Explanation Given information: The function is f ( x ) = x x 2 + 1 and the tangent line is horizontal. 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 x 2 + 1 ] = ( x 2 + 1 ) ⋅ d d x ( x ) − ( x ) ⋅ d d x ( x 2 + 1 ) ( x 2 + 1 ) 2 = ( x 2 + 1 ) ⋅ ( 1 ) − ( x ) ⋅ ( 2 x ) ( x 2 + 1 ) 2 = x 2 + 1 − 2 x 2 ( x 2 + 1 ) 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x )

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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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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Question

Chapter 3.2, Problem 48E

To determine

To find: The points on the graph of given function f(x) at which the tangent line is horizontal.

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Chapter 3.2, Problem 47E

Chapter 3.2, Problem 49E

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1. Consider the following system of equations: x13x2 + 4x3 - 5x4 = 7 -2x13x2 + x3 - 6x4 = 7 x16x213x3 - 21x4 = 28 a) Solve the system. Write your solution in parametric and vector form. b) What is a geometric description of the solution. 7 c) Is v = 7 in the span of the set S= [28. 1 HE 3 -5 3 ·6 ? If it is, write v 6 as a linear combination of the vectors in S. Justify. d) How many solutions are there to the associated homogeneous system for the system above? Justify. e) Let A be the coefficient matrix from the system above. Find the set of all solutions to Ax = 0. f) Is there a solution to Ax=b for all b in R³? Justify.

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The points on the graph of given function f ( x ) at which the tangent line is horizontal.

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) at which the tangent line is horizontal.

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Chapter 3 Solutions