Product rule

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

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.

Figure 2 shows a sketch of part of the curve C with equation y xlnx, x 0

The line l is the normal to C at the point P(e, e)

The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis.

Show that the exact area of R is Ae² B where A and B are rational numbers to be found.

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.

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.

(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.

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.

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.

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

f (x) 3xe^x 1 The curve with equation y = f (x) has a turning point P.

Find the exact coordinates of P.

The equation f (x) = 0 has a root between x = 0.25 and x = 0.3

Use the iterative formula

with x₀ = 0.25 to find, to 4 decimal places, the values of x₁, x₂ and x₃.

By choosing a suitable interval, show that a root of f (x) = 0 is x = 0.2576 correct to 4 decimal places.