Explanation Given information: The function is f ( x ) = x + 1 x − 1 and the slope of tangent line is − 1 2 . Formula used : 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 . Calculation: 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 + 1 x − 1 ] = ( x − 1 ) ⋅ d d x ( x + 1 ) − ( x + 1 ) ⋅ d d x ( x − 1 ) ( x − 1 ) 2 = ( x − 1 ) ⋅ ( 1 ) − ( x + 1 ) ⋅ ( 1 ) ( x − 1 ) 2 = x − 1 − x − 1 ( x − 1 ) 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x ) . f ′ ( x ) = − 2 ( x − 1 ) 2 ( Combine like terms ) (1) The slope of tangent line is, m = f ′ ( x ) (2) The slope of tangent line is − 1 2

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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Concept explainers

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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Question

Chapter 3.2, Problem 50E

To determine

To find: The points on the graph of given function f(x) when the slope of tangent line is −12 .

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Chapter 3.2, Problem 49E

Chapter 3.2, Problem 51E

Chapter 3 Solutions

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

Ch. 3.1 - State the following rules of differentiation in...Ch. 3.1 - Prob. 2CQ Ch. 3.1 - Prob. 3CQ Ch. 3.1 - Prob. 4CQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E

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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY

The points on the graph of given function f ( x ) when the slope of tangent line is − 1 2 .

Solution Summary: The author explains the quotient rule of differentiation for the given function to get the derivative function.

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To find: The points on the graph of given function f(x) when the slope of tangent line is −12 .

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Chapter 3 Solutions