Newton-Raphson method

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.

Given that the equation f(x) = 0 has a single root α

show that α lies in the interval [3.6, 3.7]

Find f′(x)

Using 3.7 as a first approximation for α, apply the Newton–Raphson method once to obtain a second approximation for α. Give your answer to 3 decimal places.

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 2 ln(8 x) meets the line y xat a single point, x α.

Show that 3< α <4

Figure 2 shows the graph of y 2 ln(8 x) and the graph of y x.

A student uses the iteration formula

in an attempt to find an approximation for α.

Using the graph and starting with x₁4

determine whether or not this iteration formula can be used to find an approximation for α, justifying your answer.

The equation 2x³ + x² 10 has exactly one real root.

Show that, for this equation, the Newton-Raphson formula can be written

Using the formula given in part (a) with x₁ 1

find the values of x₂ and x₃

Explain why, for this question, the Newton-Raphson method cannot be