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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4061eb31-6ba2-4539-9b51-dd7b05481e7e%2F214013b9-3362-48b4-8808-baa6c4e29e1b%2Fzrggee6_processed.png&w=3840&q=75)
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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