Final Review

Starting this Monday, we will review.

Final Topics

Topics marked [CERTAIN] are certain to show up on the final.

Topics for the final are:

Polynomial Interpolation:
- Newton’s Interpolation (only) [CERTAIN]
Derivation & Integration: [CERTAIN ONE OF THESE, INTEGRATION OR DERIVATION]
- Richardson’s
- Romberg’s (MOST-LIKELY
Least-Squared Approximation [CERTAIN]
Monte-Carlo Integration [CERTAIN]
Taylor Series (covered by below)
IVP
- Euler’s Method [CERTAIN]

Questions

There will be five questions.

Euler’s we won’t need to find the solution.
The basis of LSQA will be 4x4
Monte Carlo we’ll be given a set of points and random numbers to use.
We’ll do ten iterations of Euler’s/

Format

Calculator: If we have programmable calculator, you must still show steps.

It IS valid to use the calculator to solve the matrix in LSQA in one-step.

Time: 1h 50m

Flashcard: A 4x6 flashcard with formulae; both sides.

ONLY formulae; no examples.

Will be pair-based.

Crud I don’t know anyone in this class :sob:

Newton’s Interpolation

x_0	y_0
		f[x_0, x_1]
x_1	y_1		f[x_0, x_1, x_2]
		f[x_1, x_2]		f[x_0, x_1, x_2, x_3]
x_2	y_2		f[x_1, x_2, x_3]
		f[x_2, x_3]
x_3	y_3

Romberg’s

LSQA

Monte-Carlo

Euler’s Method

Monte Carlo

Hint: Only useful in M.V.C., when calculating volumes and hypervolumes.

We won’t be doing the easy closed surfaces with bounding box (frick my chungus life).

TODO: Clean up “Example: Multi-Variable Integration (simple area under curve)”

Example:

f(x,y) = x^2 + 3y -7

-2 \le x \le 2 -2 \le y \le 2

Random Points:

IGNORE THESE BELOW

0.2247650996  0.0097487256
0.2301878669  0.1154845843
0.8541877497  0.8150016699
0.3696478807  0.4320032957
0.1512551759  0.2040172377
0.6181553239  0.7729526075
0.8374186037  0.8417869482
0.3568608441  0.3966633329
0.8157072425  0.6248937046
0.9597630374  0.6790679323

In range (by doing TODO), points are:

x y
TODO

g(x,y) := f(x*4 - 2,y*4 - 2)

( g(0.3750063532, 0.9238014212)+ g(0.6801047421, 0.7440002147)+ g(0.4285603430, 0.5219665352)+ g(0.2380735225, 0.9999298456)+ g(0.7077894222, 0.1527320153)+ g(0.7938789609, 0.5512221053)+ g(0.5474259313, 0.3494932069)+ g(0.4752015512, 0.0892621723)+ g(0.0241782032, 0.3120241395)+ g(0.6306129428, 0.1551865790)+ g(0.0064592837, 0.8364119576)+ g(0.3760168429, 0.8772376585)+ g(0.3690376199, 0.4105594558)+ g(0.8691588920, 0.5518207140)+ g(0.2779716766, 0.2546516864)+ g(0.1522042751, 0.4125430814)+ g(0.2933060602, 0.0685110795)+ g(0.8326134208, 0.0794933096)+ g(0.4247943402, 0.0460103027)+ g(0.2247820604, 0.6518327070) )*(16/20) = -131.0950962788

Plug into:

\frac{-131.0950962788}{20} (16) = -104.876077023

This is way off, the answer was -90.66667.

I think I messed up EVEN THOUGH I was using qalc bruh.