Problem 58
Question
Lemniscate (a) The curve with equation \(y^{2}=x^{2}-x^{4}\) is shaped like the numeral eight. Find \(\frac{d y^{2}}{d x}\) at \(\left(\frac{1}{2}, \frac{1}{4} \sqrt{3}\right)\). (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately; that is, graph $$ \begin{array}{l} y_{1}=\sqrt{x^{2}-x^{4}} \\ y_{2}=-\sqrt{x^{2}-x^{4}} \end{array} $$ Choose the viewing rectangle \(-2 \leq x \leq 2,-1 \leq y \leq 1\)
Step-by-Step Solution
Verified Answer
(a) \(\frac{dy^2}{dx} = \frac{1}{\sqrt{3}}\) at \(\left(\frac{1}{2}, \frac{1}{4}\sqrt{3}\right)\). (b) Graph the upper and lower halves separately.
1Step 1: Understand the Given Equation
The equation provided is \(y^2 = x^2 - x^4\). This describes a curve resembling the numeral eight, known as a lemniscate. We are tasked with finding the derivative of \(y^2\) with respect to \(x\) at a specific point \(\left(\frac{1}{2}, \frac{1}{4} \sqrt{3}\right)\).
2Step 2: Differentiate the Equation Implicitly
To find \(\frac{d y^2}{d x}\), differentiate the equation \(y^2 = x^2 - x^4\) with respect to \(x\). This yields \(2y \frac{dy}{dx} = 2x - 4x^3\).
3Step 3: Isolate \(\frac{dy}{dx}\)
Rearrange the equation from Step 2 to express \(\frac{dy}{dx}\): \(\frac{dy}{dx} = \frac{2x - 4x^3}{2y}\).
4Step 4: Substitute the Given Point into the Derivative
Substitute \(x = \frac{1}{2}\) and \(y = \frac{1}{4}\sqrt{3}\) into the expression for \(\frac{dy}{dx}\):\[\frac{dy}{dx} = \frac{2\left(\frac{1}{2}\right) - 4\left(\frac{1}{2}\right)^3}{2\left(\frac{1}{4}\sqrt{3}\right)} = \frac{1 - \frac{1}{2}}{\frac{1}{2}\sqrt{3}} = \frac{\frac{1}{2}}{\frac{1}{2}\sqrt{3}} = \frac{1}{\sqrt{3}}.\]
5Step 5: Graph the Implicit Function or its Halves
Using a graphing calculator, plot the equations separately for the upper half \(y_1 = \sqrt{x^2 - x^4}\) and the lower half \(y_2 = -\sqrt{x^2 - x^4}\). Set the calculator's viewing rectangle to \(-2 \leq x \leq 2\) and \(-1 \leq y \leq 1\). This will allow you to visualize the lemniscate shape.
Key Concepts
implicit differentiationgraphing implicit functionsderivative calculation
implicit differentiation
To understand implicit differentiation, let's first consider what an implicit function is. Unlike explicit functions, where the relationship between variables is straightforward, implicit functions have variables tangled together in an equation. For example, the equation \( y^2 = x^2 - x^4 \) is implicit, meaning \( y \) is not isolated on one side of the equation.
When faced with implicit functions, we use implicit differentiation to compute derivatives. Instead of solving for \( y \) and then differentiating, we differentiate each term directly with respect to \( x \). During this process, use the chain rule. For instance, differentiating \( y^2 \) with respect to \( x \) involves applying the chain rule: the derivative of \( y^2 \) is \( 2y \cdot \frac{dy}{dx} \).
After differentiating the entire equation \( y^2 = x^2 - x^4 \) with respect to \( x \), we are left with \( 2y \cdot \frac{dy}{dx} = 2x - 4x^3 \). This highlights the interconnected nature of \( x \) and \( y \) in implicit equations and how they require simultaneous handling through implicit differentiation.
When faced with implicit functions, we use implicit differentiation to compute derivatives. Instead of solving for \( y \) and then differentiating, we differentiate each term directly with respect to \( x \). During this process, use the chain rule. For instance, differentiating \( y^2 \) with respect to \( x \) involves applying the chain rule: the derivative of \( y^2 \) is \( 2y \cdot \frac{dy}{dx} \).
