Explanation Given information: The given function is, f ( x ) = − 5 3 x 2 + 2 x + 2 and the specified point is ( − 1 , − 5 3 ) . Formula used: The equation of a line which passes through the point ( x 1 , y 1 ) and has slope m is given by y − y 1 = m ( x − x 1 ) . This is called as point-slope form of the equation. Calculation: The derivative of the function is the slope of the tangent of its graph at any point. The derivative of the given function is, f ′ ( x ) = d d x f ( x ) = d d x ( − 5 3 x 2 + 2 x + 2 ) Use the sum rule of differentiation , d d x [ f ( x ) ± g ( x ) ] = d d x [ f ( x ) ] ± d d x [ g ( x ) ] . d d x ( − 5 3 x 2 + 2 x + 2 ) = d d x ( − 5 3 x 2 ) + d d x 2 x + d d x 2 Use the constant multiple rule of differentiation which states that if c is any constant, then d d x [ c f ( x ) ] = c d d x [ f ( x ) ] . d d x ( − 5 3 x 2 ) + d d x 2 x + d d x 2 = − 5 3 d d x x 2 + 2 d d x x + d d x 2 Use the power rule of differentiation which states that if n is a real number, then d d x ( x ) n = n x n − 1 . − 5 3 d d x x 2 + 2 d d x x + d d x 2 = − 5 3 ( 2 x 2 − 1 ) + 2 ( 1 x 1 − 1 ) + 0 = − 5 3 ( 2 x ) + 2 ( 1 x 0 ) = − 10 3 x + 2 This implies that f ′ ( x ) = − 10 3 x + 2

Applied Calculus for the Managerial, Life, and Social Sciences

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ISBN: 9781305887831

Author: Tan

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Chapter 3.1, Problem 42E

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The slope and the equation of the tangent line of the graph of the function f at the specified point.

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Chapter 3.1, Problem 41E

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Applied Calculus for the Managerial, Life, and Social Sciences

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