79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

검증된 단계별 안내

First, understand the given equation: \((x^2 + y^2)^2 = \frac{25}{3} (x^2 - y^2)\). This is the equation of a lemniscate of Bernoulli, a type of figure-eight curve.

To find the tangent line at the point \((x_0, y_0) = (2, -1)\), we need to differentiate the given equation implicitly with respect to \(x\). This involves using the chain rule and product rule.

After differentiating, substitute \(x = 2\) and \(y = -1\) into the derivative to find the slope of the tangent line at this point.

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 \((x_1, y_1) = (2, -1)\), to write the equation of the tangent line.

The normal line is perpendicular to the tangent line. Therefore, its slope is the negative reciprocal of the tangent line's slope. Use the point-slope form again with this new slope to find the equation of the normal line.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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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. 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. For the given curve, the tangent line can be determined by calculating the derivative of the implicit function at the specified point (2, -1).

Normal Line

The normal line 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. To graph the normal line, one must first find the slope of the tangent line at the point of interest and then use this slope to determine the equation of the normal line, which will intersect the curve at the same point.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y. In this case, the equation (x²+y²)² = 25/3 (x²-y²) defines y as a function of x. By differentiating both sides with respect to x and applying the chain rule, one can find dy/dx, which is essential for determining 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.(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)