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Chapter 3 87

# Chapter 3 87 - LABORATORY PROJECT TAYLOR POLYNOMIALS U 237...

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Unformatted text preview: LABORATORY PROJECT TAYLOR POLYNOMIALS U 237 f”(<i) 2! equation we need to calculate the derivatives of f at 0: (ac — a)", To compute the coefﬁcients in this 6-Tnlvﬁl=f(a)+f’(a)(ar:—a)+ (m_a)2+...+ f(m)=cosm f(0) =cosO: f’(:c) = —smac f’(0) : ~sin0 = f"(:c) : —cosm f"(0) = —1 f’”(:c) = sum f”’(0) = 0 fl4)(a:) : cossc f(4)(0) = 1 We see that the derivatives repeat in a cycle of length 4, so f(5)(0) : 0. f(6)(0) : ~1. f(7)(0) : 0, and f(8)(0) : 1, From the original expression for Tn(m) with n = 8 and a : 0, we have ll HI (8) Tg(:c):f(0)+f’(0)(m 0)+f2(!0)(a: (1)2 l f 3f0)(\$ 0)3+~-+f 8!(0)(x—0)8 —1 2 3 1 4 ,5 —1 6 . 7 l 8 =1+0~zn+icc +0~32 +1510 +0~1 +Fat +0 2: +8l\$ 2 4 6 3 x w :1 ‘ a + I ‘ a + a 2 \$4 \$6 8 and the des1red appr0x1mat10n 1s cosy: m 1 — a + I _ a + g The Taylor polynomlals T2 T4 and T6 2 consist of the initial terms of T8 up through degree 2, 4, and 6, respectively. Therefore, T2(:c) : 1 _ %’ 2 4 2 4 6 ”7 33 a: a: a: T4(a:):1~§+E.andT6(m):1e§+E_a_ We graph T2. T4. T5. T8, and f: Notice that T2 (x) is a good approximation to cos x near 0, T4( :5) is a good approximation on a larger interval, T6 (at) is a better approximation, and T8 (:3) is better still. Each successive Taylor polynomial is a good approximation on a larger interval than the previous one. ...
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