Explanation Given information: The parabola: y = x 2 Calculation: The diagram sketched below shows two perpendicular lines that intersect on the y −axis. Notice that the two lines are tangent to the parabola y = x 2 at the point ( − 0.5 , 0.25 ) and ( 0.5 , − 0.25 ) respectively. The two lines are normal to each other at the point of intersection ( 0 , − 0.25 ) . Let’s start by supposing that the two perpendicular lines have equations given by y = m x + b 1 y = − 1 m x + b 2 Notice that the slopes of the two lines are the negative reciprocal of each other since the two lines are orthogonal. The two lines also intersect on the y -axis, therefore, the two lines have the same y -intercept, that is, b 1 = b 2 in the above pair of linear equations. So we get y = m x + b y = − 1 m x + b ( 1 ) Now since y = x 2 , then y ' = 2 x . And the two lines are both tangent to the parabola at, say, the points ( a , a 2 ) and ( d , d 2 ) respectively if each line has the same slope as that of the curve y at its corresponding point. That is, m = y ' ( a ) ⇒ m = 2 a ( 2 ) − 1 m = y ' ( d ) ⇒ − 1 m = 2 d ( 3 ) The two lines, therefore, have slopes 2a and 2d respectively, and pass through the respective tangent points ( a , a 2 ) and ( d , d 2 )

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ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 3.3, Problem 76E

To determine

To draw:a diagram showing two perpendicular lines that intersect on the y -axis and are both tangent to the parabola y=x2 . Also find point of intersection.

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Chapter 3.3, Problem 75E

Chapter 3.4, Problem 1E

Chapter 3 Solutions

Biocalculus

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To draw: a diagram showing two perpendicular lines that intersect on the y -axis and are both tangent to the parabola y = x 2 . Also find point of intersection.

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To draw:a diagram showing two perpendicular lines that intersect on the y -axis and are both tangent to the parabola y=x2 . Also find point of intersection.

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Chapter 3 Solutions