How do you use Taylor series to estimate the accuracy of approximation for #f(x)=sqrt(x)# with #a=1# and #n=3# with #0.9<=x<=1.1#?
Taylor's Theorem guarantees such an estimate will be accurate to within about 0.00000565 over the whole interval
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To estimate the accuracy of the approximation for ( f(x) = \sqrt{x} ) using Taylor series with ( a = 1 ) and ( n = 3 ) within the range ( 0.9 \leq x \leq 1.1 ), you can follow these steps:

Write down the Taylor series expansion for ( f(x) = \sqrt{x} ) centered at ( a = 1 ), up to the third term (( n = 3 )).
[ f(x) \approx f(a) + f'(a)(xa) + \frac{f''(a)}{2!}(xa)^2 + \frac{f'''(a)}{3!}(xa)^3 ]

Compute the derivatives of ( f(x) = \sqrt{x} ) up to the third order, evaluated at ( a = 1 ).
[ f'(x) = \frac{1}{2\sqrt{x}} ] [ f''(x) = \frac{1}{4x^{3/2}} ] [ f'''(x) = \frac{3}{8x^{5/2}} ]

Evaluate these derivatives at ( a = 1 ) to get ( f'(1) ), ( f''(1) ), and ( f'''(1) ).
[ f'(1) = \frac{1}{2} ] [ f''(1) = \frac{1}{4} ] [ f'''(1) = \frac{3}{8} ]

Plug these values into the Taylor series expansion formula.
[ f(x) \approx f(1) + f'(1)(x1) + \frac{f''(1)}{2!}(x1)^2 + \frac{f'''(1)}{3!}(x1)^3 ]
[ \approx 1 + \frac{1}{2}(x1)  \frac{1}{8}(x1)^2 + \frac{1}{16}(x1)^3 ]

Use the interval ( 0.9 \leq x \leq 1.1 ) to estimate the accuracy of the approximation. Calculate the maximum error by finding the maximum value of the absolute value of the fourth derivative within this interval.
[ f^{(4)}(x) = 15x^{7/2} = \frac{15}{x^{7/2}} ]
The maximum value occurs at the endpoints of the interval:
[ \frac{15}{(0.9)^{7/2}} ] and [ \frac{15}{(1.1)^{7/2}} ]

Evaluate these values to find the maximum error.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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