(a) To determine To show: That the Taylor expansion of ( a + x ) 1 / n is equivalent to a 1 / n + x a 1 / n n a , where x is very small as compare to a ,. Explanation Concepts used: If f be a function that can be differentiated n times at x = 0 . The Taylor polynomial of degree n for f at x = 0 is express as, P n ( x ) = f ( 0 ) + f ( 1 ) ( 0 ) 1 ! x + f ( 2 ) ( 0 ) 2 ! x 2 + f ( 3 ) ( 0 ) 3 ! x 3 + … + f ( n ) ( 0 ) n ! x n = ∑ i = 0 n f ( n ) ( 0 ) i ! x i Where, f n is the n t h derivative of the function f . The formula for the derivative of the function f ( x ) = ( a x + b ) n is as follows, f ′ ( x ) = n ( a x + b ) n ⋅ a Calculation: Consider f ( x ) = ( a + x ) 1 / n The first derivative for the given function f ( x ) = ( a + x ) 1 / n is calculated as follows, f ( 1 ) ( x ) = d d x ( a + x ) 1 / n = 1 n ( a + x ) 1 n − 1 ⋅ ( 1 ) = 1 n ( a + x ) 1 − n n The second derivative for the given function f ( x ) = ( a + x ) 1 / n is calculated as follows, f ( 2 ) ( x ) = d d x ( 1 n ( a + x ) 1 − n n ) = ( 1 n ) ⋅ ( 1 − n n ) ( a + x ) 1 − n n − 1 ⋅ ( 1 ) = 1 − n n 2 ( a + x ) 1 − 2 n n The third derivative for the given function f ( x ) = ( a + x ) 1 / n is calculated as follows, f ( 3 ) ( x ) = d d x ( 1 − n n 2 ( a + x ) 1 − 2 n n ) = ( 1 − n n 2 ) ⋅ ( 1 − 2 n n ) ( a + x ) 1 − 2 n n − 1 ⋅ ( 1 ) = ( 1 − n ) ( 1 − 2 n ) n 3 ( a + x ) 1 − 3 n n The fourth derivative for the given function f ( x ) = ( a + x ) 1 / n is calculated as follows, f ( 4 ) ( x ) = d d x ( ( 1 − n ) ( 1 − 2 n ) n 3 ( a + x ) 1 − 3 n n ) = ( ( 1 − n ) ( 1 − 2 n ) n 3 ) ( 1 − 3 n n ) ( a + x ) 1 − 3 n n − 1 ⋅ ( 1 ) = ( 1 − n ) ( 1 − 2 n ) ( 1 − 3 n ) n 4 ( a + x ) 1 − 4 n n In order to find the Taylor polynomial of degree 4 , use the first four derivative of f ( x ) evaluated at x = 0 as follows, Derivative of function f ( x ) Value of derivative at x = 0 f ( x ) = ( a + x ) 1 / n (b) To determine To find: The value of 66 3 up to 5 decimal places.

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 12.3, Problem 37E

(a)

To determine

To show: That the Taylor expansion of (a+x)1/n is equivalent to a1/n+xa1/nna, where x is very small as compare to a,.

(b)

To determine

To find: The value of 663 up to 5 decimal places.

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Chapter 12.3, Problem 36E

Chapter 12.3, Problem 38E

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Calculus with Applications (11th Edition)

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