What are the two methods of numerical differentiation?
Forwards difference method and central difference method
Which method generally gives a better approximation?
Central difference method
What is the equation for forward difference?
(f(c+h)-f(c))/ h (remember this is approximately equal to the gradient)
What is the equation for central difference?
(f(c+h)-f(c-h))/ 2h
(remember this is approximately equal to the gradient)
What is the proper notation for the gradient at point c
f’(c)
How do you make the approximation more accurate?
Choose a smaller step size (a smaller value of h)
How would you work out f’‘(c) (the gradient of the gradient, or the second derivative) using central difference?
(f’(c+h)- f’(c-h))/ 2h
Use normal central difference for values of f’(c+h) and f’(c-h)
What is absolute error?
The difference between the value that is measured/ calculated and the actual value
What is relative error?
The absolute difference divided by the actual value
What is percentage error?
The relative error multiplied by 100
What are the three rules for numerical integration?
Midpoint rule, Trapezium rule and Simpson’s rule
What is the most accurate rule for numerical integration?
Simpson’s rule
How do you do the midpoint rule?
The base width of the rectangles multiplied by the sum of the values of the functions at the midpoints of the rectangles
How do you do the trapezium rule?
half the sum of (values of the end functions + twice the values of the middle functions) multiplied by h (width of trapeziums)
How do you do the Simpson’s rule?
one third of the sum of (values of the end functions + four times the odd middle functions + 2 times the even middle functions) multiplied by h
How do you find h for the Simpson’s rule?
h= (b-a)/ 2n
What are the number of strips equal to
2n
