Problem 3 - Calculus Early Transcendentals

Question

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?

Step-by-Step Solution

Answer

Both the \(x\)-coordinate and the \(y\)-coordinate are needed because the derivative (slope) of an implicitly defined function depends on both variables. Since we cannot express \(y\) explicitly as a function of \(x\), we must use both coordinates to find the slope of the tangent line at a specific point, ensuring accurate calculation and representation of the function's behavior.

Step 1: Understanding Implicitly Defined Functions

An implicitly defined function is a relation between variables where the dependent variable, \(y\), cannot be explicitly expressed as a function of the independent variable, \(x\). In other words, we can't isolate \(y\) on one side of the equation. As a result, it's challenging to use traditional methods such as finding the derivative, which is needed to calculate the slope of the tangent line.

Step 2: Differentiating Implicit Functions

In order to find the slope of the tangent line for an implicitly defined function, we need to differentiate the function with respect to \(x\). However, since we can't express \(y\) explicitly as a function of \(x\), we need to use the chain rule. To differentiate the implicit function, we differentiate both sides of the equation with respect to \(x\) and use the chain rule \((\frac{dy}{dx})\) for terms containing \(y\).

Step 3: Finding the Slope of the Tangent Line

After differentiating the implicit function, we'll have an expression that includes both \(x\) and \(y\) terms. To find the slope of the tangent line at a specific point \((x_0, y_0)\), we need to substitute both the \(x\)-coordinate and the \(y\)-coordinate into the expression, since the derivative (slope) depends on both variables.

Step 4: Why Both Coordinates are Needed

Both the \(x\)-coordinate and the \(y\)-coordinate are needed to find the slope of the tangent line at a point for an implicitly defined function because the derivative (slope) depends on both variables. Since we can't express \(y\) explicitly as a function of \(x\), we must use both coordinates to find the slope of the tangent line at a specific point. Otherwise, we won't be able to fully describe the function's behavior and calculate the slope accurately.