Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

검증된 단계별 안내

First, identify the slope of the line y = 2x + 3. The slope is 2, as it is the coefficient of x.

Since we want the tangent lines to be perpendicular to y = 2x + 3, we need the negative reciprocal of 2, which is -1/2. This will be the slope of the tangent line.

Find the derivative of the function y = x/(x - 2) to determine the slope of the tangent line at any point (x, y). Use the quotient rule: if y = u/v, then y' = (v*u' - u*v')/v^2.

Apply the quotient rule: let u = x and v = x - 2. Then, u' = 1 and v' = 1. Substitute these into the quotient rule to find the derivative y'.

Set the derivative equal to -1/2 (the slope of the perpendicular line) and solve for x. This will give you the x-coordinates of the points where the tangent lines are perpendicular. Substitute these x-values back into the original function to find the corresponding y-coordinates.

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주요 개념

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Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point. To find the tangent line, we typically use the derivative of the function, which gives us the slope at any point on the curve.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m1, the other line must have a slope of m2 such that m1 * m2 = -1. In this context, we need to determine the slope of the given line (y = 2x + 3) and find the negative reciprocal to identify the slope of the tangent lines we are looking for.

Finding Derivatives

The derivative of a function provides a way to calculate the slope of the tangent line at any point on the graph. For the function y = x/(x - 2), we can use the quotient rule to find its derivative. This derivative will help us identify the points where the slope of the tangent line matches the required slope for perpendicularity to the given line.