After differentiating the entire equation \( y^2 = x^2 - x^4 \) with respect to \( x \), we are left with \( 2y \cdot \frac{dy}{dx} = 2x - 4x^3 \). This highlights the interconnected nature of \( x \) and \( y \) in implicit equations and how they require simultaneous handling through implicit differentiation.
graphing implicit functions
Graphing implicit functions can be a bit tricky since these equations, like the lemniscate shape given by \( y^2 = x^2 - x^4 \), cannot directly yield \( y \) values without manipulation. A common approach is to split the equation into functions that clearly define either the upper or lower portion of the graph.
For the lemniscate, this means defining \( y_1 = \sqrt{x^2 - x^4} \) for the upper half and \( y_2 = -\sqrt{x^2 - x^4} \) for the lower half. This method essentially converts the complex implicit relationship into two separate, standard functions that are easier to plot.
Using a graphing calculator, plot these equations within a specified viewing rectangle, such as \(-2 \leq x \leq 2\) and \(-1 \leq y \leq 1\). This setting ensures that you capture the full image of the lemniscate, helping visualize its distinctive eight-like form. While some calculators struggle with implicit curves, graphing the separated functions individually is an effective workaround.
For the lemniscate, this means defining \( y_1 = \sqrt{x^2 - x^4} \) for the upper half and \( y_2 = -\sqrt{x^2 - x^4} \) for the lower half. This method essentially converts the complex implicit relationship into two separate, standard functions that are easier to plot.
Using a graphing calculator, plot these equations within a specified viewing rectangle, such as \(-2 \leq x \leq 2\) and \(-1 \leq y \leq 1\). This setting ensures that you capture the full image of the lemniscate, helping visualize its distinctive eight-like form. While some calculators struggle with implicit curves, graphing the separated functions individually is an effective workaround.
derivative calculation
Calculating the derivative \( \frac{dy}{dx} \) of an implicit function requires you to isolate \( \frac{dy}{dx} \) after differentiation. Once you've implicitly differentiated an equation like \( y^2 = x^2 - x^4 \), you end up with an equation involving \( \frac{dy}{dx} \). In our case, it's \( 2y \cdot \frac{dy}{dx} = 2x - 4x^3 \).
To solve for \( \frac{dy}{dx} \), rearrange the terms: \( \frac{dy}{dx} = \frac{2x - 4x^3}{2y} \). It’s essential to plug in a specific point to find the slope of the tangent at that point. For example, inserting \( x = \frac{1}{2} \) and \( y = \frac{1}{4} \sqrt{3} \) makes the derivative \( \frac{1}{\sqrt{3}} \).
This result indicates the rate of change of \( y \) with respect to \( x \) at that point, representing the slope of the tangent to the curve at \( \left( \frac{1}{2}, \frac{1}{4} \sqrt{3} \right) \). Ensuring clarity in these calculations is key to understanding the role of derivatives in describing the behavior of implicit functions like the lemniscate.
To solve for \( \frac{dy}{dx} \), rearrange the terms: \( \frac{dy}{dx} = \frac{2x - 4x^3}{2y} \). It’s essential to plug in a specific point to find the slope of the tangent at that point. For example, inserting \( x = \frac{1}{2} \) and \( y = \frac{1}{4} \sqrt{3} \) makes the derivative \( \frac{1}{\sqrt{3}} \).
This result indicates the rate of change of \( y \) with respect to \( x \) at that point, representing the slope of the tangent to the curve at \( \left( \frac{1}{2}, \frac{1}{4} \sqrt{3} \right) \). Ensuring clarity in these calculations is key to understanding the role of derivatives in describing the behavior of implicit functions like the lemniscate.
Other exercises in this chapter
Problem 57
Differentiate the functions with respect to the independent variable. $$ f(x)=\log \left(x^{3}-3 x\right) $$
View solution Problem 58
Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=2-|x-3| $$
View solution Problem 58
In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\csc \left(3-x^{2}\right)}{1-x^{2}} $$
View solution Problem 58
Find the length of the subtangent to the curve \(y=\exp \left[x^{2}\right]\) at the point \(\left(2, e^{4}\right)\).
View solution