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# Class Note 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 Tuesday November 13 httpmathariz0naeducushing250ahtml First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of fx by that of its tangent at x a y tangent line aj cm pX f af axa px passes through same point as f x a and has same slope as f x First Degree Polynomial Approximation 1910 faf39aX a for x z a First degree Taylor polynomial of f x at x a First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 First Degree Polynomial Approximation 19135faf39aX azfx for Xza Examples fxex at a0 f01 First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 f0 1 f 39x ex 2 f 0 1 First Degree Polynomial Approximation p1ltxgt fa f39ltagtx a z fx for x z a Examples fxex at a0 f0 1 f 39x ex gt f 0 1 p1 x 2 1 x First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 f0 1 f 39x ex 2 f 0 1 p1xlxex for sz First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxsinx at a0 f00 First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxsinx at a0 f00 f39xcosx gt f 01 First Degree Polynomial Approximation 19135faf39aX azfx for Xza Examples fxsinx at a0 f00 f39xcosx gt f 01 p1xxsinx for x0 First Degree Polynomial Approximation 19135faf39aX azfx for Xza Examples fxlnx at al f10 First Degree Polynomial Approximation 19135faf39aX azfx for Xza Examples fxlnx at al f10 f39x1x gt f 11 First Degree Polynomial Approximation 19135faf39aX azfx for Xza Examples fxlnx at al f10 f39x1x gt f 11 P1xx 11nx for le If the graph of f x is significantly curved at x a the tangent will be a poor approximation A better approximation would try to approximate some off s curvature If the graph of f x is significantly curved at x a the tangent will be a poor approximation A better approximation would try to approximate some off s curvature Graphs of second degree polynomials are parabolas IDEA match f f 39 and the second derivative f quot with a quadratic polynomial Second Degree Polynomial Approximation Goal find a quadratic polynomial px such that 190 p39a p a equal respectively f a f 39a f quot0 Second Degree Polynomial Approximation Goal find a quadratic polynomial px such that 190 p39a p a equal respectively f a f 39a f quot0 px 2 CO C391c cz6392x cz2 Second Degree Polynomial Approximation Goal find a quadratic polynomial px such that 190 p39a p a equal respectively f a f 39a f quot0 px 2 CO 01x aczx a2 p39x c1 202 x a 9quotx 202 Second Degree Polynomial Approximation Goal find a quadratic polynomial px such that 190 p39a p a equal respectively f a f 39a f quot0 px 2 CO C391c cz6392x cz2 7a Co p39a Cl 9quot0 202 Second Degree Polynomial Approximation Goal find a quadratic polynomial px such that 190 p39a p a equal respectively f a f 39a f quot0 px 2 CO C391c cz6392x cz2 190 Co Co f a P39a 01 gt 01 f39a pquota 202 02 fquota 2 Second Degree Polynomial Approximation 19106 fa f39axa 19236 fa f39a 36 0 fquota 36 602 Second degree Taylor polynomial of f x at x a Second Degree Polynomial Approximation 19106 faf39axa 19206faf39ax afquotax a2 fx for x 62 Second degree Taylor polynomial of f x at x a Second Degree Polynomial Approximation p106 faf39axa P206 faf39axa gf39TaXx ayz x for xwa Second degree Taylor polynomial of f x at x a Note rst degree approximation is contained in second degree approximations Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 M ex 3 f 0 1 3 1 192xlx x2 fquotx ex f quot0 1 2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 fl 2 ex gt flm 1 gt 1192Jclxlx2 fquotx ex f quot0 1 2 We expect this to be a better approximation to 6 than p1x Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 f39x ex 3 f 0 1 fquotx ex f quot0 1 19235 We expect t 39s to be a better approx V ation to ex tha Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 Sequence of Taylor Polynomials 190 x 1 91 x l x l 192xlx3x2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 Sequence of Taylor Polynomials 1900021 p1xlx 192xlxx2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 Sequence of Taylor Polynomials 1900021 p1xlx 192xlxx2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxex at a0 Sequence of Taylor Polynomials 190 x 1 91 x l x 1 2 p2xlx3x Second Degree Polynomial Approximation I 1 I 19206 faf axa f 6006612 Examples fxex at a0 y 3 Sequence of Taylor Polynomials 190 x 1 xex poEx 1o p1xlx pxlxxZ2 1 2 p2x lx x Second Degree Polynomial Approximation 19206 fa f39a x a fquota x a2 Examples fxsinx at a0 f 39x cos x f 390 1 fquotx sinx fquot0 O Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxsinx at a0 f 39x cos x f 390 1 fquotx sinx fquot0 O In this case 92 x is the same as p106 gt p2xx Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxlnx at al f39x1x f3911 f x 1x2 Z fquot1 1 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxlnx at al f39x1x f3911 f x 1x2 Z fquot1 1 2 p2ltxgtltx 1 ltx 1gt2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxlnx at al y In x Sequence of Taylor Polynomials p0ltxgto 