Stationary points

Figure 1 shows a sketch of a curve C with equationy = f(x), where f(x) is a quartic expression in x. The curve

• has maximum turning points at (–1, 0) and (5, 0)

• crosses the y ‑axis at (0, –75)

• has a minimum turning point at x = 2

Find the set of values of x for which

writing your answer in set notation.

Find the equation of C. You may leave your answer in factorised form. The curve C₁ has equation y = f(x) + k, where k is a constant. Given that the graph of C₁ intersects the x‑axis at exactly four places,

find the range of possible values for k.

A racing car is driven along a straight road. Figure 2 shows a graph of the speed of the car as it travels along the road. The car starts from rest and is driven for T seconds before stopping. The speed of the car is modelled by the equation

where t seconds is the time after the car starts to move. According to the model,

find the value of T, giving your answer to one decimal place,

show that the maximum speed of the car occurs when

Using the iteration formula

find the value of t₃ to 3 decimal places,

find, by repeated iteration, the time taken for the car to reach maximum speed.

Figure 4 shows the plan view for the design of a stage. The shape of this design consists of a sector of a circle AOB joined to a rectangle OBCD. Given that

• the radius of the sector is r metres and angle AOB is θ radians

• the length and width of the rectangle are r metres and metres respectively

• the total area of the stage is 240 m²

show that the perimeter of the stage, P metres, is given by

You must make your method clear. Using algebraic differentiation,

find the value of r for which P has a stationary value.

Prove, by further differentiation, that this value of r gives the minimum perimeter of the stage.

where a is a constant. Figure 2 shows a sketch of the curve C with equation y = f (x) Given that C has a local maximum at x = –

show that a = –4

find the exact y coordinate of the local maximum. The equation f (x) = k, where k is a constant, has 4 distinct solutions.

Using algebra and showing all stages of your working, find the range of values of k. Give the answer using set notation. (Solutions relying on calculator technology are not acceptable.)

The function f is defined by

Show that

where a, b and c are constants to be found.

Hence, using algebra, find the values of x for which f is decreasing. You must show each step in your working.

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

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation y = f(x) where x

Given that

• the curve has a stationary point with x coordinate α

• α is small

use the small angle approximation for cosx to estimate the value of α to 3 decimal places.

The point P(0, 3) lies on C

Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

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

Find f ′′(x)

Solve f ′′(x) = 0

Hence find the range of values of x for which f(x) is concave.

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

and x is measured in radians. The point P, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2,

find the x coordinate of P, giving your answer to 3 significant figures. The curve crosses the x-axis at x = α, as shown in Figure 2. Given that, to 3 decimal places, f(4) = 4.274 and f(5) = −1.212

explain why α must lie in the interval [4, 5]

Taking x₀ = 5 as a first approximation to α, apply the Newton-Raphson method once to f(x) to obtain a second approximation to α. Show your method and give your answer to 3 significant figures.

The curve with equation y = f (x) where

has a single turning point at x = α

Show that α is a solution of the equation

The iterative formula

is used to find an approximate value for α.

Starting with x₁ = 0.3

calculate, giving each answer to 4 decimal places,

the value of x₂

the value of x₄

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

show that α is 0.341 to 3 decimal places.

The curve C has equation

y = 5x⁴ – 24x³ + 42x² – 32x + 11 x

Find

Verify that C has a stationary point at x = 1

Show that this stationary point is a point of inflection, giving reasons for your answer.

The curve C has equation y = 5x⁴ – 24x³ + 42x² – 32x + 11 x

Find

Verify that C has a stationary point at x = 1

Show that this stationary point is a point of inflection, giving reasons for your answer.

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

The curve C has equation

Show that

Figure 1 shows a sketch of the curve C with equation

Show that

The point P, shown in Figure 1, is the minimum turning point on C.

Show that the x coordinate of P is a solution of

Use the iteration formula

to find (i) the value of x₂ to 5 decimal places,

the x coordinate of P to 5 decimal places.

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.

[A sphere of radius r has volume ³ and surface area 4²] A manufacturer produces a storage tank. The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness. The cylinder has radius r metres and height h metres and the hemisphere has radius r metres. The volume of the tank is 6 m³.

Show that, according to the model, the surface area of the tank, in m², is given by

The manufacturer needs to minimise the surface area of the tank.

Use calculus to find the radius of the tank for which the surface area is a minimum.

Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

A curve C has equation

Verify that C has a stationary point when x 4

Determine the nature of this stationary point, giving a reason for your answer.

