Textbook Question
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
d. Verify that the results of parts (a) and (c) are consistent.
4. Applications of Derivatives
Implicit Differentiation
7:30 minutesProblem 3.8.80b
Textbook Question
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)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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