Exercise 6: Consider the function f(x) = coth(x) = cosh(x)/ sinh(x). (a) Find the domain and range of f. (b) Compute the derivative of f, by the quotient rule. (c) Prove that f is injective. Let g(x) = arccoth(x) be the inverse. (d) Compute the derivative of g(x), by thinking of it as an inverse function: let arccoth(y), and solve for the inverse using (b) and hyperbolic identities. coth(x), so x = (e) Solve for an expression of g(x) using only previoulsy known functions. (f) Compute the derivative of g(x) using the explicit expression for found in part (e). (g) (Hard) Use the expression found for the derivative to solve the following integral: dy J y – y ln(y)²* Remark: Your answer will be different dependent on | In y| < 1 or | In y| > 1. Thus to solve the integral separately on these two sets. you will have

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Chapter2: Second-order Linear Odes
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d,e,f

Exercise 6: Consider the function f(x) = coth(x) = cosh(x)/ sinh(x).
(a) Find the domain and range of f.
(b) Compute the derivative of f, by the quotient rule.
(c) Prove that f is injective. Let g(x) = arccoth(x) be the inverse.
(d) Compute the derivative of g(x), by thinking of it as an inverse function: let
arccoth(y), and solve for the inverse using (b) and hyperbolic identities.
coth(x), so x =
(e) Solve for an expression of g(x) using only previoulsy known functions.
(f) Compute the derivative of g(x) using the explicit expression for
found in part (e).
(g) (Hard) Use the expression found for the derivative to solve the following integral:
dy
J y – y ln(y)²*
Remark: Your answer will be different dependent on | In y| < 1 or | In y| > 1. Thus
to solve the integral separately on these two sets.
you
will have
Transcribed Image Text:Exercise 6: Consider the function f(x) = coth(x) = cosh(x)/ sinh(x). (a) Find the domain and range of f. (b) Compute the derivative of f, by the quotient rule. (c) Prove that f is injective. Let g(x) = arccoth(x) be the inverse. (d) Compute the derivative of g(x), by thinking of it as an inverse function: let arccoth(y), and solve for the inverse using (b) and hyperbolic identities. coth(x), so x = (e) Solve for an expression of g(x) using only previoulsy known functions. (f) Compute the derivative of g(x) using the explicit expression for found in part (e). (g) (Hard) Use the expression found for the derivative to solve the following integral: dy J y – y ln(y)²* Remark: Your answer will be different dependent on | In y| < 1 or | In y| > 1. Thus to solve the integral separately on these two sets. you will have
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