Explanation The given function is y = x − 1 x 2 − 2 x . It is known that, the points for which the graph crosses each axis is called the intercepts of the graph. Set y = 0 and obtain the x- intercept of the graph as follows: x − 1 x 2 − 2 x = 0 x − 1 = 0 x = 1 Since the above equation is not satisfied for any value of x , the curve y = x − 1 x 2 − 2 x intercept on x -axis at point x = 1 . Set x = 0 in the curve y = x − 1 x 2 − 2 x and obtain the y- intercept of the graph as follows: Since the curve y = x − 1 x 2 − 2 x is not defined for x = 0 , the curve y = x − 1 x 2 − 2 x does not intercept on y -axis. Thus, the given curve is intercept on x-axis and does not intercept on y-axis. Obtain the symmetry of the given function as follows. Substitute, x = − x in the given equation y = x − 1 x 2 − 2 x and check the symmetry of the given function as follows: Here, the curve y = x − 1 x 2 − 2 x is same as y = − x − 1 x 2 + 2 x . Hence, the curve y = − x − 1 x 2 + 2 x is not symmetric about y -axis. It is to be noted that, the curve y = x − 1 x 2 − 2 x is not same as − y = x − 1 x 2 − 2 x . Thus, the curve y = x − 1 x 2 − 2 x is not symmetric about x -axis. Obtain the asymptotes of the given function as follows. As x → 0 , 1 x → ∞ and 1 x 2 − 2 x → ∞ , the curve y = x − 1 x 2 − 2 x tends to the graph of y = 0 . Thus, the horizontal asymptote of the graph is y = 0 . Since the function y = x − 1 x 2 − 2 x is defined for x = 0 x = 2 , the vertical asymptote of the graph lies on x = 0 and x = 2 . It is to be noted that, the function y = x − 1 x 2 − 2 x is defined for all real numbers except x = 0 and, x = 2 . Thus, the domain of the function is ( − ∞ , ∞ ) − { 0 } . The function y = x − 1 x 2 − 2 x attains positive as well as negative values. Thus, the range of the function y = x − 1 x 2 − 2 x is ( − ∞ , ∞ ) . Differentiate the equation y = x − 1 x 2 − 2 x with respect to x and obtain the first derivative as follows: y ′ = d ( x − 1 x 2 − 2 x ) d x y ′ = d ( 1 x + 1 x 2 − 2 x ) d x y ′ = − 1 x 2 + ( − 1 ) ( 2 x − 2 ) ( x 2 − 2 x ) 2 y ′ = − 1 x 2 − ( 2 x − 2 ) ( x 2 − 2 x ) 2 On further simplifications, y ′ = − ( 1 x 2 + 2 ( x − 1 ) ( x 2 − 2 x ) 2 ) = − ( ( x 2 − 2 x ) 2 + 2 x 2 ( x − 1 ) x 2 ( x 2 − 2 x ) 2 ) = − ( ( x 4 + 4 x 2 − 4 x 3 ) + 2 x 3 − 2 x 2 x 2 ( x 2 − 2 x ) 2 ) = − ( x 4 − 2 x 3 + 2 x 2 x 2 ( x 2 − 2 x ) 2 ) Therefore, y ′ = − ( x 2 − 2 x + 2 ( x 2 − 2 x ) 2 )

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 24.6, Problem 16E

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Chapter 24.6, Problem 15E

Chapter 24.6, Problem 17E

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Chapter 24 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 24.1 - For the parabola y = 4 − x2, at the point (3, −5)...Ch. 24.1 - Prob. 2PE Ch. 24.1 - Prob. 1E Ch. 24.1 - Prob. 2E Ch. 24.1 - Prob. 3E Ch. 24.1 - Prob. 4E Ch. 24.1 - Prob. 5E Ch. 24.1 - Prob. 6E Ch. 24.1 - Prob. 7E Ch. 24.1 - Prob. 8E

