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If we continue to raise the degree of our approximation we can continue to add conditions. For instance a third degree approximation P(x) = A + B(x – a) + C(x – a)² + D(x - a)³ can satisfy conditions (i), (ii) and (iii) as well as a fourth condition (iv) P''(a) = f''(a) 13) Find a third degree approximation of f (x) = sin x at x = ".

If we continue to raise the degree of our approximation we can continue to add conditions. For instance a third degree approximation P(x) = A + B(x – a) + C(x – a)² + D(x - a)³ can satisfy conditions (i), (ii) and (iii) as well as a fourth condition (iv) P''(a) = f''(a) 13) Find a third degree approximation of f (x) = sin x at x = ".

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Page < 2 of 2 ZOOM + III. Higher Degree Approximations If we continue to raise the degree of our approximation we can continue to add conditions. For – a) + C(x – a)² + D(x – a)³ can satisfy instance a third degree approximation P(x) = A + B (x - conditions (i), (ii) and (iii) as well as a fourth condition (iv) P''(a) = f''(a) 13) Find a third degree approximation of f (x) = sin x at x = - 6. 14) Derive formulas for A, B, C, and D for any third degree approximation of a function f(x) centered at a. In general we would like to find an nth-degree polynomial T, (x) %3D со + c1 (х — а) + с2(х — а)2 + с3(х — а)3 + .. +Cn(x – a)" where T, and its first n derivatives have the same value at x = a as f and its first n derivatives. T, (a) = f(®)(a) (k) T,(a) = f(a), Th(a) = f'(a), T"(a) = f"(a) T(^) (a) = f(n) (a) ... ... This polynomial is called the nth-degree Taylor polynomial of f centered at a. These will be explored in further detail in Math 1C. For now we will just take a look at how these increasingly higher degree approximations relate to each other and how they provide increasingly more accurate approximations. 15) Find the 2nd, 4th, 6"™, and 8h degree Taylor polynomials for f(x) = cos x centered at x = 0. 16) Use graph paper to provide a detailed and accurate graph of f, T2, T4, T6, and Tg on the same graph with a domain of [-7, 1]. 17) In questions (12) and (14) you found formulas for the coefficients of a second degree and a third degree approximation (These are T2(x) and T3(x) respectively). Come up with a conjecture for the formula for the kth coefficient of T (x). п 18) Can you prove the conjecture you proposed in (17)?
Transcribed Image Text:Page < 2 of 2 ZOOM + III. Higher Degree Approximations If we continue to raise the degree of our approximation we can continue to add conditions. For – a) + C(x – a)² + D(x – a)³ can satisfy instance a third degree approximation P(x) = A + B (x - conditions (i), (ii) and (iii) as well as a fourth condition (iv) P''(a) = f''(a) 13) Find a third degree approximation of f (x) = sin x at x = - 6. 14) Derive formulas for A, B, C, and D for any third degree approximation of a function f(x) centered at a. In general we would like to find an nth-degree polynomial T, (x) %3D со + c1 (х — а) + с2(х — а)2 + с3(х — а)3 + .. +Cn(x – a)" where T, and its first n derivatives have the same value at x = a as f and its first n derivatives. T, (a) = f(®)(a) (k) T,(a) = f(a), Th(a) = f'(a), T"(a) = f"(a) T(^) (a) = f(n) (a) ... ... This polynomial is called the nth-degree Taylor polynomial of f centered at a. These will be explored in further detail in Math 1C. For now we will just take a look at how these increasingly higher degree approximations relate to each other and how they provide increasingly more accurate approximations. 15) Find the 2nd, 4th, 6"™, and 8h degree Taylor polynomials for f(x) = cos x centered at x = 0. 16) Use graph paper to provide a detailed and accurate graph of f, T2, T4, T6, and Tg on the same graph with a domain of [-7, 1]. 17) In questions (12) and (14) you found formulas for the coefficients of a second degree and a third degree approximation (These are T2(x) and T3(x) respectively). Come up with a conjecture for the formula for the kth coefficient of T (x). п 18) Can you prove the conjecture you proposed in (17)?
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