# How do you find the first 3 non-zero terms in the Taylor series of the function #cosh(x)# about the point #x=ln8# and use this series to estimate cosh(2)?

Taylor Series Socratic Question

The formula for a third degree Taylor Series polynomial is:

This would be tedious to simplify, so I'll plug this in to my calculator to get:

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To find the first 3 non-zero terms in the Taylor series of the function cosh(x) about the point x = ln(8), we first need to find the derivatives of cosh(x) at x = ln(8). Then, we can use these derivatives to construct the Taylor series.

The first three derivatives of cosh(x) are: cosh(x) = cosh(x) cosh'(x) = sinh(x) cosh''(x) = cosh(x)

Evaluating these derivatives at x = ln(8): cosh(ln(8)) = cosh(2 ln(2)) = cosh(2) = 8 sinh(ln(8)) = sinh(2 ln(2)) = sinh(2) cosh(ln(8)) = cosh(2 ln(2)) = cosh(2) = 8

Using these values, the Taylor series for cosh(x) about x = ln(8) is: cosh(x) ≈ cosh(ln(8)) + sinh(ln(8))(x - ln(8)) + (1/2)cosh(ln(8))(x - ln(8))^2

Now, to estimate cosh(2), we substitute x = 2 into this Taylor series: cosh(2) ≈ 8 + sinh(2)(2 - ln(8)) + (1/2)8(2 - ln(8))^2

You can compute the numerical value of this expression to estimate cosh(2).

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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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