is called the nth-degree Taylor polynomial for f at c. named after Brook Taylor, an English mathematician. f x e

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1 Taylor Polyomials ad Approimatios Polyomial fuctios ca be used to approimate other elemetary fuctios such as si, y si 3 5 y ad compare. 3! 5! e, ad l. O your calculator, graph: Defiitio of a th-degree Taylor polyomial: fc f c If f has derivatives at = c, the the polyomial P f c fcc c... c!! is called the th-degree Taylor polyomial for f at c. amed after Brook Taylor, a Eglish mathematicia. f 0 f 0 If c = 0, the P f 0 f 0... is called the th-degree Maclauri polyomial for f amed!! after aother Eglish mathematicia, Coli Maclauri. E. Fid the Maclauri polyomial of degree = 4 for f e. E. Fid a Taylor series for f e 5 cetered at c =. Give the first four ozero terms ad the geeral term. E. Fid the Maclauri polyomial of degree = 6 for f cos. The use P 6 () to approimate the value of cos (0.).

2 E. Fid the Taylor polyomial of degree = 6 for f l at c =. The use P 6 () to approimate the value of l(.). E. Suppose that g is a fuctio which has cotiuous derivatives, ad that g 3, g 4, g 7, g 5 Write the Taylor polyomial of degree 3 for g cetered at. E. The fuctio f has a Taylor series about = that coverges to f() for all i the iterval of covergece. The th derivative of f ( ) ( )! at = is give by f () for, ad f() =. Write the first four terms ad the geeral term of the Taylor series 3 for f about =.

3 3 E. Use the Taylor approimatio e! 3! for ear 0 to fid lim 0 e Taylor Series f c f c f c Taylor Series cetered at = c: f ( ) f c f c c c... c... c!! 0! If c = 0, the series is called a Maclauri series. Frequetly Used Series a , 0 b. e......,!! 0! c. 3 5 si......, 3! 5!! 0! d. 4 cos......,! 4!!! 0 We ca maipulate these three special series (or ay series we are give) to fid other series by usig the followig techiques: ) Substitute ito the series ) Multiply or divide the series by a costat ad/or a variable 3) Add or subtract two series 4) Differetiate or itegrate a series 5) Recogizig it as the sum of a geometric power series E. Fid a Maclauri series for f cos

4 E. Fid a Maclauri series for f si. E. Fid the Maclauri series for f si. Hit: use cos si E. (a) Fid a Maclauri series for f e (b) Use your aswer to (a) to fid a Maclauri series for the fuctio h, give that ad h e h 0 3 Power Series If is a variable, the a ifiite series of the form a c a0 a ca c... a c is called a power series cetered at c, where c is a costat. The iterval of covergece occurs whe 0 a lim a (Ratio Test). Oce the iterval has bee determied edpoits must be checked by substitutig the edpoits ito the origial equatio for ad determiig if the series coverges. E. Fid the radius of covergece ad the iterval of covergece. Be sure to check the edpoits.

5 (a) 0 3! (b) 3 (c) 0! (d) 5

6 3 f for all i the iterval of covergece of the power series (a) Fid the iterval of covergece for this power series. Show the work that leads to your aswer. E. A fuctio f is defied by (b) Fid lim 0 f 4 (c) Write the first three ozero terms ad the geeral term for a ifiite series that represets 0 f d (d) Fid the sum of the series determied i part (c).

7 Before we try to fid a power series by recogizig it as the sum of a geometric power series, let's do a quick review of geometric series. Geometric series are formed by multiplyig by a commo ratio r. Suppose I told you to start with a = ad to let r = 3. What geometric series would you write? E. What if a = ad r = -3? E. What if a = ad r =? E. Fid a power series for 3 f( ), cetered at = 0. Fid the first four ozero terms ad the geeral term. 4 E. Fid a power series for g 3, cetered at = 0. Fid the first four ozero terms ad the geeral term.

8 E. Fid a power series for g 3, cetered at =. Fid the first four ozero terms ad the geeral term. E. Fid a power series for g, cetered at = -3. Fid the first four ozero terms ad the geeral term. 5 E. Fid a power series for g arcta, cetered at = 0. Fid the first four ozero terms ad the geeral term.