Core 3 Numerical Methods 2

1
Core 3 Numerical Methods (2)

• Learning Objectives
• Review previous methods
• Understanding of approximating area under a
  graph using the Mid ordinate rule.

2
Numerical methods

• Sometimes a function can be too tricky or
  impossible to integrate. In cases like this we
  can use a numerical method to get an approximate
  solution.
• In Core 2, you did one of the most common methods
  of doing this is using the Trapezium Rules
• In this lesson, you will be introduced to another
  method

3
Trapezium Rule
Find the area of the purple, green, red and blue
trapezia.
Couldnt integrate
You can use your answers to approximate the area
under the curve
How could we have been more accurate?
4
Generalising..
Trapezium Rule
Area ½ h (end ordinates twice sum of
interior ordinates)
5
The mid-ordinate rule
1.5
2.5
0.5
1
2
0
The mid-ordinate rule uses rectangles to
approximate the area
6
The mid-ordinate rule
Using 4 strips
1.5
2.5
0.5
1
2
0
The height of each rectangle is taken at the
centre of each strip
The width of each rectangle is the width of the
strips
7
The mid-ordinate rule
1.5
2.5
0.5
1
2
0
Area (0.5 x 1.250) (0.5 x 1.601) (0.5 x
2.016) (0.5 x 2.462)
Area 0.5 x (1.250 1.601 2.016 2.462)
3.664
8
The mid-ordinate rule
Area width of strip x sum of mid-ordinates
You can tell if you have over estimated or
underestimated by looking at the strips
If the missed area is bigger than the bit
counted, so it is likely to be an under-estimate
9

Mid-ordinate Example2

• This is the graph of y 2x2 x 6
• Use the Mid-ordinate rule with 4 strips to
  estimate the area under the curve from x 2 to
  x 2.
• This means finding the area of the 4 rectangles
  each of width 1.

10

Mid-ordinate Example 2

• More strips give a better estimate of the area.
• Sometimes the estimate is better than an estimate
  using the trapezium rule.

11

Mid-ordinate Rule Review

• Page 139
• Core 3
• exercise C