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MATH119 Study Guide - Polynomial, Interpolation, Ibm System P

5 Pages
190 Views
Winter 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.
1
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)!
2

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Description
Calculus QuickSheet MATH 119 1 Linear Approximations L(x) = f(a) + f0(a)(x ▯ a) 2 Root Finding Bisection Find a, b where f(a) < 0 && f(b) > 0. a + b a + b ▯ If f( 2 ) < 0 then a = 2 . a + b ▯ Else, b = 2 . Newton’s Method 1. Use Bisection to ▯nd an estimated starting point (a). 2. Set L(x) = f(a) + f0(a)(x ▯ a) to 0 and solve for x. 3. Let a = x. 4. Repeat steps 2-3 until you ▯nd an acceptable estimate. Iterative Formula: x = x ▯ f(xn) n+1 n f0(xn) Fixed-Point Iteration 1. Isolate for x. 2. Solve n+1 = g(xn). 3. Repeat until answer is same twice in a row (to required decimal places). 4. If divergence occurs, try isolating x di▯erently and repeat steps 2-3. 1 3 Polynomial Stu▯ Newton Forward Di▯erence Forumla (x ▯ x0) (x ▯ x0)(x ▯ x1) 2 (x ▯ x0)(x ▯ x1):::(x ▯ n+1) n y = y0+ ▯y 0 2 ▯ y 0 ::: + n ▯ y 0 h 2!h n!h Lagrange Linear Interpolation Formula (x ▯ x1) (x ▯ x0) f(x) ▯ f1+ f2 (x0▯ x 1 (x1▯ x 0 Taylor Polynomials n X f(k(x 0 k Pn;x0(x) = (x ▯ x0) + R nx) k=0 k! Maclaurin Polynomial Xn f(k(0) Taylor Polynomial centered at 0. Pn;0(x) = x + R nx) k! k=0 00 2 000 3 0 f (0) ▯ x f (0) ▯ x f(x) = f(0) + f (0) ▯ x + 2! + 3! + ::: Remainder Theorem Z x n (x ▯ t) (n+1) jR nx)j = j f (t)dtj x0 n! n+1 ▯ jf(n+1)(t)jjx ▯ 0 j (n + 1)! 2 Common Taylor Polynomial Equations n n x X x
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