

where A and k are positive constants and t is the age of the car in years. Given that

• the initial value of the car was £20 000

• the value of the car was £12 000 when it was 2.5 years old

find a complete equation for the model, giving the exact value of A and the value of k to 3 significant figures.

Show that the rate of change in the value of the car can be expressed in the form – k (V – 1500)

State a limitation of this model.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve with equation y = f (x), where

The curve

• intersects the x ‑axis at –1 and 1

• has minimum turning points at P and Q

as shown in Figure 5.

Use calculus to find the exact coordinates of P.

Using the substitution x = tanθ show that

where α and β are constants to be found. The finite region R, shown shaded in Figure 5, is bounded by the x‑axis and the curve.

Use algebraic integration to find the area of R.

Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where

Find

f′(x)

g′(x)

Given that f′(x) = g′(x) at x = α

show that α satisfies the equation

The iterative formula

is used withx₁ = 0.6 to find an approximate value for α

Calculate, giving each answer to 4 decimal places,

the value of x₂

the value of α

A curve has equation y = f(x), where

Show that

where A and B are constants to be found.

Hence show that the x coordinates of the turning points of the curve are solutions of the equation

The equation x has two positive roots α and β where β α

A student uses the iteration formula

in an attempt to find approximations for α and β Diagram 1 shows a plot of part of the curve with equation y and part of the line with equation y = x

Using Diagram 1 on page 42

draw a staircase diagram to show that the iteration formula starting withx₁ = 1 can be used to find an approximation for β

Use the iteration formula with x₁ = 1, to find, to 3 decimal places,

the value of x₂

the value of β

Using a suitable interval and a suitable function that should be stated

show thatα = 0.432 to 3 decimal places.

Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

Diagram 1
copy of Diagram 1

The function f is defined by

where k is a positive constant.

Show that

where g(x) is a function to be found. Given that the curve with equation y = f(x) has at least one stationary point,

find the range of possible values of k.

Figure 2 shows a sketch of the curve C with equation y = f (x) where

Show that f ʹ(x) = 8(2 + x − x²)e^−2x

Hence find, in simplest form, the exact coordinates of the stationary points of C. The function g and the function h are defined by g (x) = 2f (x) x h (x) = 2f (x) − 3 x 0

(c) Find (i) the range of g

the range of h

f(x) = 10e^–0.25x sin x, x 0

Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4

Figure 3 shows a sketch of part of the curve with equation y = f(x).

Sketch the graph of H against t where

showing the long-term behaviour of this curve.

The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find

the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

The curve C, in the standard Cartesian plane, is defined by the equation

The curve C passes through the origin O

Find the value of at the origin.

Use the small angle approximation for sin 2 y to find an equation linking x and y for points close to the origin.

Explain the relationship between the answers to (a) and (b)(i).

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

Find the number of mice in the population at the start of the study.

The rate of growth is a maximum after T months.

Find, according to the model, the value of T.

According to the model, the maximum number of mice on the island is P.

State the value of P.

The point P lies on the curve with equation

Given that P has (x, y) coordinates p, , where is a constant,

find the exact value of p.

The tangent to the curve at P cuts the y- axis at the point A.

Use calculus to find the coordinates of A.

Figure 2 shows a sketch of part of the curve with equation

Show that g' (x) = f(x)e^–2x, where f(x) is a cubic function to be found.

Hence find the range of g.

State a reason why the function g^–1(x) does not exist.

The curve C has equation x = 8 y tan 2 y

Verify that P lies on C.

Find the equation of the tangent to C at P in the form ay = x + b, where the constants a and b are to be found in terms of .

Given that

show that

Given that y = (x² + x³)ln 2x find the exact value of at x = 2, giving your answer in its simplest form.

Given that

show that

where g(x) is an expression to be found.

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

Given that

Hence show that

Find an expression for in terms of x. Give your answer in its simplest form.

Differentiate

Show that (sec? 3x) can be written in the form

where yu is a constant. (3)

The curve cuts the x-axis at points A and B as shown in Figure 2.

Calculate the x coordinate of A and the x coordinate of B, giving your answers to 3 decimal places. (2)

Find f/(x). (3) The curve has a minimum turning point at the point P as shown in Figure 2.

Show that the x coordinate of P is the solution of

Use the iteration formula

to calculate the values of x), x. and x3, giving your answers to 3 decimal places. (3)

The x coordinate of P is a.

The population of a town is being studied. The population P, at time ¢ years from the start of the study, is assumed to be

wherek is a positive constant. The graph of P against is shown in Figure 3. Use the given equation to

find the population at the start of the study, (2)

find a value for the expected upper limit of the population. (1)

Given also that the population reaches 2500 at 3 years from the start of the study,

calculate the value of k to 3 decimal places. (5) Using this value for k,

find the population at 10 years from the start of the study, giving your to 3 significant figures. (2)

Find, using the rate at which the population is growing at 10 years from the start of the study. (3)

The curve C has equation

The point P lies on C and has coordinates (w, – 32). Find

the value of w,

the equation of the tangent to C at the point P in the form y = mx c, where m and c are constants.

(i) Differentiate with respect to x

Figure 1 shows a sketch of the curve C which has equation

Find the x coordinate of the turning point P on C, for which x 0 Give your answer as a multiple of .

Find an equation of the normal to C at the point where x 0

(a) Differentiate with respect to x,

, giving your answer in its simplest form.

Given that x = 3tan 2y find in terms of x.

Differentiate with respect to x

Show that

The curve C has equation y = f (x). The point P −1, − lies on C.

Find an equation of the normal to C at P.

Express 2cos 3x – 3sin 3x in the form R cos (3x + ), where R and are constants, R 0 and 0 α . Give your answers to 3 significant figures.

Show that f (x) can be written in the form

where R and are the constants found in part (a).

Hence, or otherwise, find the smallest positive value of x for which the curve with equation y = f (x) has a turning point.

A curve C has equation

The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers.

Figure 1 shows a sketch of the curve C with the equation y (2x² 5x 2)e^x.

Find the coordinates of the point where C crosses the y-axis.

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.

Hence find the exact coordinates of the turning points of C.

Given that x = tan y, show that =

By writing sec x as , show that sec x tan x.

Given that y = e^2x sec 3x,

The curve with equation y = e^2x sec 3x, − x , has a minimum turning point at (a, b).

Find the values of the constants a and b, giving your answers to 3 significant figures.

Differentiate with respect to x

A curve C has the equation

The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form ax by c = 0, where a, b and c are integers.

Find the value of at the point where x = 2 on the curve with equation

Find the equation of the tangent to the curve x cos(2y ) at 0 . Give your answer in the form y = ax + b, where a and b are constants to be found.

The functions f and g are defined by

Write down the range of g.

Show that the composite function fg is defined by

Write down the range of fg.