Explanation Domain: The given function is, f ( x ) = x + tan x = x + sin x cos x = x cos x + sin x cos x Notice that, the function is not defined at x = ± π 2 . Therefore, the domain of the given function is ( − π 2 , π 2 ) . Symmetry: The function is even if f ( − x ) = f ( x ) and odd if f ( − x ) = − f ( x ) . Thus, f ( − x ) = − x + tan ( − x ) = − x − tan ( x ) = − ( x + tan ( x ) ) = − f ( x ) Notice that f ( − x ) = − f ( x ) . That is, the function is odd function. Thus, the graph is symmetric. Derivatives: Differentiate f ( x ) = x + tan x with respect to x on both sides as, f ′ ( x ) = d d x ( x + tan x ) = 1 + sec 2 x Differentiate f ′ ( x ) = 1 + sec 2 x with respect to x on both sides as, d d x ( f ′ ( x ) ) = d d x ( 1 + sec 2 x ) f ″ ( x ) = 2 sec x sec x ⋅ tan x = 2 sec 2 x ⋅ tan x Critical points and possible inflection points: Set the first derivative to 0 and find the critical points as follows. f ′ ( x ) = 0 1 + sec 2 x = 0 The above equation is always greater than one for any value of x. Therefore, the function has no extreme points. Set the second derivative to 0 and find the inflection points as follows. f ′ ′ ( x ) = 0 2 sec 2 x ⋅ tan x = 0 tan x = 0 x = n π Since the given domain is ( − π 2 , π 2 ) , the possible points of inflection are x = 0 . Substitute x = 0 in f ( x ) = x + tan x as, f ( 0 ) = 0 + tan 0 = 0 Thus, the inflection points are ( 0 , 0 )

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 4.4, Problem 45E

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To sketch: The complete graph of the polynomial f(x)=x+tanx on (−π2,π2).

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Chapter 4.4, Problem 44E

Chapter 4.4, Problem 46E

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

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The complete graph of the polynomial f ( x ) = x + tan x on ( − π 2 , π 2 ) .

Solution Summary: The author illustrates the graph of the polynomial f(x)=x+mathrmtanx.

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To sketch: The complete graph of the polynomial f(x)=x+tanx on (−π2,π2).

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Chapter 4 Solutions