Explanation Given information: The given function is f ( x ) = x 2 x + 1 and the point is ( 2 , 4 3 ) . Formula used : (1) The quotient rule of differentiation is, f ′ ( x ) = d d x [ f ( x ) g ( x ) ] = g ( x ) ⋅ d d x f ( x ) − f ( x ) ⋅ d d x g ( x ) [ g ( x ) ] 2 The above formula is defined when the denominator g ( x ) ≠ 0 . (2) The point-slope form of equation is, y − y 1 = m ( x − x 1 ) (1) In the formula, m is the slope of line and x 1 , y 1 , x 2 and y 2 are coordinates of two points. Calculation: Section1: The derivative of the given function gives the slope of the tangent line. Apply the quotient rule of differentiation for the given function to get the derivative function f ′ ( x ) . f ′ ( x ) = d d x [ x 2 x + 1 ] = ( x + 1 ) ⋅ d d x ( x 2 ) − ( x 2 ) ⋅ d d x ( x + 1 ) ( x + 1 ) 2 = ( x + 1 ) ⋅ ( 2 x ) − ( x 2 ) ⋅ ( 1 ) ( x + 1 ) 2 = 2 x 2 + 2 x − x 2 ( x + 1 ) 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x ) . f ′ ( x ) = x 2 + 2 x ( x + 1 ) 2 ( Combine like terms ) (1) The slope of tangent line is, m = f ′ ( x ) Substitute 2 for x above equation to get the slope of tangent line from the point ( 2 , 4 3 )

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3.2, Problem 40E

To determine

To find: The slope and an equation of the tangent line of the given function that passes through a given point.

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Chapter 3.2, Problem 39E

Chapter 3.2, Problem 41E

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

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The slope and an equation of the tangent line of the given function that passes through a given point.

Solution Summary: The author explains how to find the slope and an equation of the tangent line of a given function.

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To find: The slope and an equation of the tangent line of the given function that passes through a given point.

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Chapter 3 Solutions