Can you help me with this please,Thank you! 1.(a) Use Taylor's theorem to find the fourth degree expansion of In x about x = 1.(b) Using the fourth degree Taylor polynomial you found in (a), evaluate K, the fourth degreenumerical approximation to ln(4/3).(c) Use your answers to (a) and (b) to find an upper bound on |K – In(4/3)|.

Answered: Image /qna-images/answer/63b7680b-a5e6-4103-8ce6-cac448224b1d.jpg

Answered: 1. (a) Use Taylor's theorem to find the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section: Chapter Questions

Problem 14T

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Find the second Taylor polynomial P2(x) for the function f(x) = excos(x) expanded about x0 = 0(a) Use P2(0:5) to approximate f(0.5). Find an upper bound for the error |f(0.5)-P2(0.5)| using theerror formula and compare it to the actual error.
b3
Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (a) at the point a In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 3 is 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T₂ (2) is less than 8/375 for all € (2,4). 4. If (3,4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.
If you use the second Taylor polynomial of f(x)=√x centered at x = 1 to estimate √1.1 as √1.1=1+ (1.1-1)-(1.1-1)², which of the following estimates the error? 1 16(0.1) 5/2 -3 8(1.1) 5/2 (0.1) ³| 16 -(0.1)³ (0.1) ³ 1 - (0.1) ³ 16(1.1) 5/2
Let f(x) = ln(2x+1), x = (2,4). 1. The Taylor polynomial T₂ (2) at the point a = 3 is In(7)+(2/7)*(x-3)-(2/49)*(x-3)^ 2. The smallest value of M that occurs in Taylor's inequality is 16/125 3. With M having the above value, Taylor's inequality assures that the error in the approximation f(x) ≈ T2(x) is less than 8/375 for all x € (2,4). 4. If Є (3, 4) the Alternate Series Estimation Theorem assures that the error in the approximation f(x) T₂(2) is less than 16/343 Notice: Your input in 3. and 4. should contain the smallest possible value, as indicated by Taylor Inequality and ASET.
Find the degree 2 Taylor polynomial of f centered at x = 2 when f(x) = 5x ln x . 5 1. 10ln 2+5 ln 2(x − 2) + − (x − 2)² 2. 5 3. 10 In 2+5(ln 2 + 1)(x − 2) + ¹ + (x − 2)² 5 10+5(ln2+1)(x − 2) + (x 5 4. 10+2 ln 5(x - 2) + (x − 2)² 4 5. 10 In 2+5(ln 2 + 1)(x − 2) + 5 6. 10+5 ln 2(x − 2) + + (x − 2)² - − 2)² (x - 2)²
Degree 8.
4. Let I be an open interval. Let f be a function defined on I. Let a E I. Assume f is C² on I. Assume f'(a) = 0. As you know, a is a candidate for a local extremum for f. The "2nd Derivative Test" says that, under these circumstances: • IF f"(a) > 0, THEN ƒ has a local minimum at a. • IF f"(a) < 0, THEN ƒ has a local maximum at a. This theorem is easier to justify, and to generalize, using Taylor polynomials.
Exercise 2. (a) Write the Taylor polynomial of degree 6 centered at π for cosx. (b) Use ≈ 3.14159 and the above polynomial to find an approximate value of cos(3). Round your answer to 5 digits after the decimal sign. (c) Use a computing device to find the value of cos(3). Round the output to 5 digits after the decimal sign. (d) Write the difference between the values that you obtained in parts (b) and (c).

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1. (a) Use Taylor's theorem to find the fourth degree expansion of In r about x = 1. (b) Using the fourth degree Taylor polynomial you found in (a), evaluate K, the fourth degree numerical approximation to lIn(4/3). (c) Use your answers to (a) and (b) to find an upper bound on |K – In(4/3)|-