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Chapter 3: Problem 3

Why are both the \(x\) -coordinate and the \(y\) -coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function?

Short Answer

Expert verified
Answer: Both x and y coordinates are needed to find the slope of the tangent line at a point for an implicitly defined function because they impact the values of the partial derivatives when evaluating the expression for dy/dx. The slope of the tangent line is determined by evaluating the expression at a specific point (x, y), which requires both the x and y coordinates to accurately compute the slope.

Step by step solution

01

Differentiate both sides with respect to x

Start by differentiating both sides of the implicitly defined function F(x, y) = 0 concerning x. Remember to use the chain rule while differentiating with respect to y. \[\frac{dF}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y}\frac{dy}{dx}\] In this step, we implicitly differentiate the function; however, the values of x and y are not needed yet.
02

Rearrange the equation to find dy/dx

Next, rearrange the equation to find the explicit expression for dy/dx, the slope of the tangent line. \[\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}\] In this step, we have found the expression for the slope. However, we still need to evaluate this expression at a specific point (x, y) to find the slope of the tangent line.
03

Evaluate the expression at the point (x, y)

Now, evaluate the expression for dy/dx at the point (x, y) to get the slope of the tangent line at that specific point. \[\left.\frac{dy}{dx}\right|_{(x,y)} = -\frac{\frac{\partial F}{\partial x}(x,y)}{\frac{\partial F}{\partial y}(x,y)}\] In this final step, we needed both the x and y coordinates to compute the slope of the tangent line at the point (x, y). This step is crucial as both the \(x\)-coordinate and the \(y\)-coordinate impact the values of the partial derivatives, leading to the tangent line's slope.

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