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Chapter 3: Problem 71

The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots.\) c. Use the functions found in part (b) to graph the given equation. \(x^{4}=2\left(x^{2}-y^{2}\right)\) (eight curve)

Short Answer

Expert verified
Answer: The graph of the implicitly defined equation \(x^4 = 2(x^2 - y^2)\) has a shape resembling the number 8, hence the name "eight curve".

Step by step solution

01

(a) Implicit Differentiation

Given the equation \(x^4 = 2(x^2 - y^2)\), we need to find the derivative of y with respect to x using implicit differentiation. To do this, we'll differentiate both sides of the equation with respect to x, treating y as a function of x. Differentiating the left-hand side with respect to x: \(\frac{d}{dx}(x^4) = 4x^3\) Argument:The chain rule is \(\frac{d(u^2)}{dx} = 2u \cdot \frac{du}{dx}\) Differentiating the right-hand side with respect to x: \(\frac{d}{dx}(2(x^2 - y^2)) = 4x - 4y \cdot \frac{dy}{dx}\) Now, we have: \(4x^3=4x - 4y\frac{dy}{dx}\) Solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx}=\frac{4x^3-4x}{4y}\) \(\frac{dy}{dx}=\frac{x^3-x}{y}\)
02

(b) Solve for y

Now we need to solve the given equation for y to identify the implicitly defined functions. \(x^4 = 2(x^2 - y^2)\) Rearrange the equation to get y-squared terms on one side of the equation: \(y^2= x^2 - \frac{x^4}{2}\) Now, take the square root to isolate y: \(y = \pm\sqrt{x^2 - \frac{x^4}{2}}\) So, we have the implicitly defined functions \(y=f_1(x)\) and \(y=f_2(x)\), where: \(f_1(x) = \sqrt{x^2 - \frac{x^4}{2}}\) \(f_2(x) = -\sqrt{x^2 - \frac{x^4}{2}}\)
03

(c) Graph the implicitly defined functions

To graph the equation \(x^4 = 2(x^2 - y^2)\) using the functions \(f_1(x)\) and \(f_2(x)\), plot the functions \(y = f_1(x)\) and \(y = f_2(x)\) on the same set of axes. 1. For \(f_1(x) = \sqrt{x^2 - \frac{x^4}{2}}\), the graph will be the upper part of the eight curve, and will be symmetric with respect to the x-axis. 2. For \(f_2(x) = -\sqrt{x^2 - \frac{x^4}{2}}\), the graph will be the lower part of the eight curve, and it will also be symmetric with respect to the x-axis. Combine the graphs of these two functions on the same axes to get the entire graph of the implicitly defined equation. The final graph will have a shape resembling the number 8, hence the name "eight curve".

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Implicitly Defined Functions
An implicitly defined function arises when a relationship between variables is represented by an equation, rather than expressing one variable explicitly in terms of another.
This means that rather than solving for one variable, such as y in terms of x, the function remains tucked within an equation.

For example, in the equation \(x^4 = 2(x^2 - y^2)\), y is not isolated or expressed as a neat function of x, like \(y = f(x)\). Instead, x and y are entangled within the equation.
Solving this often requires sophisticated techniques like implicit differentiation or algebraic manipulation to follow a function's behavior within these confines.
  • Equation Appearance: Both variables (x and y) appear intertwined in a single equation.
  • Complexity: Solving or graphing such relationships can require advanced calculus or algebra.
Graphing Implicit Functions
Graphing implicit functions involves plotting points that satisfy the equation, even if the function isn鈥檛 explicitly defined.
For the equation \(x^4 = 2(x^2 - y^2)\), it defines a shape that resembles an "eight" or infinity symbol after calculation.

In practice, once we solve for y, we find functions like \(y=\sqrt{x^2-\frac{x^4}{2}}\) and its negative counterpart.
The graph of these functions is symmetrical and lies in the plane such that the two parts reflect around the x-axis.
  • Symmetry: Often, these shapes are symmetrical in nature.
  • Compound Graphs: A single equation may yield multiple graphs or shapes.
Chain Rule
The chain rule is a fundamental derivative technique used in calculus to differentiate compositions of functions.
It states that if you have a composition of two or more functions, you take the derivative of the outer function, keeping the inner function unchanged, and multiply it by the derivative of the inner function.

In implicit differentiation, the chain rule helps us differentiate any implicit function, especially when y is treated as a function of x.
For example: Differentiating \(y^2\) with respect to x gives \(2y \cdot \frac{dy}{dx}\) by the chain rule. This finds the rate of change of y in terms of x even when this direct relationship isn't explicit.
  • Composition Focus: Useful when functions are nested within each other, like \(u\) in \(u^2\).
  • Essential for Implicit Derivatives: Differentiation in implicit equations often relies on it.
Solving Equations for Functions
To transform an implicitly defined equation into explicit functions, you must rearrange and manipulate the equation.
You'll isolate one of the variables, often by moving terms around and taking roots or applying inverses.

In the exercise's equation, \(x^4 = 2(x^2 - y^2)\), solving for y involved rearranging to \(y^2 = x^2 - \frac{x^4}{2}\), followed by taking the square root.
This yields explicit forms, \(y = \pm\sqrt{x^2 - \frac{x^4}{2}}\), representing two functions: one positive and one negative.
  • Manipulation: Different techniques might be needed: factoring, substation, or simplification.
  • Multiple Functions: Equations can yield more than one function.

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Most popular questions from this chapter

a. Differentiate both sides of the identity \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\) to prove that \(\sin 2 t=2 \sin t \cos t\) b. Verify that you obtain the same identity for \(\sin 2 t\) as in part (a) if you differentiate the identity \(\cos 2 t=2 \cos ^{2} t-1\) c. Differentiate both sides of the identity \(\sin 2 t=2 \sin t \cos t\) to prove that \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\)

Recall that \(f\) is even if \(f(-x)=f(x),\) for all \(x\) in the domain of \(f,\) and \(f\) is odd if \(f(-x)=-f(x),\) for all \(x\) in the domain of \(f\) a. If \(f\) is a differentiable, even function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither. b. If \(f\) is a differentiable, odd function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither.

Suppose an object of mass \(m\) is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position \(y=0\) when the mass hangs at rest. Suppose you push the mass to a position \(y_{0}\) units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$ where \(k>0\) is a constant measuring the stiffness of the spring (the larger the value of \(k\), the stiffer the spring) and \(y\) is positive in the upward direction. A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by \(y=10 e^{-t / 2} \cos \frac{\pi t}{8}\) a. Graph the displacement function. b. Compute and graph the velocity of the mass, \(v(t)=y^{\prime}(t)\) c. Verify that the velocity is zero when the mass reaches the high and low points of its oscillation.

Tangency question It is easily verified that the graphs of \(y=x^{2}\) and \(y=e^{x}\) have no point of intersection (for \(x>0\) ), while the graphs of \(y=x^{3}\) and \(y=e^{x}\) have two points of intersection. It follows that for some real number \(2 < p < 3,\) the graphs of \(y=x^{p}\) and \(y=e^{x}\) have exactly one point of intersection (for \(x > 0) .\) Using analytical and/or graphical methods, determine \(p\) and the coordinates of the single point of intersection.

Assume \(f\) is a differentiable function whose graph passes through the point \((1,4) .\) Suppose \(g(x)=f\left(x^{2}\right)\) and the line tangent to the graph of \(f\) at (1,4) is \(y=3 x+1 .\) Find each of the following. a. \(g(1)\) b. \(g^{\prime}(x)\) c. \(g^{\prime}(1)\) d. An equation of the line tangent to the graph of \(g\) when \(x=1\)
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