(a) Explanation Given information: The given function is y = tan − 1 x (1) The range lies between [ 0 , ∞ ) Definition used: (a) If ∫ a t f ( x ) d x exists for every number t ≥ a , then ∫ a ∞ f ( x ) d x = lim t → ∞ ∫ a t f ( x ) d x Provided this limit exists (as a finite number). (b) If ∫ t b f ( x ) d x exists for every number t ≤ b , then ∫ − ∞ b f ( x ) d x = lim t → − ∞ ∫ t b f ( x ) d x Provided this limit exists (as a finite number). The integral functions ∫ a ∞ f ( x ) d x and ∫ − ∞ b f ( x ) d x are called convergent if the corresponding limit exists. The improper function ∫ a ∞ f ( x ) d x and ∫ − ∞ b f ( x ) d x are called divergent if the corresponding limit does not exist. (c) If both the integral function ∫ a ∞ f ( x ) d x and ∫ − ∞ a f ( x ) d x are convergent, then ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ a f ( x ) d x + ∫ a ∞ f ( x ) d x Here, a is any real number. Calculation: To find the average value of function, express it as follows: f ( x ) = lim t → ∞ 1 t − a ∫ a t f ( x ) d x (2) Apply Part (a) of the definition of improper integral in Equation (1). ( tan − 1 x ) a v g = lim t → ∞ 1 t ∫ 0 t tan − 1 x d x (3) Show the table of integrals as shown below: ∫ tan − 1 u d u = u tan − 1 u − 1 2 ln ( 1 + u 2 ) + C Apply the table of integrals in Equation (2), with u = x (b) To determine To show: That the average value of f on the interval [ a , ∞ ) is lim x → ∞ f ( x ) . (c) To determine To Find: the average value of f on the interval [ a , ∞ ). (d) To determine To find: the average value of f of function of y = sin x on the interval [0, ∞ ).

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ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 7, Problem 78RE

(a)

To determine

To find: the average value of function y=tan−1x on the interval [0,∞).

(b)

To determine

To show: That the average value of f on the interval [a,∞) is limx→∞f(x).

(c)

To determine

To Find: the average value of f on the interval [a,∞).

(d)

To determine

To find: the average value of f of function of y=sinx on the interval [0, ∞).

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Chapter 7, Problem 77RE

Chapter 7, Problem 79RE

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Chapter 7 Solutions

Calculus: Early Transcendentals

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the average value of function y = tan − 1 x on the interval [ 0 , ∞ ).

Solution Summary: The author explains the average value of the function y=mathrmtan-1x on the interval [0,infty ].

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To find: the average value of function y=tan−1x on the interval [0,∞).

To show: That the average value of f on the interval [a,∞) is limx→∞f(x).

To Find: the average value of f on the interval [a,∞).

To find: the average value of f of function of y=sinx on the interval [0, ∞).

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Chapter 7 Solutions