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Chapter 7: Problem 2

Let \(g(x, y)=\sqrt{x^{2}+2 y^{2}}\). Compute \(g(1,1), g(0,-1)\), and \(g(a, b)\).

Short Answer

Expert verified
g(1, 1) = \sqrt{3}, g(0, -1) = \sqrt{2}, g(a, b) = \sqrt{a^2 + 2b^2}

Step by step solution

01

Understanding the Function

Given the function, we need to compute the values at different points. The function is defined as \( g(x, y) = \sqrt{x^{2} + 2y^{2}} \). We will use this formula to compute each required value.
02

Compute \( g(1,1) \)

Substitute \( x = 1 \) and \( y = 1 \) into the function: \( g(1, 1) = \sqrt{1^{2} + 2 \cdot 1^{2}} \). Simplify: \[ g(1, 1) = \sqrt{1 + 2} = \sqrt{3} \].
03

Compute \( g(0,-1) \)

Substitute \( x = 0 \) and \( y = -1 \) into the function: \( g(0, -1) = \sqrt{0^{2} + 2 \cdot (-1)^{2}} \). Simplify: \[ g(0, -1) = \sqrt{0 + 2} = \sqrt{2} \].
04

Compute \( g(a, b) \)

Substitute \( x = a \) and \( y = b \) into the function: \( g(a, b) = \sqrt{a^{2} + 2 b^{2}} \). Simplify: \[ g(a, b) = \sqrt{a^{2} + 2 b^{2}} \].

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Key Concepts

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

Function Evaluation
Function evaluation is the process of finding the output of a function given its inputs. In our exercise, we deal with the function defined by \( g(x, y) = \sqrt{x^2 + 2y^2} \). To evaluate this function at specific points, we substitute the given values for \( x \) and \( y \) into the function formula.

For example, to find \( g(1, 1) \), substitute \( x = 1 \) and \( y = 1 \) into the formula and simplify:

\( g(1, 1) = \sqrt{1^2 + 2 \times 1^2} = \sqrt{1 + 2} = \sqrt{3} \).

This process is straightforward but requires careful substitution and simplification. It's essential to correctly follow these steps to ensure accurate function evaluations at any given point.
Square Root Function
The square root function is a very important concept in mathematics. It is denoted as \( \sqrt{...} \) and represents a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

In the context of our function \( g(x, y) = \sqrt{x^2 + 2y^2} \), the square root is used to combine the squares of the variables \( x \) and \( y \) in a meaningful way.

Important properties of the square root function include:
  • It only produces non-negative results. For instance, \( \sqrt{4} \) is 2, not -2.
  • The argument of the square root (the expression inside it) must be non-negative since the square root of negative numbers involves complex numbers.
  • It's crucial for simplifying many mathematical and physical problems, especially in equations involving distance or magnitude.
Symmetry in Functions
Symmetry in functions refers to the property where the function exhibits a consistent pattern or invariance when input values are changed in specific ways.

There are various types of symmetry, but two common types are:
  • Even symmetry: A function \( f(x) \) is even if \( f(x) = f(-x) \). This means the function is symmetric about the y-axis.
  • Odd symmetry: A function \( f(x) \) is odd if \( f(x) = -f(-x) \). This means the function is symmetric about the origin.
Regarding our example \( g(x, y) = \sqrt{x^2 + 2y^2} \), we can explore if it has any symmetry. Since \( g(-x, y) = \sqrt{(-x)^2 + 2y^2} = \sqrt{x^2 + 2y^2} = g(x, y) \), the function is symmetric with respect to changing the sign of \( x \).

This property can be useful in simplifying calculations or understanding the behavior of the function in various regions.

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Most popular questions from this chapter

The productivity of a country is given by \(f(x, y)=300 x^{2 / 3} y^{1 / 3}\), where \(x\) and \(y\) are the amount of labor and capital. (a) Compute the marginal productivities of labor and capital when \(x=125\) and \(y=64\). (b) Use part (a) to determine the approximate effect on productivity of increasing capital from 64 to 66 units. while keeping labor fixed at 125 units. (c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?

Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for each of the following functions. $$ f(x, y)=\frac{x-y}{x+y} $$

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of \(f(x, y)\) at each of these points. If the second-derivative test is inconclusive, so state. $$ f(x, y)=x^{3}-y^{2}-3 x+4 y $$

A monopolist manufactures and sells two competing products, I and II, that cost \(\$ 30\) and \(\$ 20\) per unit, respectively, to produce. The revenue from marketing \(x\) units of product \(\mathrm{I}\) and \(y\) units of product \(\mathrm{II}\) is \(98 x+112 y-.04 x y-.1 x^{2}-.2 y^{2} .\) Find the values of \(x\) and \(y\) that maximize the monopolist's profits.

Find the point on the parabola \(y=x^{2}\) that has minimal distance from the point \(\left(16, \frac{1}{2}\right)\). [See Fig. 2(b).] [Suggestion: If \(d\). denotes the distance from \((x, y)\) to \(\left(16, \frac{1}{2}\right)\), then \(d^{2}=(x-16)^{2}+\left(y-\frac{1}{2}\right)^{2} .\) If \(d^{2}\) is minimized, then \(d\) will be minimized.]
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