Ch. 24.1 - Prob. 9E Ch. 24.1 - Prob. 10E Ch. 24.1 - Prob. 11E Ch. 24.1 - Prob. 12E Ch. 24.1 - In Exercises 11–14, find the equations of the...Ch. 24.1 - Prob. 14E Ch. 24.1 - Prob. 15E Ch. 24.1 - Prob. 16E Ch. 24.1 - Prob. 17E Ch. 24.1 - Prob. 18E Ch. 24.1 - Prob. 19E Ch. 24.1 - Prob. 20E Ch. 24.1 - Prob. 21E Ch. 24.1 - Where does the normal line to the parabola y = x —...Ch. 24.1 - Prob. 23E Ch. 24.1 - Prob. 24E Ch. 24.1 - A certain suspension cable with supports on the...Ch. 24.1 - Prob. 26E Ch. 24.1 - Prob. 27E Ch. 24.1 - Prob. 28E Ch. 24.1 - Prob. 29E Ch. 24.1 - Prob. 30E Ch. 24.2 - In Example 1, let x1 = 0.3, and find x2. 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Prob. 20E Ch. 24.4 - Prob. 21E Ch. 24.4 - Prob. 22E Ch. 24.4 - Prob. 23E Ch. 24.4 - Prob. 24E Ch. 24.4 - Prob. 25E Ch. 24.4 - Prob. 26E Ch. 24.4 - Prob. 27E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 31E Ch. 24.4 - Prob. 32E Ch. 24.4 - Prob. 33E Ch. 24.4 - Prob. 34E Ch. 24.4 - Prob. 35E Ch. 24.4 - Prob. 36E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 38E Ch. 24.4 - Prob. 39E Ch. 24.4 - Prob. 40E Ch. 24.4 - Prob. 41E Ch. 24.4 - Prob. 42E Ch. 24.5 - Prob. 1PE Ch. 24.5 - Prob. 2PE Ch. 24.5 - Prob. 1E Ch. 24.5 - Prob. 2E Ch. 24.5 - Prob. 3E Ch. 24.5 - Prob. 4E Ch. 24.5 - Prob. 5E Ch. 24.5 - Prob. 6E Ch. 24.5 - Prob. 7E Ch. 24.5 - Prob. 8E Ch. 24.5 - Prob. 9E Ch. 24.5 - Prob. 10E Ch. 24.5 - Prob. 11E Ch. 24.5 - Prob. 12E Ch. 24.5 - Prob. 13E Ch. 24.5 - Prob. 14E Ch. 24.5 - Prob. 15E Ch. 24.5 - Prob. 16E Ch. 24.5 - 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A microprocessor chip is being designed with a...Ch. 24.7 - Prob. 18E Ch. 24.7 - What are the dimensions of the largest rectangular...Ch. 24.7 - A rectangular storage area is to be constructed...Ch. 24.7 - Prob. 21E Ch. 24.7 - Prob. 22E Ch. 24.7 - Prob. 23E Ch. 24.7 - Prob. 24E Ch. 24.7 - Prob. 25E Ch. 24.7 - Prob. 26E Ch. 24.7 - Prob. 27E Ch. 24.7 - Prob. 28E Ch. 24.7 - Prob. 29E Ch. 24.7 - Prob. 30E Ch. 24.7 - Prob. 31E Ch. 24.7 - Prob. 32E Ch. 24.7 - Prob. 33E Ch. 24.7 - What is the minimum slope of the curve y = x5 −...Ch. 24.7 - Prob. 35E Ch. 24.7 - Prob. 36E Ch. 24.7 - Prob. 37E Ch. 24.7 - Prob. 38E Ch. 24.7 - Prob. 39E Ch. 24.7 - Prob. 40E Ch. 24.7 - Prob. 41E Ch. 24.7 - Computer simulation shows that the drag F (in N)...Ch. 24.7 - Prob. 43E Ch. 24.7 - The potential energy E of an electric charge q due...Ch. 24.7 - An open box is