Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.

검증된 단계별 안내

Identify the slope of the given line y = -x/2. The slope is -1/2.

Find the derivative of the curve y = tan(x) to determine the slope of the tangent line at any point x. The derivative is dy/dx = sec^2(x).

Since the normal line is perpendicular to the tangent line, the slope of the normal line is the negative reciprocal of the tangent line's slope. Set the slope of the normal line equal to -1/2 to find the x-values where this condition holds.

Solve the equation -1/(sec^2(x)) = -1/2 to find the x-values. This simplifies to sec^2(x) = 2, which further simplifies to cos^2(x) = 1/2.

Solve for x in the interval -π/2 < x < π/2 where cos^2(x) = 1/2. These x-values are the points where the normal line is parallel to y = -x/2. Use these x-values to find the corresponding y-values on the curve y = tan(x).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative of a Function

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. For the function y = tan(x), the derivative is y' = sec^2(x). Understanding this concept is crucial for finding where the tangent line has a specific slope, which is necessary for determining where the normal line is parallel to a given line.

Normal Line to a Curve

A normal line to a curve at a given point is a line perpendicular to the tangent line at that point. If the slope of the tangent line is m, the slope of the normal line is -1/m. In this problem, we need to find where the normal line has a slope of -1/2, which means the tangent line must have a slope of 2.

Parallel Lines

Parallel lines have the same slope. In this problem, the normal line must be parallel to the line y = -x/2, which has a slope of -1/2. By setting the slope of the normal line equal to -1/2, we can find the points on the curve where this condition is satisfied, helping us solve the problem.