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The exponential function exp can be defined by the Taylor series (centred at 0) exp x = ∞ n=0 xn n! (†) (a) Determine the radius of convergence of (†). (b) Show that (†) implies exp' x = exp x. 1 (c) The function log can be defined as the inverse of exp, so that exp(log x) = x, where x > 0. Apply the chain rule to this relation, and use the result from (b), to prove that log' x = Ꮖ dt -. Xx Hence show that log x =

The exponential function exp can be defined by the Taylor series (centred at 0) exp x = ∞ n=0 xn n! (†) (a) Determine the radius of convergence of (†). (b) Show that (†) implies exp' x = exp x. 1 (c) The function log can be defined as the inverse of exp, so that exp(log x) = x, where x > 0. Apply the chain rule to this relation, and use the result from (b), to prove that log' x = Ꮖ dt -. Xx Hence show that log x =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you please do a, b ,c , please provide explanations 

B2. The exponential function exp can be defined by the Taylor series (centred at 0) Hence show that log x = exp x = 1 Hence evaluate log x dx. n=0 x" .ท n! (a) Determine the radius of convergence of (†). (b) Show that (†) implies exp' x = exp x. (c) The function log can be defined as the inverse of exp, so that exp(log x) = x, where x > 0. Apply the chain rule to this relation, and use the result from (b), to prove that log' x = dt X = [² d ² (d) By making the substitution x = expu, or otherwise, evaluate [₁ (†) log x dx. (e) Define cosh in terms of exp, and thus prove that cosh 2x = 2 cosh² x - 1. Using (†), or otherwise, derive the first three non-zero terms of the Taylor series (centred at 0) for cosh.
Transcribed Image Text:B2. The exponential function exp can be defined by the Taylor series (centred at 0) Hence show that log x = exp x = 1 Hence evaluate log x dx. n=0 x" .ท n! (a) Determine the radius of convergence of (†). (b) Show that (†) implies exp' x = exp x. (c) The function log can be defined as the inverse of exp, so that exp(log x) = x, where x > 0. Apply the chain rule to this relation, and use the result from (b), to prove that log' x = dt X = [² d ² (d) By making the substitution x = expu, or otherwise, evaluate [₁ (†) log x dx. (e) Define cosh in terms of exp, and thus prove that cosh 2x = 2 cosh² x - 1. Using (†), or otherwise, derive the first three non-zero terms of the Taylor series (centred at 0) for cosh.
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