79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

검증된 단계별 안내

First, understand the problem: We need to find the equations of the tangent and normal lines to the curve given by \(x^4 = 2x^2 + 2y^2\) at the point \((x_0, y_0) = (2, 2)\).

To find the tangent line, we need the derivative of the curve. Start by implicitly differentiating the equation \(x^4 = 2x^2 + 2y^2\) with respect to \(x\). Use the chain rule for \(y^2\) since \(y\) is a function of \(x\).

After differentiating, solve for \(\frac{dy}{dx}\), which represents the slope of the tangent line at any point \((x, y)\) on the curve.

Substitute \((x_0, y_0) = (2, 2)\) into the derivative to find the slope of the tangent line at this specific point.

Use the point-slope form of a line, \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in the previous step, to write the equation of the tangent line. The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent line's slope. Use this to write the equation of the normal line.

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주요 개념

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

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. It represents the instantaneous rate of change of the function at that point, which can be found using the derivative. For the curve defined by the equation, the slope of the tangent line can be calculated by differentiating the equation implicitly.

Normal Line

The normal line at a point on a curve is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the slope of the tangent line. Understanding the normal line is essential for visualizing how the curve behaves at that point, and it can also be derived from the tangent line's slope.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated. In this case, the equation x⁴ = 2x² + 2y² involves both x and y. By differentiating both sides with respect to x, we can find dy/dx, which is necessary to determine the slopes of the tangent and normal lines.

79–82. {Use of Tech} Visualizing tangent and normal linesb. Graph the tangent and normal lines on the given graph.x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)