Single mark scheme specifics

If x<\frac13 or x>2

Set notation: {x\colon x<\frac13}$\cup${x\colon x>2 }

Find the set of possible values of (integer) n

{1,2,3,…,n} - write the set listing all possible unique integers like so

Obtained using Newton-Raphson:

$$x_1=3.9285714$$

$$x_2=3.9249928$$

Joe states that the root must be 3.92 to 2 d.p. because both start with 3.92. Comment on the validity of Joe’s argument.

Not valid as the sequence might decrease further, far enough to change the first 3 sig. fig.

$x_0$ is a stationary point. Explain why Newton-Raphson fails.

The tangent doesn’t meet the x-axis, and so the sequence doesn’t converge

$x_0$ doesn’t give the specific root required using the Newton-Raphson method. (It’s close to a stationary point)

Close to a stationary point means that the tangent meets the x-axis far from the required (closer) root, and the sequence converges to something else.