Exercises: Working with Taylor Series (from 10.3 and 10.4) Name: _______________________ Execute this command first! QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMmSShTdHVkZW50R0YnNiNJKkNhbGN1bHVzMUdGKCEiIg== Section 10.3 Exercise 30 (page 610) a) Using the Binomial Series Formula, find the first four terms of the Taylor series centered at 0 for 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 ________________________________________________________ Check your answer with Maple's: 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 Section 10.4 Exercise 31 (page 618) 1. Attempt to compute this definite integral in the standard way we might in Calculus II, using The Fundamental Theorem of Calculus: Compute the antiderivative and then evaluate at the endpoints. First ask for Maple's help to find the antiderivative (Enter from the Expression Palette on the left) JSFH What is Maple telling us? If you want to know what erf is highlight erf and select Help - Help on erf _____________________________________________________________________________________ We proceed to use Taylor series to approximate the definite integral, assuring an error of less than 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 Step 1. Find the Taylor Series for 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. Substituting ________ for x in the Taylor Series for 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 We get ____________________________________________. Ask Maple to verify that you are correct by entering your series (Use the expression pallate again) JSFH Step 2: Find an antiderivative of this the Taylor Series (find an antiderivative term by term) ____________________________________________________ Step 3: Use this series representing an antiderivative to compute the desired definite integral (Evaluate from limits 0 to 0.25) _____________________________________________________ Note that this gives you an alternating series. By the Alternating Series Remainder Theorem, we can approximate the true infinite sum by truncating to the sum of the 0'th through n'th terms and the error will be less than the first neglected (n+1'st) term. Use trial and error below to find the appropriate value of n that will make this error less then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtb0dGJDYtUSomdW1pbnVzMDtGJ0Y4LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlItRjU2JFEiNEYnRjhGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOEYrRjg= . n = _________________. JSFH Now use Maple compute the sum of these truncated number of terms: JSFH Check your answer asking Maple to approximate the definite integral below. Does your answer agree to within 0.0001? _______________ JSFH JSFH Repeat these steps for Exercise 32 and 36, page 618. JSFH