Use Descartes' method from Problem 91 to find an equation of the line tangent to each graph at t

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Use Descartes' method from Problem 91 to find an equation of...

Key Concepts

Implicit Differentiation

Implicit differentiation is the process used to differentiate equations where y is not isolated on one side, such as curves defined by relations between x and y. By differentiating both sides of an equation with respect to x and applying the chain rule to terms involving y, one can solve for dy/dx, which represents the slope of the curve at any point.

Tangent Line

A tangent line to a curve at a given point is a line that just touches the curve at that point and has the same instantaneous slope as the curve itself. Finding the equation of the tangent line typically involves determining the slope through differentiation and then applying the point-slope form of a line.

Derivative at a Point

After obtaining the general derivative using implicit differentiation, evaluating it at a specific point on the curve provides the slope of the tangent line at that point. This step is crucial for accurately determining the tangent line's direction.

Equation of a Line

The point-slope form of the equation of a line is used to determine the tangent line once the slope and a point of tangency are known. The form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope derived from differentiation.

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