Explanation Results used: Basic Taylor series Function f ( x ) Taylor series at 0 Interval of convergence 1 1 − x 1 + x + x 2 + ⋅ ⋅ ⋅ + x n + ⋅ ⋅ ⋅ − 1 < x < 1 Table 1 Calculation: Consider the equation f ( x ) = 1 5 − 2 x . Given that t = x − a . Then, t = x − a t = x − 2 ( ∵ a = 2 ) x = t + 2 Substitute the value x = t + 2 in the equation f ( x ) = 1 5 − 2 x and simplify it as follows. f ( x ) = 1 5 − 2 x f ( t + 2 ) = 1 5 − 2 ( t + 2 ) = 1 5 − 2 t − 4 = 1 1 − 2 t Rewrite the function f ( t ) = 1 1 − 2 t as g ( z ) = 1 1 − z , where z = 2 t . From Table 1, the Taylor series at 0 for 1 1 − z is 1 + z + z 2 + ⋅ ⋅ ⋅ + z n + ⋅ ⋅ ⋅ . Substitute the value z = 2 t in the expression 1 + z + z 2 + ⋅ ⋅ ⋅ + z n + ⋅ ⋅ ⋅ and simplify it as follows. 1 + z + z 2 + ⋅ ⋅ ⋅ + z n + ⋅ ⋅ ⋅ = 1 + 2 t + ( 2 t ) 2 + ⋅ ⋅ ⋅ + ( 2 t ) n + ⋅ ⋅ ⋅ = 1 + 2 t + 4 t 2 + ⋅ ⋅ ⋅ + 2 n t n + ⋅ ⋅ ⋅ Substitute the value t = x − 2 in the expression 1 + 2 t + 4 t 2 + ⋅ ⋅ ⋅ + 2 n t n + ⋅ ⋅ ⋅

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Chapter 10.3, Problem 46E

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To find: The Taylor series for f(x)=15−2x at a=2 by the use of the substitution t=x−a.

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Chapter 10.3, Problem 45E

Chapter 10.3, Problem 47E

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Chapter 10 Solutions

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Ch. 10.1 - Find the nth derivative of f(x)=lnx.Ch. 10.1 - Prob. 2MP Ch. 10.1 - Prob. 3MP Ch. 10.1 - Find the second-degree Taylor polynomial at a = 8...Ch. 10.1 - Prob. 5MP Ch. 10.1 - Prob. 1ED Ch. 10.1 - (A)Let p(x) be a polynomial of degree n 1....Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E

Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Problems 1316, find f(3)(x). 15.f(x)=ex Ch. 10.1 - In Problems 1316, find f(3)(x). 16.f(x)=x Ch. 10.1 - Prob. 17E Ch. 10.1 - In Problems 1720, find f4(x). 18.f(x)=e5x Ch. 10.1 - Prob. 19E Ch. 10.1 - In Problems 1720, find f4(x). 20.f(x)=12+x Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - In Problems 2128, find the indicated Taylor...Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Use the third-degree Taylor polynomial at 0 for...Ch. 10.1 - Prob. 41E Ch. 10.1 - Use the third-degree Taylor polynomial at 4 for...Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - Prob. 59E Ch. 10.1 - Prob. 60E Ch. 10.1 - Prob. 61E Ch. 10.1 - Prob. 62E Ch. 10.1 - Prob. 63E Ch. 10.1 - Prob. 64E Ch. 10.1 - Prob. 65E Ch. 10.1 - Prob. 66E Ch. 10.1 - Prob. 67E Ch. 10.1 - Prob. 68E Ch. 10.1 - Prob. 69E Ch. 10.1 - Prob. 70E Ch. 10.1 - Prob. 71E Ch. 10.1 - Consider f(x) = ln (1 + x) and its third-degree...Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Average price. 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The rate of healing for a skin wound (in...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE

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The Taylor series for f ( x ) = 1 5 − 2 x at a = 2 by the use of the substitution t = x − a .

Solution Summary: The author explains how to find the Taylor series for f(x)=15-2x

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To find: The Taylor series for f(x)=15−2x at a=2 by the use of the substitution t=x−a.

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Chapter 10 Solutions