Quotient rule

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.

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.

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.

Given that

show that

where A is a constant to be found.

Given that

show that

where A is a constant to be found.

The function g is defined by

where k is a constant.

Deduce the value of k.

Prove that

for all values of x in the domain of g.

Find the range of values of a for which

Show that = where A and n are constants to be found.

Hence deduce the range of values for x for which 0

Given that

show that

where A is a rational constant to be found.

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.

Given that k is a negative constant and that the function f(x) is defined by

Hence find f ' (x), giving your answer in its simplest form.

State, with a reason, whether f(x) is an increasing or a decreasing function. Justify your answer.

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

Show that

Given that P is a point on C such that f (x) = –1,

find the coordinates of P.

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

Hence, or otherwise, find h(x) in its simplest form.

Figure 2 shows a graph of the curve with equation y = h(x).

Calculate the range of h(x).

(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.

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.

The function f is defined by

The function g is defined by

Find the exact values of x for which g ( x) = 1

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

Express f (x) as a single fraction in its simplest form.