Taylor and Maclaurin Series

More Practice Using Taylor Series, Maclaurin Series, and their Corresponding Polynomials

1. What is the approximation for sin1 using the fifth-degree Maclaurin polynomial for đ?‘Ś = sin (đ?‘Ľ)? , , (A) 1 − + ,

-/ ,

,

/ ,

0 ,

1 ,

/ ,

2 ,

3

,-4

(B) 1 − + (C) 1 − + (D) 1 − + (E) 1 − +

The Maclaurin polynomial for sine is one of our must knows: 0 1 đ?‘Ľ đ?‘Ľ đ?‘Ľâˆ’ + âˆ’â‹Ż 3! 5!

We use it to approximate sin1 through the fifth-degree: 10 11 1 1 1− + =1− + 3! 5!

6 120 The correct answer is E!

2. Which of the following is a power series expansion for <=

?

(A) 1 + đ?&#x2018;Ľ - + đ?&#x2018;Ľ / + â&#x2039;Ż (B) đ?&#x2018;Ľ - + đ?&#x2018;Ľ 0 + đ?&#x2018;Ľ / + â&#x2039;Ż 0 / (C) đ?&#x2018;Ľ + 2đ?&#x2018;Ľ + 3đ?&#x2018;Ľ + â&#x2039;Ż (D) đ?&#x2018;Ľ - + đ?&#x2018;Ľ / + đ?&#x2018;Ľ 3 + â&#x2039;Ż (E) đ?&#x2018;Ľ - â&#x2C6;&#x2019; đ?&#x2018;Ľ / + đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; â&#x2039;Ż ,>< =

This expression looks a lot like ?@ , which is the sum of a ,>A geometric series!

-

This means that đ?&#x2018;&#x17D;, = đ?&#x2018;Ľ and đ?&#x2018;&#x; = đ?&#x2018;Ľ-.

-

/

Our terms would be đ?&#x2018;Ľ + đ?&#x2018;Ľ + đ?&#x2018;Ľ3 + â&#x2039;Ż The correct answer is D!

3. What is the coefficient of đ?&#x2018;Ľ in the Maclaurin Series , expansion for =?

(A)

, 3 ,

(B) 0 (C) 1 (D) 3 (E) 6

(,D<)

-

-

The đ?&#x2018;Ľ term would relate to the second derivative, so letâ&#x20AC;&#x2122;s find that first: >đ?&#x2018;&#x201C; đ?&#x2018;Ľ = (1 + đ?&#x2018;Ľ) đ?&#x2018;&#x201C; F đ?&#x2018;Ľ = â&#x2C6;&#x2019;2(1 + đ?&#x2018;Ľ)>0 đ?&#x2018;&#x201C; FF đ?&#x2018;Ľ = 6(1 + đ?&#x2018;Ľ)>/ Since this is a Maclaurin Series, it would be centered at 0: đ?&#x2018;&#x201C; FF 0 = 6 Now, we plug in to our standard term and simplify: (G) G đ?&#x2018;&#x201C; (đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;?)

6đ?&#x2018;Ľ = đ?&#x2018;&#x203A;!

2!

The leading coefficient would be 3, so the answer is D!