Y2, C10 - Numerical Methods

Question 1

How would you show that f(x) = e^x +2x - 3 has a root between x = 0.5 and x = 0.6

Answer 1

Solve f(0.5) and f(0.6)
= -0.35127
= 0.02211
Because CHANGE IN SIGN and f(x) CONTINUOUS, root for f(x) is between 0.5 and 0.6

Question 2

When is a function continuous

Answer 2

When you can trace it without taking your pen off the page

Question 3

Does a change in sign always mean there isn’t a root

Answer 3

NO
The sign change method fails to detect a root if there were an even number of roots in that interval

Question 4

g(x) = e^x-1 + x - 6
The root of g(x) = 0 is A
What is a suitable interval to show that A = 2.307 to 3 decimal places

Answer 4

[2.3065, 2.3075]

Question 5

How do you solve f(x) = 0 by an iterative method

Answer 5

Rearrange into form x = g(x) and use the iterative formula x(n+1) = g(x(n))
Where n+1 and n are subscripts

Question 6

The iterative formula:
x(n+1) = ln(6 - x(n)) + 1
x(0) = 2
is used to find an approximate value for A, calculate the values of x1, x2 and x3

Answer 6

x1 = ln(6 - x(0)) + 1
x1 = ln(6 - 2) + 1 = 2.3863
x2 = ln(6 - x(1)) + 1 = …
x3 = ln(6 - x(2)) + 1 = …

Question 7

What does it mean if the x(n) values get closer and closer together

Answer 7

The iteration is successful and convergent

Question 8

What does it mean if the x(n) values get further and further apart

Answer 8

The iteration has failed and is divergent

Question 9

What does it mean if the x(n) values bounce back and forth between values

Answer 9

The iteration oscillates and is periodic / non-convergent

Question 10

Do staircase diagrams always converge

Answer 10

NO, they can diverge

Question 11

How do you draw staircase / cobweb diagrams

Answer 11

From x0, draw up to curve, across to line (y=x), then to curve etc.

Question 12

What is the Newton-Raphson iteration formula for solving f(x) = 0

Answer 12

x(n+1) = xn - (f(xn)) / (f’(xn))
Where xn is the first x coordinate and f(xn) is the first y coordinate

Question 13

Using the Newton-Raphson process, state the recurrence relation for f(x) = x^2 - x - 1

Answer 13

x(n+1) = x(n) - (f(xn)) / (f’(xn))
x(n+1) = x(n) - (xn^2 - xn - 1) / (2xn - 1)

Question 14

When can the Newton-Raphson method fail (3 ways)

Answer 14

When the starting x0 value is the stationary point
It can diverge (oscillate)
Can give approximation for a different (wrong) root

Question 15

Why doesn’t the Netwon-Raphson method work when the starting x0 point is stationary

Answer 15

f’(x0) = 0
You cannot divide by 0 and f’(x0) is the denominator of the formula
The tangent will never meet the x-axis

Question 16

How would you draw / follow a Newton-Raphson diagram

Answer 16

Draw a tangent from the point x0 and follow it to the x axis
Go vertically up or down from the x axis point to find the next value on the curve
Repeat

Question 17

Mark scheme: explain why the Newton-Raphson method cannot be used if the denominator = 0

Answer 17

Tangent to the curve would not meet the x-axis
OR
There is a stationary point at x = 0
OR
Tangent to the curve is horizontal

Question 18

What is f’(x) of 15000(0.85)^x - 1000sinx

Answer 18

Product rule
f’(x) = 15000 * (0.85)^x * ln(0.85) - 1000cos(x)