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# Review Sheet for MATH 250A with Professor Bergevin at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday November 15 httprnathariz0naeducushing250ahtrnl Sorne Basic Power Series Expansions x 00 1 n 00 1 n l e Z 39x forallx lnxZlt x l for Oltxlt2 n0 71 quot 1 n IX 1n IX 1 l Slnx x2nl for all x 005x Zlt x2 for all x n0 2nl quot0 Zn The Binomial Series 1xp lZWxn for 1ltxltl nl quot p is any constant not necessarily an integer pp1 x2 pp1p2 x3 2 3 Example 1xp1px for 1ltxlt1 1x 1x12 p12 p1quotlx2wx3 2 3 Example 1xp1px for 1ltxlt1 1a 1lesz x1x1x121lx2 2 x22 2 2 3 2 2 6 p12 pp1 x2 pp1p2 x3 2 3 Example 1xp1px for 1ltxlt1 1 1 1 1 1 1 if 36 x1x1x121 x x2 3 2 6 1x 1x x2x3m for 1ltxlt1 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x l l x2 x22 x23 x2quot for x2llt1 lx 2 Substitute x2 for x 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x l lx l lx2 2 l x2 x22 x23 x2quot for x2llt1 l x2x4 x6 l x2 m for xltl 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l lx l l x2x4 x6 l x2 m for xltl lx2 equivalent 2 l x2 x22 x23 x2quot for x2llt1 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x 1 3 1 5 Slnxx x x 3 5 forall x Find the 3rd degree Taylor polynomial 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n l Rmmt T 1xx2x3ax 1ml lt1 x 1 3 1 5 Slnxx x x 3 5 forall x l l sinx 3 lsinxsinx2sinx for lsinxltl 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l l Sinxzx x3 x5 forall x 3 5 1 1 1 1 1 2 l x x3 x5 x x3 x5 l s1nx 3 5 3 5 3 l l x x3 x5 mj 3 5 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x 1 15 forall x sinx x x3 x 3 5 3r01 degree or less 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l l Sinxzx x3 x5 forall x 3 5 1 1 1 1 2 zl x x3 x x3 x5 l s1nx 3 3 5 3 l l x x3 x5 mj 3 5 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x sinxzx ix3ix5 m forall x 3 5 2 l 1 1 1 l x x3 x x3 x5 l s1nx 3 3 5 3 1 1 x x3 x5 mj 3 5 Only term 3r01 degree or less is x2 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x 1 15 sinxx x3 x forall x 3 5 l 1 l x x3 x2 l s1nx 3 3 l l x x3 x5 mj 3 5 Only term 3r01 degree or less is x2 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x 1 15 forall x sinx x x3 x 3 5 Only term 3r01 degree or less is x3 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l l smxx x3 x5 forall x 3 5 l l 1 x x3 x2 l s1nx 3 x3 Only term 3r01 degree or less is x3 103 Other Ways of Calculating Tavlor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l l smxx x3 x5 forall x 3 5 l l 1 x x3 x2 l s1nx 3 x3 No terms 3r01 degree or less 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions n 1 Recall 1 1xx2x3x forxlt1 x l l Sinxzx x3 x5 forall x 3 5 l l l x x3 x2 l sinx 3 X 103 Other Ways of Calculating Taylor Polynomials Substitution composite functions 1 Recall 1 1xx2x3x m for xltl x 1 3 1 5 Slnxx x x 3 5 forall x l 5 lxx2 x3 p3x l s1nx 6 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence 1 lx2 Recall l x2 x4 x6 lquotx2quot for xltl 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence 1 lx2 Recall l x2x4 x6 l x2 for xltl 1 l l 1 1quot Thus I1 dex x3 x57x7m 1x2nim x n 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence 1 Recall 21 X2x4 x6 1 x2 for xlt1 lx 1 l l 1 1quot Thus I 2dx2xx3x5x7n x2nlnl 1x 3 5 7 2nl 00 2nl arctanx Z x for xltl n20 2nl 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence 1 lx2 d 1 Th 2x4x3 6x52n l quotx2 1 us abclx2 Recall l x2x4 x6 l x2 for xltl 103 Other Ways of Calculating Tavlor Polynomials Calculus of Power Series Power series can be integrated and differentiated terrnWise on their intervals of convergence Recall 12 21 38x4x61 x2 for xlt1 lx Thus 1 1 2x4x3 6x52n lquotx2 1 39 abclx2 i22 ZZZ458 1 for lxl ltl 11 n21 x M8 n 1 1x2 1 for xltl 1x22 n Question How good an approximation is pnx fx z fa Maxx a waxx aquot For example how accurate are these approximations lnxzx lzplbc SiI IXNXZPICX Question How good an approximation is pnx fx z fa Maxx a waxx aquot For example how accurate are these approximations lnxzx lzplbc SiI IXNXZPICX 1 lnxzx l Ex 12p2x sinxzx x3p3x Question How good an approximation is pnx fx z fa Maxx a waxx aquot For example how accurate are these approximations lnxzx lzplbc sinxxp1x 1 lnxzx l Ex 12p2x sinxzx x3p3x