In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

(y - x)² = 2x + 4, (6, 2)

검증된 단계별 안내

First, recognize that the given equation \((y - x)^2 = 2x + 4\) is implicitly defined. To find the tangent line, we need to differentiate this equation with respect to \(x\).

Apply implicit differentiation to \((y - x)^2 = 2x + 4\). Differentiate both sides with respect to \(x\), remembering to use the chain rule for the left side: \(2(y - x)(\frac{dy}{dx} - 1) = 2\).

Solve the differentiated equation for \(\frac{dy}{dx}\) to find the slope of the tangent line at any point \((x, y)\). Substitute \(x = 6\) and \(y = 2\) into the differentiated equation to find the specific slope at the point \((6, 2)\).

Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope found in the previous step and \((x_1, y_1)\) is the point \((6, 2)\), to write the equation of the tangent line.

To find the normal line, use the fact that the slope of the normal line is the negative reciprocal of the slope of the tangent line. Use the point-slope form again with this new slope to write the equation of the normal line at the point \((6, 2)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of functions that are not explicitly solved for one variable in terms of another. In this problem, the equation (y - x)² = 2x + 4 involves both x and y, requiring implicit differentiation to find dy/dx, which is essential for determining the slope of the tangent line at the given point.

Tangent Line

The tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Its slope is equal to the derivative of the curve at that point. For the equation (y - x)² = 2x + 4 at the point (6, 2), the tangent line can be found using the derivative obtained through implicit differentiation.

Normal Line

The normal line to a curve at a given point is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the slope of the tangent line. Once the slope of the tangent line is determined, the normal line can be calculated, providing insight into the geometric properties of the curve at the specified point.

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.(y - x)² = 2x + 4, (6, 2)