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The first four Taylor polynomials of the function f(x) = e* at a = 0 are given as follows: Tо (х) 1 Ti (х) 1+x T2(x) 1+x + T3(x) 1+x + %3D It is known that the Taylor polynomials of the function f(x) = e* at a = 0 satisfy the following property: If h is a very small positive number then, for any real number x and any positive integer n, T„(x + h) – T,(x) - Tn-1(x). h (x means approximately equal to, when h is very small) You are asked to prove this for specific values of x and n as follows: (a) For x = 1 and n = 1: Using the Taylor polynomials, show that if h is very small, then T;(1 + h) – T¡(1) z To(1). h (b) For x = 1 and n = 2: Using the Taylor polynomials, show that if h is very small, then T2(1+ h) – T2(1) - T¡(1). h

The first four Taylor polynomials of the function f(x) = e* at a = 0 are given as follows: Tо (х) 1 Ti (х) 1+x T2(x) 1+x + T3(x) 1+x + %3D It is known that the Taylor polynomials of the function f(x) = e* at a = 0 satisfy the following property: If h is a very small positive number then, for any real number x and any positive integer n, T„(x + h) – T,(x) - Tn-1(x). h (x means approximately equal to, when h is very small) You are asked to prove this for specific values of x and n as follows: (a) For x = 1 and n = 1: Using the Taylor polynomials, show that if h is very small, then T;(1 + h) – T¡(1) z To(1). h (b) For x = 1 and n = 2: Using the Taylor polynomials, show that if h is very small, then T2(1+ h) – T2(1) - T¡(1). h

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The first four Taylor polynomials of the function f(x) = e* at a = 0 are given as follows: Tо (х) 1 Ti (х) 1+x T2(x) 1+x + T3(x) 1+x + %3D It is known that the Taylor polynomials of the function f(x) = e* at a = 0 satisfy the following property: If h is a very small positive number then, for any real number x and any positive integer n, T„(x + h) – T,(x) - Tn-1(x). h (x means approximately equal to, when h is very small) You are asked to prove this for specific values of x and n as follows: (a) For x = 1 and n = 1: Using the Taylor polynomials, show that if h is very small, then T;(1 + h) – T¡(1) z To(1). h (b) For x = 1 and n = 2: Using the Taylor polynomials, show that if h is very small, then T2(1+ h) – T2(1) - T¡(1). h
Transcribed Image Text:The first four Taylor polynomials of the function f(x) = e* at a = 0 are given as follows: Tо (х) 1 Ti (х) 1+x T2(x) 1+x + T3(x) 1+x + %3D It is known that the Taylor polynomials of the function f(x) = e* at a = 0 satisfy the following property: If h is a very small positive number then, for any real number x and any positive integer n, T„(x + h) – T,(x) - Tn-1(x). h (x means approximately equal to, when h is very small) You are asked to prove this for specific values of x and n as follows: (a) For x = 1 and n = 1: Using the Taylor polynomials, show that if h is very small, then T;(1 + h) – T¡(1) z To(1). h (b) For x = 1 and n = 2: Using the Taylor polynomials, show that if h is very small, then T2(1+ h) – T2(1) - T¡(1). h
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