Revision sheet 4

Question 1

Taylors expansion for 3 for f(x+ah)

Answer 1

f(x)+ahf’(x)/1!+((ah)^2 f’‘(X))/2!+((ah)^3f’’‘(x))/3!+((ah)^4f’’’‘(x))/4!+….+O(h^n)

Question 2

Given a sufficiently smooth function 𝑓: ℝ ↦ ℝ, derive the best possible approximation to 𝑓^n(𝑥) in terms of f (x), f (x + ah), and f (x + bh).

Answer 2

1) write out taylor expansion of each
f(x+ah) = f(x)+ahf’(x)/1!+((ah)^2 f’‘(X))/2!+((ah)^3f’’‘(x))/3!+((ah)^4f’’’‘(x))/4!+….+O(h^n)
2) minus f (x + ah) from f (x + bh) to eliminate f^(n+1)(x)
3)rearrange so you get f^n(x) = ….. and get rid of all terms larger than f^n(x) and replace with O(h^n+1)

orrrrrr

4) make matrix
(1,….1 (0
a1,a2,…,an …
a1^2,a2^2,….,an^2. = n!/h^n (a^nth row,h is unknown)
. …
.
.
a1^n-1,….,an^n-1) 0)
5) guassian to find different values (c)
6) Put in form c1f(x) + c2 f (x + ah) + c3f (x + bh).

Question 3

method of undetermined co efficients

Answer 3

1) set up equations
(x_1^n)w1 + (x_2^n)w2 + (x_3^n)*w3 = integral of x^n from n = 0 to 2
2) put in matrix to solve

Question 4

method of lagrange

Answer 4

1) set up equation similar to other Lagrange (each one is equal to l_n(x))
2) work out the corresponding w_n by integrating l_n(x) by the interval

Question 5

relative error

Answer 5

actual error/|true value|

Question 6

Simpson’s rule for integral of f(x) by dx from a to b

Answer 6

((b-a)/6)[f(a)+4f((a+b)/2)+f(b)]