ylnx p7x ysinx p9x p11x 3 fx hix p006 o p1xx1 p2xx1 x1Z2 Question How good an approximation is pnx fx z fa f ax a ifltquotgtax aquot n Clearly the answer depends on both n and x Question How good an approximation is pnx 1 fx z fa f ax a f ax aquot n Clearly the answer depends on both n and x Lagrange s Error Bound Theorem 101 text If xPnx S Question How good an approximation is pnx 1 fx z fa f ax a f ax aquot n Clearly the answer depends on both n and x Lagrange s Error Bound Theorem 101 text M n1 n1 If xPnx S lx a Question How good an approximation is pnx 1 fx z fa f ax a f ax aquot n Clearly the answer depends on both n and x Lagrange s Error Bound Theorem 101 text M n1 where M is a number satisfying n1 If xPnx S lx a S M on the interval between a and x fnl fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example sinxxp1x fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 39 M 2 SIHXNxszCx lSlnX XISEX fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example EX2 2 sinxxp1x lsinx xls fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example M 2 SIHXNxszCx lSlnX XISEX fa sinx and so fm l fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example M 2 SIHXNxszCx lSlnX XISEX f2 sinx andso f2 SIM fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 SiHXNxszCx lSll lX XISEJC2 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 2 smxzxszOC lsmx xlSEX For example take x 01 lsinO1 01SO12 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 2 smxzxszOC lsmx xlSEX For example take x 01 lsin01 01 s 012 0005 sin 01 01i 0005 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example sinx z x if p3x fx z fa f ax a f ax a M n1 S M n the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 1 Slnxzx 3ilx3 p3x Slnx x 3 lx3 34 x4 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 3 1 3 1 4 z lt smx x 3x p3x Slnx x 3x 4x For example take x 01 1 4 S 01 24 sinO1 O1 O13 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 3 1 3 1 4 z lt smx x 3x p3x Slnx x 3x 4x For example take x 01 sin 01 009983 g 4167 gtlt106 fx z fa f ax a f ax a M n1 S M on the interval between a and x n1 If xPnxl S lx a Where M is a number satisfying I f n1 Example 1 3 1 3 1 4 z lt smx x 3x p3x Slnx x 3x 4x For example take x 01 sin 01 009983 s 4167gtlt106 sin01 0099833 i 4167x10 6 Application Q A thin disk of radius a and mass M A particle of mass m height h above center C A thin disk of radius a and mass M Application m Application 0 I 2ngh 1 l h F a2 I a I I Gravitational force exerted by the disk on the particle m A lication O I I 2ngh 1 1 39h F 2 12 I a h a2h2 I I I Problem approximate F for large k in particular for h gt a m A lication O I 2ngh 1 1 h F a2 7W I a h I I Problem approximate F for large k in particular for h gt a Method consider F as a function of x a h and calculate low order Taylor polynomials in x centered at O o Application 2ngh 1 a2 h x2 gt h h Application m C I I I I I I I I 12 o Application 2ngh 1 1 F 2 12 a h a2h2 xZgth h 2ng2 1 1 x F 2 a 2 2 2 12 a ax a o Application 2ngh 1 1 F 2 12 a h a2h2 x2 gt h h 2ng2 1 F x o Application m Application 0 I 2ngh 1 1 I F 2 12 39 h a h a2h2 I I I M x2 gt h2 h x 2ng 1 F 2 1 212 a 1x pp1 x2 pp1p2 x3 2 3 for 1ltxlt1 1xp 1px m Application I 2ngh 1 1 h F 2 Z W a a h I I M x2 gt h2 h x Fzzg fm 1 1212 a 1x 1x2p1px2p11x4 ppfp2x6m for 1ltxlt1 substitution of x2 o Application 2ngh 1 1 F 2 12 a h a2h2 x2 gt h h 2ng 1 F 2 1 212 a 1x o Application 2ngh 1 1 F 2 12 a h a2h2 x22 h h 2ng 1 F 2 1 212 a 1x 2 12 1 2 3 4 1x 1 x x for 1ltxlt1 2 o 1 1 x212 x Application 2 8 x4 for 1ltxlt1 o Application 2ngh 1 1 F 2 12 a h a2h2 x2 gt h h F 2g fm if 2351 a 2 8 l962 3964 for 1ltxlt1 o Application 2ngh 1 1 F 2 2 2 212 a a h xZgth h Fzzg fm if 2351 a 2 8 2ng 1a2 3614 F 2 2 4 61 2h 8h o Application F 2ngh 1 2 13a h for Olt lt1 h o Application 2ngh l l T0 lowest order ng hZ F for h large m Am C I 2ngh 1 1 I h F 2 W a h a h I M 39 2 M 3 F28 2r 1a2 for Olt lt1 G h 4h h To lowest order M DOCS DOt Fzg m forhlarge depend on a hZ m Application 0 I 2 ngh l l l h F 2 12 I a h a2 hz I I I 2 h 4h h To lowest order ng hZ F for h large To next order approximation hz depends on a higher order ng 3 a2 4 h2 l j for h large

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