A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y = 2/x; P(1, 2)

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Step 1: Find the derivative of the function y = \(\frac{2}{x}\) to determine the slope of the tangent line at point P(1, 2). The derivative, y', can be found using the power rule and the fact that \(\frac{d}{dx}\)(x^{-1}) = -x^{-2}.

Step 2: Evaluate the derivative at x = 1 to find the slope of the tangent line at point P(1, 2). Substitute x = 1 into the derivative to get the slope of the tangent line.

Step 3: Determine the slope of the line perpendicular to the tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent line.

Step 4: Use the point-slope form of a line equation, y - y_1 = m(x - x_1), where m is the slope of the normal line and (x_1, y_1) is the point P(1, 2), to write the equation of the normal line.

Step 5: Simplify the equation from Step 4 to get the final equation of the normal line in slope-intercept form, y = mx + b, or any other preferred form.

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주요 개념

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Tangent Line

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, which can be found using the derivative of the function.

Normal Line

A normal line is a line that is perpendicular to the tangent line at a given point on a curve. Its slope is the negative reciprocal of the slope of the tangent line. This relationship is crucial for finding the equation of the normal line, as it allows us to determine its slope based on the tangent line's slope.

Finding Derivatives

Finding the derivative of a function is essential for determining the slope of the tangent line. For the function y = 2/x, we can use the power rule or quotient rule to compute the derivative. Evaluating this derivative at the point P(1, 2) gives us the slope needed to find both the tangent and normal lines.

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.y = 2/x; P(1, 2)