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MATH119 Study Guide - Polynomial, Interpolation, Ibm System P

5 pages82 viewsWinter 2013

Department
Mathematics
Course Code
MATH119
Professor
Michael Dunphy

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Calculus QuickSheet
MATH 119
1 Linear Approximations
L(x) = f(a) + f0(a)(xa)
2 Root Finding
Bisection
Find a, b where f(a)<0 && f(b)>0.
If f(a+b
2)<0 then a=a+b
2.
Else, b=a+b
2.
Newton’s Method
1. Use Bisection to find an estimated starting point (a).
2. Set L(x) = f(a) + f0(a)(xa) to 0 and solve for x.
3. Let a=x.
4. Repeat steps 2-3 until you find an acceptable estimate.
Iterative Formula: xn+1 =xnf(xn)
f0(xn)
Fixed-Point Iteration
1. Isolate for x.
2. Solve xn+1 =g(xn).
3. Repeat until answer is same twice in a row (to required decimal places).
4. If divergence occurs, try isolating x differently and repeat steps 2-3.
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3 Polynomial Stuff
Newton Forward Difference Forumla
y=y0+(xx0)
hy0+(xx0)(xx1)
2!h22y0+... +(xx0)(xx1)...(xxn+1)
n!hnny0
Lagrange Linear Interpolation Formula
f(x)(xx1)
(x0x1)f1+(xx0)
(x1x0)f2
Taylor Polynomials
Pn,x0(x) =
n
X
k=0
f(k)(x0)
k!(xx0)k+Rn(x)
Maclaurin Polynomial
Taylor Polynomial centered at 0. Pn,0(x) =
n
X
k=0
f(k)(0)
k!xk+Rn(x)
f(x) = f(0) + f0(0) ·x+f00(0) ·x2
2! +f000(0) ·x3
3! +...
Remainder Theorem
|Rn(x)|=|Zx
x0
(xt)n
n!f(n+1)(t)dt|
≤ |f(n+1)(t)||xx0|n+1
(n+ 1)!
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