Figure 4 shows a sketch of the curve with equation x² 2xy 3y² 50

The curve is used to model the shape of a cycle track with both x and y measured in km.

The points P and Q represent points that are furthest west and furthest east of the origin O, as shown in Figure 4.

Using part (a),

find the exact coordinates of the point P.

Explain briefly how to find the coordinates of the point that is furthest north of the origin O. (You do not need to carry out this calculation).

Find the values of the constants A, B and C.

Prove that f (x) is a decreasing function.

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.

Figure 3 shows a sketch of part of the curve with equation

The curve has a turning point at the point A, where x 0, as shown in Figure 3.

Using calculus, find the coordinates of the point A. The curve crosses the x-axis at the point B, as shown in Figure 3.

Use algebra to find the x coordinate of the point B. The finite region R, shown shaded in Figure 3, is bounded by the curve, the line through A parallel to the x-axis and the line through B parallel to the y-axis.

Use integration to find the area of the region R, giving your answer to 2 decimal places.

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

The curve has a turning point at the point A.

Using calculus, show that the x coordinate of A is 1

The curve crosses the x-axis at the points B (2, 0) and C 0 The finite region R, shown shaded in Figure 2, is bounded by the curve, the line AB, and the x-axis.

Use integration to find the area of the finite region R, giving your answer to 2 decimal places.

Figure 4 shows a plan view of a sheep enclosure. The enclosure ABCDEA, as shown in Figure 4, consists of a rectangle BCDE joined to an equilateral triangle BFA and a sector FEA of a circle with radius x metres and centre F. The points B, F and E lie on a straight line with FE = x metres and 10 x ≤ 25

Find, in m², the exact area of the sector FEA, giving your answer in terms of x, in its simplest form.

Given that BC = y metres, where y 0, and the area of the enclosure is 1000 m²,

show that

Hence show that the perimeter P metres of the enclosure is given by

Use calculus to find the minimum value of P, giving your answer to the nearest metre.

Justify, by further differentiation, that the value of P you have found is a minimum.

A B A 4x cm B

Figure 4 shows a closed letter box ABFEHGCD, which is made to be attached to a wall of a house. The letter box is a right prism of length y cm as shown in Figure 4. The base ABFE of the prism is a rectangle. The total surface area of the six faces of the prism is S cm². The cross section ABCD of the letter box is a trapezium with edges of lengths DA = 9 x cm, AB = 4 x cm, BC = 6 x cm and CD = 5 x cm as shown in Figure 5. The angle DAB = 90° and the angle ABC = 90°. The volume of the letter box is 9600 cm³.

Show that

Hence show that the surface area of the letter box, S cm², is given by

Use calculus to find the minimum value of S.

Justify, by further differentiation, that the value of S you have found is a minimum.

Using calculus, find the coordinates of the stationary point on the curve with equation

The curve with equation y = x² – 32x) x has a stationary point P. Use calculus

to find the coordinates of P,

to determine the nature of the stationary point P.

The curve C has equation y = 6 − 3x , x 0

Use calculus to show that the curve has a turning point P when x =

Find the x-coordinate of the other turning point Q on the curve.

Hence or otherwise, state with justification, the nature of each of these turning points P and Q.

A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius x mm and height h mm, as shown in Figure 3. Given that the volume of each tablet has to be 60 mm³,

express h in terms of x,

show that the surface area, A mm², of a tablet is given by A = 2² + The manufacturer needs to minimise the surface area A mm², of a tablet.

Use calculus to find the value of x for which A is a minimum.

Calculate the minimum value of A, giving your answer to the nearest integer.

Show that this value of A is a minimum.

A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm, as shown in Figure 2. The volume of the cuboid is 81 cubic centimetres.

Show that the total length, L cm, of the twelve edges of the cuboid is given by

Use calculus to find the minimum value of L.

Justify, by further differentiation, that the value of L that you have found is a minimum.

Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height h cm. The cross section is a sector of a circle. The sector has radius r cm and angle 1 radian. The volume of the box is 300 cm³.

Show that the surface area of the box, S cm², is given by

Use calculus to find the value of r for which S is stationary.

Prove that this value of r gives a minimum value of S.

Find, to the nearest cm², this minimum value of S.

Figure 2 shows a sketch of part of the curve with equation.y 10 8x x² x³ The curve has a maximum turning point A.

Using calculus, show that the x-coordinate of A is 2.

The region R, shown shaded in Figure 2, is bounded by the curve, the y-axis and the line from O to A, where O is the origin.

Using calculus, find the exact area of R.