# How do you find MacLaurin's Formula for #f(x)=sinhx# and use it to approximate #f(1/2)# within 0.01?

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To find MacLaurin's formula for ( f(x) = \sinh x ), we first need to compute the derivatives of ( f(x) ) at ( x = 0 ).

- The first derivative of ( f(x) = \sinh x ) is ( f'(x) = \cosh x ), evaluated at ( x = 0 ), ( f'(0) = \cosh 0 = 1 ).
- The second derivative of ( f(x) = \sinh x ) is ( f''(x) = \sinh x ), evaluated at ( x = 0 ), ( f''(0) = \sinh 0 = 0 ).
- The third derivative of ( f(x) = \sinh x ) is ( f'''(x) = \cosh x ), evaluated at ( x = 0 ), ( f'''(0) = \cosh 0 = 1 ).
- The fourth derivative of ( f(x) = \sinh x ) is ( f''''(x) = \sinh x ), evaluated at ( x = 0 ), ( f''''(0) = \sinh 0 = 0 ).

Since the even derivatives are ( 0 ) and the odd derivatives are ( 1 ) at ( x = 0 ), the Maclaurin series for ( f(x) = \sinh x ) is:

[ \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \ldots ]

To approximate ( f\left(\frac{1}{2}\right) ) within ( 0.01 ), we use the first few terms of the Maclaurin series:

[ f\left(\frac{1}{2}\right) \approx \frac{1}{2} + \frac{\left(\frac{1}{2}\right)^3}{3!} = \frac{1}{2} + \frac{1}{48} ]

[ = \frac{24}{48} + \frac{1}{48} = \frac{25}{48} ]

Therefore, ( f\left(\frac{1}{2}\right) ) is approximately ( \frac{25}{48} ), which is within ( 0.01 ) of the actual value.

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MacLaurin's formula for ( f(x) = \sinh(x) ) is:

[ \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \frac{x^9}{9!} + \ldots ]

To approximate ( \sinh\left(\frac{1}{2}\right) ) within ( 0.01 ), we'll use the first few terms of the Maclaurin series:

[ \sinh\left(\frac{1}{2}\right) \approx \frac{1}{2} + \frac{\left(\frac{1}{2}\right)^3}{3!} + \frac{\left(\frac{1}{2}\right)^5}{5!} ]

[ \sinh\left(\frac{1}{2}\right) \approx \frac{1}{2} + \frac{1}{48} + \frac{1}{3840} ]

[ \sinh\left(\frac{1}{2}\right) \approx 0.5 + 0.0208333333 + 0.0002604167 ]

[ \sinh\left(\frac{1}{2}\right) \approx 0.52009375 ]

This approximation is within (0.01) of the actual value of ( \sinh\left(\frac{1}{2}\right)).

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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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