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Chapter 15: Problem 8

Given the function \(g(x, y, z)=\frac{x+y}{z},\) evaluate \(g(1,5,3)\) and \(g(3,7,2)\).

Short Answer

Expert verified
Answer: The values of the function at the given points are \(g(1, 5, 3) = 2\) and \(g(3, 7, 2) = 5\).

Step by step solution

01

Write down the given function

The function we need to evaluate is \(g(x, y, z)=\frac{x+y}{z}\).
02

Substitute the first set of values

To find the value of \(g(1, 5, 3)\), substitute x = 1, y = 5, and z = 3 into the function: $$g(1, 5, 3)=\frac{1+5}{3}$$
03

Calculate the value for the first set

Now, we'll calculate the value: $$g(1, 5, 3)=\frac{6}{3}=2$$ Therefore, \(g(1, 5, 3) = 2\).
04

Substitute the second set of values

Next, to find the value of \(g(3, 7, 2)\), substitute x = 3, y = 7, and z = 2 into the function: $$g(3, 7, 2)=\frac{3+7}{2}$$
05

Calculate the value for the second set

Now, we'll calculate the value: $$g(3, 7, 2) = \frac{10}{2}=5$$ Therefore, \(g(3, 7, 2) = 5\). In conclusion, \(g(1, 5, 3) = 2\) and \(g(3, 7, 2) = 5\).

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Key Concepts

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

Evaluating Functions
Evaluating functions is a fundamental concept in mathematics, especially when dealing with multivariable functions. In simple terms, evaluating a function means finding its value for specific inputs. For the given problem, we have the function \( g(x, y, z) = \frac{x+y}{z} \). This tells us how to calculate a result when we know the values of \(x\), \(y\), and \(z\).

When evaluating a function:
  • Identify the variables in your function. Here, the variables are \(x\), \(y\), and \(z\).
  • Substitute the given numerical values into the function. For example, replacing \(x = 1\), \(y = 5\), \(z = 3\) gets us \( g(1, 5, 3) = \frac{1+5}{3} \).
  • Finally, perform the arithmetic operations to find the result.
Evaluating ensures you accurately determine a function鈥檚 output given particular inputs, an essential skill in mathematics and beyond.
Algebraic Operations
Algebraic operations involve mathematical procedures used to manipulate algebraic expressions. In this exercise, addition and division are the operations needed to evaluate the function \(g(x, y, z) = \frac{x+y}{z}\).

Here's how algebraic operations work in this context:
  • **Addition:** Sum the values of \(x\) and \(y\). For \(g(1,5,3)\), we first add: \(1 + 5 = 6\).

  • **Division:** Divide the sum by \(z\). So, \(\frac{6}{3} = 2\) is calculated next.
In algebra, remember the order of operations often referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). These principles guide how we handle multiple operations in a single expression, ensuring results are correctly computed.
Substitution Method
The substitution method is a technique used to simplify the evaluation of functions by replacing variables with their corresponding values. It鈥檚 widely used in solving equations and evaluating expressions.

Let's break down the substitution method:
  • **Identify Variables:** Start by identifying the function's variables, like \(x\), \(y\), and \(z\) in our exercise.
  • **Insert Given Values:** Substitute each identified variable with its specific value. For example, for \(g(3, 7, 2)\), replace \(x = 3\), \(y = 7\), and \(z = 2\), getting \(\frac{3+7}{2}\).

  • **Solve:** Proceed to solve the substituted expression to get your final result.
By substituting values effectively, you transform a generic function into a specific numerical problem, making it easier to compute and understand. This technique simplifies complex problems by turning them into straightforward calculations.

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Most popular questions from this chapter

Heron's formula The area of a triangle with sides of length \(a, b\) and \(c\) is given by a formula from antiquity called Heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where \(s=\frac{1}{2}(a+b+c)\) is the semiperimeter of the triangle. a. Find the partial derivatives \(A_{\sigma}, A_{b},\) and \(A_{c}\) b. A triangle has sides of length \(a=2, b=4, c=5 .\) Estimate the change in the area when \(a\) increases by \(0.03, b\) decreases by \(0.08,\) and \(c\) increases by 0.6 c. For an equilateral triangle with \(a=b=c,\) estimate the percent change in the area when all sides increase in length by \(p \% .\)

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective on the surface. $$p(x, y)=1-|x-1|+|y+1|$$

Looking ahead- tangent planes Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) .\) A point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The set of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=x^{2}+y^{2}+z^{2}-3=0 ; P(1,1,1)$$

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. $$w=f(u, x, y, z)=\frac{u+x}{y+z}$$

Find the points at which the following surfaces have horizontal tangent planes. $$z=\cos 2 x \sin y \text { in the region }-\pi \leq x \leq \pi,-\pi \leq y \leq \pi$$
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