Explanation Definition used: The n th degree Taylor polynomial at a for a function f is p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + … + f n ( a ) n ! ( x − a ) n = ∑ k = 0 n f k ( a ) k ! ( x − a ) k provided f has n derivatives at a . Calculation: The air pollution level (in parts per million) is given as P ( x ) = 10 4 x 2 + 36 − 50 . Obtain the first derivative of P ( x ) = 10 4 x 2 + 36 − 50 as, P ′ ( x ) = d d x ( 10 4 x 2 + 36 − 50 ) = ( 10 ) d d x ( 4 x 2 + 36 ) − d d x ( 50 ) = ( 10 ) ( 1 2 ( 4 x 2 + 36 ) − 1 2 d d x ( 4 x 2 + 36 ) ) = ( 10 ) ( 1 2 ( 4 x 2 + 36 ) − 1 2 ( d d x ( 4 x 2 ) + d d x ( 36 ) ) ) Simplify further to obtain the value of P ′ ( x ) . P ′ ( x ) = 5 4 x 2 + 36 ( 8 x ) = 40 x 4 x 2 + 36 Thus, the value of P ′ ( x ) = 40 x 4 x 2 + 36 . Obtain the second derivative of P ( x ) = 10 4 x 2 + 36 − 50 as, P ″ ( x ) = 4 x 2 + 36 d d x ( 40 x ) − ( 40 x ) d d x ( 4 x 2 + 36 ) ( 4 x 2 + 36 ) 2 = ( 4 x 2 + 36 ) 40 − 40 x ( 4 x 4 x 2 + 36 ) ( 4 x 2 + 36 ) 2 = ( 4 x 2 + 36 ) 40 − ( 160 x 2 ) ( 4 x 2 + 36 ) ( 4 x 2 + 36 ) Simplify the terms and obtain the value of P ″ ( x ) . P ″ ( x ) = 160 x 2 + 1440 − 160 x 2 ( 4 x 2 + 36 ) 3 2 = 1440 ( 4 x 2 + 36 ) 3 2 Thus, the value of P ″ ( x ) = 1440 ( 4 x 2 + 36 ) 3 2

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Chapter 10.1, Problem 90E

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To approximate: The average pollution level during the 8 hour period with the use of second degree Taylor polynomial at a=4.

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Chapter 10.1, Problem 89E

Chapter 10.1, Problem 91E

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Chapter 10 Solutions

CALCULUS FOR BUSINESS...-MYLAB ACESS

Ch. 10.1 - Find the nth derivative of f(x)=lnx.Ch. 10.1 - Prob. 2MP Ch. 10.1 - Prob. 3MP Ch. 10.1 - Find the second-degree Taylor polynomial at a = 8...Ch. 10.1 - Prob. 5MP Ch. 10.1 - Prob. 1ED Ch. 10.1 - (A)Let p(x) be a polynomial of degree n 1....Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E

Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Problems 1316, find f(3)(x). 15.f(x)=ex Ch. 10.1 - In Problems 1316, find f(3)(x). 16.f(x)=x Ch. 10.1 - Prob. 17E Ch. 10.1 - In Problems 1720, find f4(x). 18.f(x)=e5x Ch. 10.1 - Prob. 19E Ch. 10.1 - In Problems 1720, find f4(x). 20.f(x)=12+x Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - In Problems 2128, find the indicated Taylor...Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Use the third-degree Taylor polynomial at 0 for...Ch. 10.1 - Prob. 41E Ch. 10.1 - Use the third-degree Taylor polynomial at 4 for...Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - Prob. 59E Ch. 10.1 - Prob. 60E Ch. 10.1 - Prob. 61E Ch. 10.1 - Prob. 62E Ch. 10.1 - Prob. 63E Ch. 10.1 - Prob. 64E Ch. 10.1 - Prob. 65E Ch. 10.1 - Prob. 66E Ch. 10.1 - Prob. 67E Ch. 10.1 - Prob. 68E Ch. 10.1 - Prob. 69E Ch. 10.1 - Prob. 70E Ch. 10.1 - Prob. 71E Ch. 10.1 - Consider f(x) = ln (1 + x) and its third-degree...Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Average price. 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To approximate: The average pollution level during the 8 hour period with the use of second degree Taylor polynomial at a = 4 .

Solution Summary: The author calculates the average pollution level during the 8 hour period with the use of second degree Taylor polynomial at a=4.

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To approximate: The average pollution level during the 8 hour period with the use of second degree Taylor polynomial at a=4.

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Chapter 10 Solutions