11.10 – Taylor and Maclaurin Series

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Presentation transcript:

11.10 – Taylor and Maclaurin Series Math 181 11.10 – Taylor and Maclaurin Series

If a function has a power series representation, what does it look like?

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

𝑓 𝑥 = 𝑐 0 + 𝑐 1 𝑥+ 𝑐 2 𝑥 2 + 𝑐 3 𝑥 3 +… Find 𝑐 0 by plugging in 0: 𝒇 𝟎 = 𝒄 𝟎 𝑓 ′ 𝑥 = 𝑐 1 +2 𝑐 2 𝑥+3 𝑐 3 𝑥 2 +4 𝑐 4 𝑥 3 +… Find 𝑐 1 by plugging in 0: 𝒇 ′ 𝟎 = 𝒄 𝟏 𝑓 ′′ 𝑥 =2 𝑐 2 +3⋅2 𝑐 3 𝑥+4⋅3 𝑐 4 𝑥 2 +… Find 𝑐 2 by plugging in 0: 𝑓 ′′ 0 =2 𝑐 2 → 𝒄 𝟐 = 𝒇 ′′ 𝟎 𝟐 𝑓 ′′′ 𝑥 =3⋅2 𝑐 3 +4⋅3⋅2 𝑐 4 𝑥+… Find 𝑐 3 by plugging in 0: 𝑓 ′′′ 0 =3⋅2 𝑐 3 → 𝒄 𝟑 = 𝒇 ′′′ 𝟎 𝟑⋅𝟐 Find 𝑐 4 : 𝑓 4 0 =4⋅3⋅2 𝑐 4 → 𝒄 𝟒 = 𝒇 𝟒 𝟎 𝟒⋅𝟑⋅𝟐 In general, 𝒄 𝒏 = 𝒇 𝒏 𝟎 𝒏!

So, 𝒇 𝒙 = 𝒏=𝟎 ∞ 𝒇 𝒏 𝟎 𝒏. 𝒙 𝒏 =𝑓 0 + 𝑓 ′ 0 1. 𝑥+ 𝑓 ′′ 0 2 So, 𝒇 𝒙 = 𝒏=𝟎 ∞ 𝒇 𝒏 𝟎 𝒏! 𝒙 𝒏 =𝑓 0 + 𝑓 ′ 0 1! 𝑥+ 𝑓 ′′ 0 2! 𝑥 2 + 𝑓 ′′′ 0 3! 𝑥 3 +… This is called the Taylor series of 𝒇 at 𝟎. (Also called the Maclaurin series of 𝒇.)

We could also put the center at 𝑥=𝑎: 𝒇 𝒙 = 𝒏=𝟎 ∞ 𝒇 𝒏 𝒂 𝒏! 𝒙−𝒂 𝒏 𝒇 𝒙 = 𝒏=𝟎 ∞ 𝒇 𝒏 𝒂 𝒏! 𝒙−𝒂 𝒏 =𝑓 𝑎 + 𝑓 ′ 𝑎 1! (𝑥−𝑎)+ 𝑓 ′′ 𝑎 2! 𝑥−𝑎 2 + 𝑓 ′′′ 𝑎 3! 𝑥−𝑎 3 +… This is called the Taylor series of 𝒇 at 𝒂.

Ex 1. Find the Maclaurin series of 𝑓 𝑥 = 𝑒 𝑥 using the definition of a Maclaurin series. Also find the radius of convergence. Ex 2. Find the Taylor series for 𝑓 𝑥 = 𝑒 𝑥 at 𝑎=2, given your results from the previous example.

Ex 3. Find the Maclaurin series for sin 𝑥 using the definition of a Maclaurin series. Ex 4. Find the Maclaurin series for cos 𝑥 , given your results from the previous example. Ex 5. Find the Maclaurin series for 𝑥 cos 𝑥 , given your results from the previous example.

Here is the Taylor series generated by 𝑓 𝑥 = 1+𝑥 𝑘 with convergence on −1<𝑥<1: 𝟏+𝒙 𝒌 = 𝒏=𝟎 ∞ 𝒌 𝒏 𝒙 𝒏 =𝟏+𝒌𝒙+ 𝒌 𝒌−𝟏 𝟐! 𝒙 𝟐 + 𝒌 𝒌−𝟏 𝒌−𝟐 𝟑! 𝒙 𝟑 +… (the Binomial Series) The above is true where 𝑘 is any real number. Also, 𝑘 𝑛 = 𝑘 𝑘−1 𝑘−2 … 𝑘−𝑛+1 𝑛!

Ex 6. Find the Maclaurin series for 1 4−𝑥 .

Here are some frequently-used Taylor Series Here are some frequently-used Taylor Series. 1 1−𝑥 =1+𝑥+ 𝑥 2 + 𝑥 3 +…= 𝑛=0 ∞ 𝑥 𝑛 𝑥 <1 𝑒 𝑥 =1+𝑥+ 𝑥 2 2! + 𝑥 3 3! +…= 𝑛=0 ∞ 𝑥 𝑛 𝑛! sin 𝑥 =𝑥− 𝑥 3 3! + 𝑥 5 5! − 𝑥 7 7! +…= 𝑛=0 ∞ −1 𝑛 𝑥 2𝑛+1 2𝑛+1 ! cos 𝑥 =1− 𝑥 2 2! + 𝑥 4 4! − 𝑥 6 6! +…= 𝑛=0 ∞ −1 𝑛 𝑥 2𝑛 2𝑛 ! ln 1+𝑥 =𝑥− 𝑥 2 2 + 𝑥 3 3 − 𝑥 4 4 +…= 𝑛=1 ∞ −1 𝑛−1 𝑥 𝑛 𝑛 −1<𝑥≤1 tan −1 𝑥 =𝑥− 𝑥 3 3 + 𝑥 5 5 − 𝑥 7 7 +…= 𝑛=0 ∞ −1 𝑛 𝑥 2𝑛+1 2𝑛+1 𝑥 ≤1

Taylor series can help us find integrals and evaluate limits. Ex 7 Taylor series can help us find integrals and evaluate limits. Ex 7. Find an infinite series representation for 𝑒 − 𝑥 2 𝑑𝑥. Ex 8. Evaluate 0 1 𝑒 − 𝑥 2 𝑑𝑥 correct to within an error of 0.001.

Ex 9. Evaluate lim 𝑥→0 tan −1 𝑥 −𝑥 𝑥 3 .

Note: We can multiply and divide power series as follows. Ex 10 Note: We can multiply and divide power series as follows. Ex 10. Find the first three nonzero terms in the Maclaurin series for 𝑒 𝑥 sin 𝑥 . Ex 11. Find the first three nonzero terms in the Maclaurin series for tan 𝑥 .