/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 For \(f(x, y, z)=x^{2}-y^{2}+z^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 7

For \(f(x, y, z)=x^{2}-y^{2}+z^{2},\) find \(f(-1,2,3)\) and \(f(2,-1,3)\)

Short Answer

Expert verified
For \( f(-1, 2, 3) = 6 \) and for \( f(2, -1, 3) = 12 \).

Step by step solution

01

- Understand the function

First, it's important to understand the given function: \[ f(x, y, z) = x^2 - y^2 + z^2 \]
02

- Substitute values for the first set of points

Substitute \(x = -1\), \(y = 2\), and \(z = 3\) into the function. The function becomes: \[ f(-1, 2, 3) = (-1)^2 - 2^2 + 3^2 \]
03

- Calculate the first set of points

Calculate the value: \[ (-1)^2 = 1, \]\[ 2^2 = 4, \]\[ 3^2 = 9. \]Then, \[ 1 - 4 + 9 = 6. \]
04

- Final value for the first set

Thus, \( f(-1, 2, 3) = 6 \).
05

- Substitute values for the second set of points

Substitute \(x = 2\), \(y = -1\), and \(z = 3\) into the function. The function becomes: \[ f(2, -1, 3) = 2^2 - (-1)^2 + 3^2 \]
06

- Calculate the second set of points

Calculate the value: \[ 2^2 = 4, \]\[ (-1)^2 = 1, \]\[ 3^2 = 9. \]Then, \[ 4 - 1 + 9 = 12. \]
07

- Final value for the second set

Thus, \( f(2, -1, 3) = 12 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Function Evaluation
Evaluating a function means you plug in specific values for the variables in the function and calculate the result. For example, in our original function \[ f(x, y, z) = x^2 - y^2 + z^2 \], we evaluated the function by substituting the values of -1, 2, and 3 for x, y, and z respectively. To do this, replace the variables in the function with the given numbers. Next, follow the arithmetic operations to solve the function step-by-step. This process is especially useful in multivariable calculus, as it allows you to find specific function values at given points.
Partial Derivatives
Partial derivatives are used to find the rate at which a function changes with respect to one of its variables, holding the others constant. Though this exercise didn't explicitly involve partial derivatives, understanding them is crucial for deeper understanding. To find a partial derivative, differentiate the function with respect to one variable while treating the other variables as constants. For our function \[ f(x, y, z) = x^2 - y^2 + z^2 \], the partial derivative with respect to x is \[ \frac{\text{\text{鈭倉}f}{\text{\text{鈭倉}x} = 2x \]. Similarly, the partial derivatives with respect to y and z can be found by differentiating with respect to these variables.
Coordinate Substitution
Coordinate substitution involves plugging specific values for the variables into a function to evaluate it at a particular point. In our given function \[ f(x, y, z) = x^2 - y^2 + z^2 \], you performed coordinate substitution twice: once with the coordinates (-1, 2, 3) and another time with (2, -1, 3). The substitution steps make it simple to understand how changing the input values affects the output of the function. This technique is straightforward but powerful in multivariable calculus as it helps to visualize how functions behave over their domains.
Algebraic Manipulation
Algebraic manipulation is about rearranging and simplifying expressions to make them easier to work with. When you substituted (-1, 2, 3) into the function \[ f(x, y, z) = x^2 - y^2 + z^2 \], the function became \[ f(-1, 2, 3) = (-1)^2 - 2^2 + 3^2 \]. You simplified this step-by-step: \[ (-1)^2 = 1, \ 2^2 = 4, \ 3^2 = 9, \ 1 - 4 + 9 = 6 \]. This method involves basic arithmetic operations and ensures the problem stays manageable. Simplifying at each step helps prevent mistakes and makes it easier to understand the process.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that \(a\) continuous random variable has a joint probability density function given by $$f(x, y)=x^{2}+\frac{1}{3} x y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$. Find the probability that a point \((x, y)\) is in the region bounded by \(0 \leq x \leq \frac{1}{2}, 1 \leq y \leq 2,\) by evaluating the integral $$\int_{1}^{2} \int_{0}^{1 / 2} f(x, y) d x d y$$

Evaluate. $$\int_{-1}^{3} \int_{1}^{2} x^{2} y d y d x$$

Nursing facilities. A study of Texas nursing homes found that the annual profit \(P\) (in dollars) of profit-seeking, independent nursing homes in urban locations is modeled by the function $$P(w, r, s, t)=0.007955 w^{-0.638} r^{1.038} s^{0.873} t^{2.468}$$ In this function, \(w\) is the average hourly wage of nurses and aides (in dollars), \(r\) is the occupancy rate (as a percentage), s is the total square footage of the facility, and \(t\) is the Texas Index of Level of Effort (TILE), a number between 1 and 11 that measures state Medicaid reimbursement. (Source: K.J. Knox, E. C. Blankmeyer, and J. R. Stutzman, "Relative Economic Efficiency in Texas Nursing Facilities," Journal of Economics and Finance, Vol. \(23,199-213(1999) .\) Use the preceding information for Exercises 43 and 44 A profit-seeking, independent Texas nursing home in an urban setting has nurses and aides with an average hourly wage of \(\$ 20\) an hour, a TILE of \(8,\) an occupancy rate of \(70 \%,\) and \(400,000 \mathrm{ft}^{2}\) of space. a) Estimate the nursing home's annual profit. b) Find the four partial derivatives of \(P\) c) Interpret the meaning of the partial derivatives found in part (b).

Temperature-humidity heat index. In the summer, humidity interacts with the outdoor temperature, making a person feel hotter because of reduced heat loss from the skin caused by higher humidity. The temperature-humidity index, \(T_{\mathrm{h}},\) is what the temperature would have to be with no humidity in order to give the same heat effect. One index often used is given by $$T_{\mathrm{h}}=1.98 T-1.09(1-H)(T-58)-56.9$$ where \(T\) is the air temperature, in degrees Fahrenheit, and H is the relative humidity, expressed as a decimal. Find the temperature-humidity index in each case. Round to the nearest tenth of a degree. $$T=85^{\circ} \mathrm{F} \text { and } \mathrm{H}=60 \%$$

Do some research on the Cobb-Douglas production function, and explain how it was developed.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.