to be made from a square piece of...Ch. 24.7 - Prob. 46E Ch. 24.7 - Prob. 47E Ch. 24.7 - Prob. 48E Ch. 24.7 - An oil pipeline is to be built from a refinery to...Ch. 24.7 - 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Prob. 43E Ch. 24.8 - Prob. 44E Ch. 24 - Prob. 1RE Ch. 24 - Prob. 2RE Ch. 24 - Prob. 3RE Ch. 24 - Prob. 4RE Ch. 24 - Prob. 5RE Ch. 24 - Prob. 6RE Ch. 24 - Prob. 7RE Ch. 24 - Prob. 8RE Ch. 24 - Prob. 9RE Ch. 24 - Prob. 10RE Ch. 24 - Prob. 11RE Ch. 24 - Prob. 12RE Ch. 24 - Prob. 13RE Ch. 24 - Prob. 14RE Ch. 24 - Prob. 15RE Ch. 24 - Prob. 16RE Ch. 24 - Prob. 17RE Ch. 24 - Prob. 18RE Ch. 24 - Prob. 19RE Ch. 24 - Prob. 20RE Ch. 24 - Prob. 21RE Ch. 24 - Prob. 22RE Ch. 24 - Prob. 23RE Ch. 24 - Prob. 24RE Ch. 24 - Prob. 25RE Ch. 24 - Prob. 26RE Ch. 24 - Prob. 27RE Ch. 24 - In Exercises 25–32, sketch the graphs of the given...Ch. 24 - Prob. 29RE Ch. 24 - Prob. 30RE Ch. 24 - Prob. 31RE Ch. 24 - Prob. 32RE Ch. 24 - Prob. 33RE Ch. 24 - Prob. 34RE Ch. 24 - Prob. 35RE Ch. 24 - Prob. 36RE Ch. 24 - Prob. 37RE Ch. 24 - Prob. 38RE Ch. 24 - Prob. 39RE Ch. 24 - Prob. 40RE Ch. 24 - Prob. 41RE Ch. 24 - Prob. 42RE Ch. 24 - Prob. 43RE Ch. 24 - Prob. 44RE Ch. 24 - Prob. 45RE Ch. 24 - Prob. 46RE Ch. 24 - Prob. 47RE Ch. 24 - Prob. 48RE Ch. 24 - Prob. 49RE Ch. 24 - Prob. 50RE Ch. 24 - Prob. 51RE Ch. 24 - Prob. 52RE Ch. 24 - In Exercises 49–94, solve the given problems. 53....Ch. 24 - Prob. 54RE Ch. 24 - Prob. 55RE Ch. 24 - Prob. 56RE Ch. 24 - The deflection y (in m) of a beam at a horizontal...Ch. 24 - Prob. 58RE Ch. 24 - Prob. 59RE Ch. 24 - Prob. 60RE Ch. 24 - Prob. 61RE Ch. 24 - Prob. 62RE Ch. 24 - In Fig. 24.75, the tension T supports the 40.0-N...Ch. 24 - Prob. 64RE Ch. 24 - Prob. 65RE Ch. 24 - Prob. 66RE Ch. 24 - An analysis of the power output P (in kW/m3) of a...Ch. 24 - The altitude h (in ft) of a certain rocket as a...Ch. 24 - Prob. 69RE Ch. 24 - Prob. 70RE Ch. 24 - Prob. 71RE Ch. 24 - Prob. 72RE Ch. 24 - Prob. 73RE Ch. 24 - A special insulation strip is to be sealed...Ch. 24 - Prob. 75RE Ch. 24 - Prob. 76RE Ch. 24 - Prob. 77RE Ch. 24 - Prob. 78RE Ch. 24 - Prob. 79RE Ch. 24 - Prob. 80RE Ch. 24 - Prob. 81RE Ch. 24 - Prob. 82RE Ch. 24 - Prob. 83RE Ch. 24 - Prob. 84RE Ch. 24 - Prob. 85RE Ch. 24 - Prob. 86RE Ch. 24 - 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The graph of the function y = x − 1 x 2 − 2 x .

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Chapter 24 Solutions