Question

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

Accepted Answer

First, understand the problem: We need to find the equations of the tangent and normal lines to the curve at a specific point. The curve is given by the equation $$(x^2 + y^2)^2 = \frac{25}{3}(x^2 - y^2)$$, and the point is $$(x_0, y_0) = (2, -1). To $$find the tangent line, we need the derivative of the curve at the point $$(2, -1). $$Start by differentiating the given equation implicitly with respect to $$x. $$This involves using the chain rule and implicit differentiation techniques. After differentiating, substitute $$x = 2$$ and $$y = -1$$ into the derivative to find the slope $$m of $$the tangent line at the point $$(2, -1). $$Use the point-slope form of the equation of a line, $$y - y_0 = m(x - x_0)$$, to write the equation of the tangent line. Here, $$m is $$the slope found in the previous step, and $$(x_0, y_0) is$$ $$(2, -1). $$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 write the equation of the normal line at the point $$(2, -1).$$

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Tangent Line

Normal Line

Implicit Differentiation

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Tangent Line

Normal Line

Implicit Differentiation