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Chapter 4: Problem 20

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$f(x, y)=x^{2}-y ; c=1,2$$

Short Answer

Expert verified
The level curves are \( y = x^2 - 1 \) for \( c=1 \) and \( y = x^2 - 2 \) for \( c=2 \).

Step by step solution

01

Understand Level Curves

Level curves of a function \( f(x, y) \) are sets of points \( (x, y) \) such that \( f(x, y) = c \) for some constant \( c \). These curves display where the function takes on certain values and are useful for visualizing the behavior of the function in the plane.
02

Set the Function Equal to \( c \)

For \( f(x, y) = x^2 - y \), we need to find the level curves for \( c=1 \) and \( c=2 \). This means setting \( x^2 - y = c \).
03

Solve for \( y \) in Terms of \( x \) for \( c=1 \)

Set the function equal to 1: \( x^2 - y = 1 \). Solve for \( y \):\[ y = x^2 - 1 \]This represents the level curve for \( c=1 \).
04

Solve for \( y \) in Terms of \( x \) for \( c=2 \)

Set the function equal to 2: \( x^2 - y = 2 \). Solve for \( y \):\[ y = x^2 - 2 \]This represents the level curve for \( c=2 \).
05

Describe the Level Curves

The level curve for \( c=1 \) is the curve \( y = x^2 - 1 \), a parabola opening upwards with its vertex at \( (0, -1) \). The level curve for \( c=2 \) is \( y = x^2 - 2 \), which is also a parabola opening upwards with its vertex at \( (0, -2) \).

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Key Concepts

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

Contour Plots
Contour plots are a way to represent three-dimensional surfaces on a two-dimensional plane. Imagine a topographic map: instead of showing hills and valleys in 3D, it uses lines to represent points of equal elevation. Similarly, in mathematics, contour plots use level curves to describe how a function behaves over a range of values.

These plots are especially useful when dealing with multivariable functions. For the function given in the exercise, the level curves are found where the equation takes a constant value, such as 1 or 2. The contour plot would illustrate these curves, showing how the function's output changes over the plane.

Key points for contour plots:
  • Each contour line indicates where the function has the same output.
  • The spacing between lines can indicate the steepness of the function. Closer lines mean steeper changes.
  • They are essential for visualizing complex functions without needing a 3D graph.
Multivariable Functions
Multivariable functions are functions with more than one input. They often look like this: \( f(x, y) \), where the function depends on variables \( x \) and \( y \). These functions are prevalent in many fields, including engineering, physics, and economics because they model scenarios where multiple factors influence an outcome.

In the exercise, the function \( f(x, y) = x^2 - y \) is a simple multivariable function. It illustrates how changes in both \( x \) and \( y \) affect the function's overall output. By using level curves, we can see constant output values across different inputs.

Here are some basics of multivariable functions:
  • They often require more advanced math techniques, like calculus, for analyzing.
  • They are represented graphically by surfaces rather than lines or curves.
  • Level curves are a handy way to simplify their analysis.
Parabolas
Parabolas are a type of curve that you'll often encounter in mathematics, particularly in quadratic equations. They have a characteristic "U" shape and can open either upwards or downward depending on the equation.

In this specific exercise, both level curves \( y = x^2 - 1 \) and \( y = x^2 - 2 \) are parabolas. Here's what's interesting:
  • Both parabolas open upwards since the terms \( x^2 \) are positive.
  • These curves are similar, just shifted vertically on the graph. For \( c=1 \), the vertex is at \( (0, -1) \), and for \( c=2 \), it's at \( (0, -2) \).
  • Parabolas are symmetrical along their vertex, meaning if you fold the graph at the vertex, both sides would match perfectly.
Understanding parabolas in the context of level curves helps visualize how changes in \( c \) shift the graph. The lower the vertex, the higher the value of \( c \). It's a simple yet powerful way to transform our understanding of algebraic functions into visual insights.

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Most popular questions from this chapter

Find the point in the plane \(2 x-y+2 z=16\) that is closest to the origin.

Use the information provided to solve the problem. If \(f(x, y)=x y, x=r \cos \theta, \quad\) and \(\quad y=r \sin \theta,\) find \(\frac{\partial f}{\partial r}\) and express the answer in terms of \(r\) and \(\theta\).

If \(\quad w=\sin (x y z), x=1-3 t, y=e^{1-t}, \quad\) and \(z=4 t,\) find \(\frac{\partial w}{\partial t} .\)

The volume of a right circular cylinder is given by \(V(x, y)=\pi x^{2} y,\) where \(x\) is the radius of the cylinder and \(y\) is the cylinder height. Suppose \(x\) and \(y\) are functions of \(t\) given by \(x=\frac{1}{2} t\) and \(y=\frac{1}{3} t\) so that \(x\) and \(y\) are both increasing with time. How fast is the volume increasing when \(x=2\) and \(y=5 ?\)

Let \(z=e^{1-x y}, x=t^{1 / 3},\) and \(y=t^{3} .\) Find \(\frac{d z}{d t}\)
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