TaylorPolynomial

TAYLOR POLY NOMIAL f(x) = sinx tangeht line EQ +(E) cos(E) 0 T(x) = f () f()x y 1 0(x ) polynomial : y 1 0 =fix e P - () = 1 + 0x+ L-1) x Pc(x) 1 - x for a function centered &0 (x = 0 f(x = + * + xx+ ... -x ex p , (x) 4 + x in ! ) + 0x q + + x = x D , (x) = (10) + ! x + x + x Otto Y 1 . find the Sit degree approximation of f(x) = sinx hear x = 0 f(x) Sin x +" (xj0) sincos +coscoix - x" -x" in x" x f(x) cox 2 6 120 +"(X)= sinx ↑ =0 + y 0 jx + 0 + TX S Taylor f(x) = -cOSX polynomial x x + o * +1 * (x) = Sinx f "(x) = 10) X

2 find the 61 degree taylot polynomial for fix) + * for x near 0 f(x) e 4 Tj0) = e- ce"x+ x x x" e 64e ! S t - x 2x f(x) = ze 24 120 720 S S ↑ "(X) He * f () = 8 e 2x T(x ; 0) = 1 + 2x + 2x ! x + x" 4x Is xS f(x) 16e 2x f (x) = 32e * fl"(x) 64e * for a functions eat e its agedy now i is -x e 3 find, the si Gegter Taylor a no missi j * e f(x) = cos x 6 f"(x) = Sinx 1 + 0 (x ) + 0 f(x) = 203X -1 - (x - - 4 · Find the y degree taylor polynomial for fix) + * for x nears f(x) e 4 2x f "(X) He ex TYx ; 3) Fetzer entertain " f(x) = ze 24 2x f () = 8 e f(x) 16e2 * S I , exc ? -> Use Taylor approx . e x = 0 f(x) e x T ( ; 0) = e + 2)eix + ex /nee + (e) x f(x) 2x 2 6 f"(x) ze * + 4xe * 2 = 1 + 0 + x 0 1 + xx f(x) = 4 ye * + x e * + xex use this to approximate

Se * ( , xdx = x ! = conve ent Taylor (series) Polynomials n 0 e Y E n 0 n ! · X - * If you have a very close to zero , higher powers go to ZerG f(x) = 3 + x x x +(8 01) 3 + 0 01 0 012 + 0 01 3 + 0 01 0 . 0001 0 000001 3 0101001 ⊥ f(x) = 31x when x is near 0 * Taylor polynomial I X near 8 , it's also called a Maclaurin polynomial m fin (0) e n' 6 Taylor Polynomial degree 3 for f(x) e * x near f(x) = ex f(x) - ze * 2 ! f(x) 4e * +x ; 0) = fo + fox or * +z e f"(x) 8 2x -> find an approximate val for Ho 1) +(x , 0)(0 1) = 1 210 1) + 210 172 + = (0 1) 1 22133