Explanation Given: The equation y 4 + 1 = y 2 + x 2 Formula used: Equation of tangent line is given as: y − y 1 = m ( x − x 1 ) Key Concepts Used: Derivative of a function y = f ( x ) gives the rate of change of y with respect to x . Thus, the derivative of a function at a point is equivalent to slope of the tangent line at that point. Using that, equation of tangent line is given by y − y 1 = m ( x − x 1 ) Calculations: We have been given the equation of the curve as y 4 + 1 = y 2 + x 2 On taking derivative of this equation on both the sides, we get: ⇒ d d x ( y 4 + 1 ) = d d x ( y 2 + x 2 ) ⇒ 4 y 3 d y d x = 2 y d y d x + 2 x ⇒ 4 y 3 d y d x − 2 y d y d x = 2 x ⇒ 2 y ( 2 y 2 − 1 ) d y d x = 2 x ⇒ d y d x = x y ( 2 y 2 − 1 ) In order to find the points where tangent lines are vertical, we set reciprocal of derivative equal to zero and solve for x and y

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 3.8, Problem 51E

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The derivative of the given equation as well as the points where tangent lines are vertical.

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Chapter 3.8, Problem 50E

Chapter 3.8, Problem 52E

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The derivative of the given equation as well as the points where tangent lines are vertical.

Solution Summary: The author explains that the derivative of a function is equivalent to slope of the tangent line at that point.

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The derivative of the given equation as well as the points where tangent lines are vertical.

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Chapter 3 Solutions