Numerical Methods - [Pure 4].

Question 1

What two things do you have to state to show there is a root between two points?

Answer 1

• f(x) is continuous.
• There is a change of sign.

Question 2

How would you show that an equation had a root between ‘a’ and ‘b’ ?

Answer 2

Sub x = ‘a’ and x = ‘b’ into f(x), replacing x in the equation with your roots and one answer should be +ve and the other -ve for there to be a root between them. Then you state f(x) is continuous and there is a change of sign.

Question 3

What do you need to do to show there is a root between the intersection of two equations?

Answer 3

Sub one equation into the other and rearrange to make that equal to zero. Then sub in your values in which the root supposedly lies in between. Look for a +ve and -ve answer.

Question 4

How do you show there is a sign change in the interval (a,b) for example?

Answer 4

Sub ‘a’ and ‘b’ into the equation. Look for a sign change and use the appropriate greater than, less than or greater to or less than and equal to symbols.

Question 5

How is a graph of: f(x) = 1/x+a for example, not appropriate?

Answer 5

Because it is not a continuous graph.

Question 6

What is the iterative formula?

Answer 6

x_n+1 = g_x(n) where the starting point (x₁) is close to the root but x₂, x₃… converges to the root.

For solving numerically.

Question 7

How do you use the Iterative Formula?

Answer 7

Type the value of x₁ into your calc and press EXE. Sub in ‘Ans’ to the ‘x’ position in the equation and keep pressing EXE until the required x_n.

Question 8

How do you know if a graph is divergent?

Answer 8

The values of ‘x₁, x₂, x₃…’ keep getting bigger.

Question 9

How do you know if a graph is convergent?

Answer 9

The values of ‘x₁, x₂, x₃…’ keep getting smaller and closer to the root (limit).

Question 10

If a sequence is convergent how do you find the limit?

Answer 10

Keep pressing EXE on the calculator until the value doesnt change anymore. Round to the amount of sig fig/ decimal places stated in the question.

Question 11

How do you make Cobweb/ Staircase Diagrams?

Answer 11

The value of x₁ goes vertically up until it touches your curve, then it goes right until it touches the line y = x and so on, hitting the curve then line.

ALWAYS up/down to CURVE, then across to the LINE, then CURVE and so on…

Question 12

Where do the curve and line converge to in a cobweb/ staircase diagram?

Answer 12

To the point of intersection of the curve and line.

Question 13

When will a sequence converge in a cobweb/ staircase diagram?

Answer 13

When the first intersection to the curve has a gradient at that point between 1 and -1.

Question 14

When won’t a sequence converge in a cobweb/ staircase diagram?

Answer 14

When the first intersection to the curve has a gradient at that point more than 1 or less than -1.

Question 15

How does the Newton-Raphson method work?

Answer 15

It uses the gradient of a curve at a point to find a tangent from that point to give you the root of that tangent.

Question 16

In the Newton-Raphson method, what is the root of the tangent?

Answer 16

An approximation of the actual root.

Question 17

What is the formula for the Newton-Raphson method, solving f(x) = 0?

Answer 17

x_n+1 = x_n - f(x_n)/f’(x_n).

Question 18

What two things may explain why the Newton-Raphson Method may fail?

Answer 18

1. If the function can’t be differentiated.
2. The first approximation is too close to or on the x coordinate of a stationary point, so it diverges or converges to a different root.

Question 19

Explain how to use the Newton-Raphson method?

Answer 19

Firstly, find f(x) and f’(x). Then sub these into the formula and then sub in the given value of x1.

Question 20

Why might you need to read the question again?

Answer 20

To check amount of significant figures, or whether it wants x₂, x₃ or so on…