The FCF (Functional Continued Fraction) #cosh_(cf) (x;a) = cosh(x+a/cosh(x+a/cosh(x+...)))#. How do you prove that #cosh_(cf) (0;1) = 1.3071725#, nearly and the derivative #(cosh_(cf) (x;1))'=0.56398085#, at x = 0?

Answer 1

See the explanation and the Socratic graph for y = cosh(x+1/y).

Let #y = cosh_(cf)(x;1)=cosh(x+1/cosh(x+1/cosh(x+...)))#.

This FCF is produced by

#y=cosh(x+1/y)#

At x=0,

y is equal to cosh(1/y).

Making use of the discrete analog iteration

#y_n=cosh(1/y_(n-1)), n=1, 2, 3, ...,#
with starter #y_0=cosh (1)#.

8-sd approximation

y is equal to 1.3071725.

Now,

#y'=(cosh(x+1/y))'#
#=sinh(x+1/y)(x+1/y)'#
#=sinh(x+1/y)(1-1/y^2y')#

At x = 0,

#y'=sinh(1/y)(1-1/y^2y')#

Finding y' by substituting y = 1.3071725 and solving

#y'=sinh(1/1.3071725)/(1+sinh(1/1.3071725)/(1.3071725.^2))#
#=0.56398068#, nearly.

Graph utilizing the inversion for y = cosh(x+1/y)

#x = ln(y+sqrt(y^2 - 1))-1/y#:

x=ln(y+(y^2-1)^0.5)-1/y} graph

Observe that #x >=-1 and y >=1#

The tangent at x = 0 is found on the second graph.

graph{(y-1.307-0.564x)=0}(x-ln(y+(y^2-1)^0.5)+1/y)

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Answer 2

To prove that cosh_(cf) (0;1) = 1.3071725, we can use the definition of the functional continued fraction and evaluate it numerically. By substituting x = 0 and a = 1 into the formula, we can calculate the value of the functional continued fraction.

To find the derivative (cosh_(cf) (x;1))' at x = 0, we can differentiate the functional continued fraction with respect to x and then substitute x = 0 into the derivative expression. By evaluating this expression numerically, we can find the value of the derivative at x = 0.

Please note that the numerical values provided (1.3071725 and 0.56398085) are specific approximations and may vary depending on the level of precision used in the calculations.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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