9105 x 1 p2ltxgtltx 1gt lltx 1gt2 x 2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxlnx at al y In x Sequence of Taylor Polynomials p0ltxgto p0x 0 p1xx1 p2ltxgtltx 1gt lltx 1gt2 x 2 Second Degree Polynomial Approximation 192 x fa f39a x a fquota x 602 Examples fxlnx at al y In x Sequence of Taylor Polynomials 19000 0 p1xx1 p2ltxgtltx 1gt ltx 1gt2 I fx In x p0x 0 p1x xl x Second Degree Polynomial Approximation I 1 I 19206 faf axa f 6006612 Examples fxlnx at al y In x Sequence of Taylor Polynomials 39 21 Po 0 0 p1ltxltx1 p2xxlxlZ2 91X x p2ltxgtltx 1gt ltx 1gt2 x Higher Degree Polynomial Approximation IDEA Find a polynomial px that matches f x and its first n derivatives at x a Higher Degree Polynomial Approximation IDEA Find a polynomial px that matches f x and its first n derivatives at x a Having n 1 things to match means we need a polynomial with n 1 coefficients Higher Degree Polynomial Approximation IDEA Find a polynomial px that matches f x and its first n derivatives at x a Having n 1 things to match means we need a polynomial with n 1 coefficients ie a polynomial of degree n pxcoclx aczx a2cnx an Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an Higher Degree Polynomial Approximation PXCo01x aczx a2Cnxan dpx dx 61262xa3c3X a2ncnxan 1 Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an dpx dx d2px dx2 61262x a3c3x a2ncn ac a 1 22023263x ann 1cnx an2 Higher Degree Polynomial Approximation Pxcoclx a02xa2quot39Cnxan d1lx 61262xa3c3xa2ncn ac a 1 x 2 djgx 263263x a 39nn 1Cn 36 61 2 x 3 do 2 3203 nn 1Vl 2Cn 35 60114 x Higher Degree Polynomial Approximation Pxcoclx a02xa2quot39Cnxan d1lx 61262xa3c3xa2ncn ac a 1 x 2 djgx 263263x a 39nn 1Cn 36 61 2 x 3 do 2 3203 nn 1Vl 2Cn 35 60114 x dHPEX nn1n226n dx Higher Degree Polynomial Approximation Pxcoclx a02xa2quot39Cnxan d1lx 61262xa3c3xa2ncn ac a 1 x 2 djgx 263263x a 39nn 1Cn 36 61 2 x 3 do 2 3203 nn 1Vl 2Cn 35 60114 x n dHPEX n dx Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an d1lx 61262xa3c3xa2ncn ac a 1 x 2 djgx 263263x a 39nn 1Cn 36 61 2 x d3px n 3 dx3 23263nn 1n 2cnx a Matching with f x at x a n re uires settm x a d mm C q g dxn n and equating the results to the derivatives of f x Higher Degree Polynomial Approximation 170 2 Co dpm C dx 1 6121906 dxz 202 3 d 6 3203 x2 Matching with f x at x a 39 requires settingx a d px n 39C and equating the results dx quot to the derivatives of f x Higher Degree Polynomial Approximation paCO dpm C dx 1 d2 x 02 2c2 d3 x 6ng 32c3 dig ncn fa Z dfx dx xa Z d fx dx xa Matching with f x at x 61 requires settingx a and equating the results to the derivatives of f x Higher Degree Polynomial Approximation pa Co f a dpm 6 Zero dx xa 1 dx xa dzpm 26 d2fx 2 2 2 dx xza dx xza d3 x d3 x 023 3203 fg This formula tells us how to calculate the coefficients on of the approximating polynomial Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an id fx n dx for n12n CO fa9 Cn xa Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an id fx n dx for n12n CO fa9 Cn xa A shorthand notation fa for n20 f 61 d fx dx for n12n xa Higher Degree Polynomial Approximation pxcoclx aczx a2cnx an Cn zi39f a for n012n n Higher Degree Polynomial Approximation 1w fa f ax a f2ax a2 39 nif ax a The Taylor polynomial of degree n of f x at x a Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxex at a0 f xex gtf 01 gtp1xlx f20ex gtf201 gtp2xlxl2x2 Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxex at a0 f xex gtf 01 gtp1xlx f20ex gtf201 gtp2xlxl2x2 f3x ex 3 f30 1 Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxex at a0 f xex gtf 01 gtp1xlx f20ex gtf201 gtp2xlxl2x2 f3xex gtf30l gt 93xlxl2x2l3x3 Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxex at a0 f1xexgtf101gtP1x1x f2xexgtf201gtp2xlxl2x2 f3xex gtf30l gt 93xlxl2x2l3x3 l fquotxex gtf 0lgtpnxlxx2 39x n Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 10 ax a Examples fxex at a0 f1xexgtf101gtP1x1x f2xexgtf201gtp2xlxl2x2 f3xex gtf30l gt 93xlxl2x2l3x3 l fquotxex gtf 0lgtpnxlxx2 39x n nth degreeTaylor polynomial at x O Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxlnx at al f1x1x gtf 11 gt P1XX 1 f2x lx2 gtf21 1gt p2xx l l2x l2 Higher Degree Polynomial Approximation pnxfaf axaf2axa2f ax a Examples fxlnx at al f x1x gtf111 gt P1Xx1 f2x lx2 gtf21 1gt p2xx l l2x l2 flt3gtx2x3 gt flt3gt12 gt Higher Degree Polynomial Approximation pnxfaf axaf2axa2f ax a Examples fxlnx at al f x1x gtf111 gt P1Xx1 f2x lx2 gtf21 1gt p2xx l l2x l2 f3x2x3 gtf3l2 gt p300x l 12x l213x l3 Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxlnx at al f1x1x gtf111 gt P1xX 1 f2x lx2 gtf21 1gt p2xx l l2x l2 flt3gtx2x3 gtflt3gt12 gt p3xx l l2x l2 l3x l3 flt4gtx 3 x4 gt flt4gt1 3 gt Higher Degree Polynomial Approximation 1w fa f ax 61 f2ax a2 39 nif ax a Examples fxlnx at al f1x1x gtf111 gt P1xX 1 f2x lx2 gtf21 1gt p2xx l l2x l2 flt3gtx2x3 gtflt3gt12 gt p3xx l l2x l2 l3x l3 flt4gtx 3 x4 gt flt4gt1 3 gt p4x x 1 12x 12 l3x l3 l4x l4

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