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5. Consider the functions e² + e- a(r) = 2 2 In this question, you may freely use the fact that 32(x) - a²(x) = <= 1. Verify that a'(x) = B(r) and B'(x) = a(r). (i) (ii) and B(x) = The function a is invertible. Use the Inverse Function Theorem to compute the derivative of a¹. Simplify as much as possible, using the fact that 8²(x)-a²(x) = 1 to write your answer without any a's or 3's. The inverse of a can be explicitly computed to be a ¹(x) = ln(x + √²+1). Compute the derivative of a¹ (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii).

5. Consider the functions e² + e- a(r) = 2 2 In this question, you may freely use the fact that 32(x) - a²(x) = <= 1. Verify that a'(x) = B(r) and B'(x) = a(r). (i) (ii) and B(x) = The function a is invertible. Use the Inverse Function Theorem to compute the derivative of a¹. Simplify as much as possible, using the fact that 8²(x)-a²(x) = 1 to write your answer without any a's or 3's. The inverse of a can be explicitly computed to be a ¹(x) = ln(x + √²+1). Compute the derivative of a¹ (this time without using the Inverse Function Theorem) and confirm that you get the same answer as part (ii).

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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5. Consider the functions a(x) = e -e- and B(x) = e +e- %3D %3D In this question, you may freely use the fact that 8 (x) - a (x) = 1. (i) %3D Verify that a'(z) = B(x) and B'(z) = 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 B(x) - a(x) =1 to write your answer without any a's or B's. %3D (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(x+ v + 1). Compute %3D
Transcribed Image Text:5. Consider the functions a(x) = e -e- and B(x) = e +e- %3D %3D In this question, you may freely use the fact that 8 (x) - a (x) = 1. (i) %3D Verify that a'(z) = B(x) and B'(z) = 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 B(x) - a(x) =1 to write your answer without any a's or B's. %3D (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(x+ v + 1). Compute %3D
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