Differentiating standard 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.

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

Coffee is poured into a cup. The temperature of the coffee, H°C, t minutes after being poured into the cup is modelled by the equation

where A and B are constants. Initially, the temperature of the coffee was 85 °C.

State the value of A. Initially, the coffee was cooling at a rate of 7.5 °C per minute.

Find a complete equation linking H and t, giving the value of B to 3 decimal places.

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.

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.

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.

Figure 8 shows a sketch of the curve C with equation y = x^x, x 0

Find, by firstly taking logarithms, the x coordinate of the turning point of C. ((6ROXWLRQV EDVHG HQWLUHO\ RQ JUDSKLFDO RU QXPHULFDO PHWKRGV DUH QRW DFFHSWDEOH ))

The point P(, 2) lies on C.

Show that 1.5 < < 1.6

A possible iteration formula that could be used in an attempt to find α is

Using this formula with x₁ = 1.5

find x₄ to 3 decimal places,

describe the long-term behaviour of x_n

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.

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.

f(x) = 25x²e^2x – 16, x

Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x).

Show that the equation f(x) = 0 can be written as x = ± e^–x The equation f(x) = 0 has a root , where = 0.5 to 1 decimal place.

Starting with x₀ = 0.5, use the iteration formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 3 decimal places.

Give an accurate estimate for to 2 decimal places, and justify your answer.

(i) Differentiate with respect to x

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.

Differentiate with respect to x

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.

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.

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.

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.