Explanation Given information: The function is f ( x ) = x 4 − 3 x 3 + 2 x 2 − x + 1 and the specified point is ( 2 , − 1 ) . 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 ( x 4 − 3 x 3 + 2 x 2 − x + 1 ) 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 ( x 4 − 3 x 3 + 2 x 2 − x + 1 ) = d d x x 4 − d d x 3 x 3 + d d x 2 x 2 − d d x x + d d x 1 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 x 4 − d d x 3 x 3 + d d x 2 x 2 − d d x x + d d x 1 = d d x x 4 − 3 d d x x 3 + 2 d d x x 2 − d d x x + d d x 1 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

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 3.1, Problem 43E

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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 42E

Chapter 3.1, Problem 44E

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22. Given the function f(x) = -2x²-16x-29, a) find the slope of the line tangent to the function at the point (-2,-5), then b) write the equation of the function in slope-intercept form. Tangent Slope: Equation of Tangent Line: Show Your Work 口

In Exercises 37-40, for each function find all values of x where the tangent line is horizontal. 38. f(x) = x³ + 15x² + 63x – 10

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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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 Solutions