Consider the functions a(z) and B(x) = e +e- In this question, you may freely use the fact that 8 (x)- a (x) = 1. (i) %3D Verify that a'(r) = B(r) and B'(r) = a(r). The function a is invertible. Use the Inverse Function Theorem to compute the derivative (ii) of a Simplify as much as possible, using the fact that B2(x) – a (r) =1 to write your answer without any a's or B's. (iii) the derivative of a (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii). |The inverse of a can be explicitly computed to be a(x) = In(r+ v + 1). Compute %3D
Consider the functions a(z) and B(x) = e +e- In this question, you may freely use the fact that 8 (x)- a (x) = 1. (i) %3D Verify that a'(r) = B(r) and B'(r) = a(r). The function a is invertible. Use the Inverse Function Theorem to compute the derivative (ii) of a Simplify as much as possible, using the fact that B2(x) – a (r) =1 to write your answer without any a's or B's. (iii) the derivative of a (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii). |The inverse of a can be explicitly computed to be a(x) = In(r+ v + 1). Compute %3D
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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Transcribed Image Text:5. Consider the functions
a(x) =
e - e-
and B(x) =
e +e-z
%3D
%3D
In this question, you may freely use the fact that 82(x)- a²(x) = 1.
= 1.
(i)
Verify that a' (x) = B(x) and B'(x) = a(z).
The function a is invertible. Use the Inverse Function Theorem to compute the derivative
%3D
(ii)
of a. Simplify as much as possible, using the fact that 82(x) - a2(x) = 1 to write your answer
without any a's or B's.
(iii)
the derivative of a (this time without using the Inverse Function Theorem) and confirm that you
get the same answer as part (ii).
The inverse of a can be explicitly computed to be a(r) = In(x+ v + 1). Compute
%3D
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