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(i) Use the linear approximation formula or Ay≈ f'(x) Ar f(x+Ax) = f(x) + f'(z) Ar with a suitable choice of f(z) to show that log(1+30) 30 for small values of 0. (ii) Use the result obtained in part (a) above to approximate * log(1+30) do. 1/6 log(1+30) de exactly using inte- (iii) Check your result in (b) by evaluating gration by parts.

(i) Use the linear approximation formula or Ay≈ f'(x) Ar f(x+Ax) = f(x) + f'(z) Ar with a suitable choice of f(z) to show that log(1+30) 30 for small values of 0. (ii) Use the result obtained in part (a) above to approximate * log(1+30) do. 1/6 log(1+30) de exactly using inte- (iii) Check your result in (b) by evaluating gration by parts.

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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(a) (i) Use the linear approximation formula or Ay≈ f'(x) Ar f(x+Ax) f(x) + f'(x) Ax with a suitable choice of f(x) to show that log(1+30) 30 for small values of 0. 1/6 Use the result obtained in part (a) above to approximate ¹ log(1+30) de. log(1+36) de exactly using inte- (iii) Check your result in (b) by evaluating gration by parts. (b) (i) Given that -2+3i is a complex root of the cubic polynomial ³-3x - 52, determine the other two roots (without using a calculator). (ii) Hence, (and without using a calculator) determine 12x + 42 ³-3x - 52 (Hint: Use the result of part (a) to write dx. x³3x-52=(x-a) (x²+bx+c) for some a, b and c, and use partial fractions.)
Transcribed Image Text:(a) (i) Use the linear approximation formula or Ay≈ f'(x) Ar f(x+Ax) f(x) + f'(x) Ax with a suitable choice of f(x) to show that log(1+30) 30 for small values of 0. 1/6 Use the result obtained in part (a) above to approximate ¹ log(1+30) de. log(1+36) de exactly using inte- (iii) Check your result in (b) by evaluating gration by parts. (b) (i) Given that -2+3i is a complex root of the cubic polynomial ³-3x - 52, determine the other two roots (without using a calculator). (ii) Hence, (and without using a calculator) determine 12x + 42 ³-3x - 52 (Hint: Use the result of part (a) to write dx. x³3x-52=(x-a) (x²+bx+c) for some a, b and c, and use partial fractions.)
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