5. Consider the functions et - e-* et + e-* a(x) = and B(x) = In this question, you may freely use the fact that B² (x) – a² (x) = 1. Verify that a'(x) = B(x) and B'(x) = a(x). (ii) of a-1. Simplify as much as possible, using the fact that 32 (x) - a2 (x) = 1 to write your answer without any a's or B's. The function a is invertible. Use the Inverse Function Theorem to compute the derivative The inverse of a can be explicitly computed to be a-'(x) = lIn(x + /x² + 1). Compute (ii) the derivative of a-1 (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii).
5. Consider the functions et - e-* et + e-* a(x) = and B(x) = In this question, you may freely use the fact that B² (x) – a² (x) = 1. Verify that a'(x) = B(x) and B'(x) = a(x). (ii) of a-1. Simplify as much as possible, using the fact that 32 (x) - a2 (x) = 1 to write your answer without any a's or B's. The function a is invertible. Use the Inverse Function Theorem to compute the derivative The inverse of a can be explicitly computed to be a-'(x) = lIn(x + /x² + 1). Compute (ii) the derivative of a-1 (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii).
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
et - e-r
et + e-*
a(x)
and B(x)
2
2
In this question, you may freely use the fact that B² (x) – a²(x) = 1.
(i)
Verify that a'(x) = B(x) and B'(x) = a(x).
(ii)
of a-1. Simplify as much as possible, using the fact that 32 (x) - a2 (x) = 1 to write your answer
without any a's or B's.
The function a is invertible. Use the Inverse Function Theorem to compute the derivative
The inverse of can be explicitly computed to be a-1(x) = ln(x + Væ² + 1). Compute
(ii)
the derivative of a-1 (this time without using the Inverse Function Theorem) and confirm that you
get the same answer as part (ii).
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