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Chapter 13: Problem 64

Let \(f(x, y)=2 \sqrt{x y}\) (a) Use a graphing utility to generate the level curve that passes through the point \((2,2) .\) (b) Generate the level curve of height 8 .

Short Answer

Expert verified
The level curve through (2,2) is \(xy=4\); the level curve of height 8 is \(xy=16\).

Step by step solution

01

Understand Level Curves

A level curve for the function \(f(x, y)\) is a set of points \((x, y)\) where \(f(x, y)\) is equal to a constant. For this function, \(f(x, y) = 2 \sqrt{x y}\). This means a level curve with a constant \(c\) is expressed as \(2 \sqrt{x y} = c\).
02

Determine the Level Curve through (2,2)

At the point \((2,2)\), calculate \(f(2,2)\) by substituting \(x=2\) and \(y=2\) into the equation \(f(x, y) = 2 \sqrt{x y}\):\[f(2,2) = 2 \sqrt{2 \times 2} = 2 \times 2 = 4\]Thus, the level curve passing through \((2,2)\) is \(2 \sqrt{x y} = 4\).
03

Simplify the Level Curve Equation

Starting from \(2 \sqrt{x y} = 4\), divide both sides by 2 to get \(\sqrt{x y} = 2\). Then, square both sides to eliminate the square root:\[x y = 4\].Therefore, the equation of the level curve is \(x y = 4\), forming a hyperbola.
04

Generate the Level Curve of Height 8

For the curve where \(f(x, y) = 8\), start with the equation \(2 \sqrt{x y} = 8\). Divide both sides by 2 to get \(\sqrt{x y} = 4\), and then square both sides to obtain:\[x y = 16\].This is another hyperbola, different from the previous one.

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Key Concepts

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

Multivariable Calculus
Multivariable calculus extends the concepts of single-variable calculus into two or more dimensions. This involves functions with several input variables, like our function here, \(f(x, y) = 2 \sqrt{x y}\).
Understanding such functions requires considering how changes in one variable impact the entire function, given the context of another variable.
This is where the concept of level curves becomes invaluable.
  • **Partial Derivatives:** In multivariable calculus, you often focus on the rate of change regarding a single variable by using partial derivatives while keeping the other variable constant.
  • **Level Curves:** A level curve represents the set of points for which the function holds a constant value, giving you a 'contour' of the function at that level.
  • **Applications:** Level curves are crucial in fields like physics and engineering, where they help in mapping elevation, pressure, or temperature distributions.
Hyperbolas
A hyperbola is a type of conic section obtained by slicing a double cone. In the context of our exercise involving level curves, hyperbolas represent the sets of solutions for the level curve equations \(x y = 4\) and \(x y = 16\).
These equations are in the standard form of a hyperbola.
  • **Definition:** A hyperbola consists of two curves mirroring each other, typically separated and extending indefinitely.
  • **Equation Forms:** A standard hyperbola oriented in the cartesian plane can be expressed as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(xy = k\), as in our example.
  • **Characteristics:** Hyperbolas have asymptotes, and these are straight lines that the hyperbola approaches infinitely.
Using algebraic manipulations like those we performed, you can transform the level curve formula into a more recognizable form, revealing the hyperbolic nature of these curves.
Graphing Utilities
Graphing utilities are tools that let you visualize the graphical representation of mathematical equations like level curves. These can be software applications or online tools that provide a visual platform to understand complex mathematical concepts.
For our exercise, these utilities are essential.
  • **Purpose:** They help in visualizing functions of two or more variables, effectively displaying multiple level curves for different heights.
  • **Features:** Modern graphing software can update graphs dynamically with input changes, provide 3D representations, and offer interactive exploration.
  • **Usage:** To graph our level curves \(x y = 4\) and \(x y = 16\), input the equations into a graphing utility, and it will render the hyperbolic curves.
By using such utilities, students gain deeper insights into how these curves look and behave in a 2D plane, enhancing their comprehension of multivariable functions.

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Most popular questions from this chapter

Confirm that the mixed second-order partial derivatives of \(f\) are the same. $$ f(x, y)=\ln \left(x^{2}+y^{2}\right) $$

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