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Chapter 7: Problem 12

For each function, evaluate the given expression. $$ g(x, y)=\ln \left(x^{3}-y^{2}\right), \text { find } g(e, 0) $$

Short Answer

Expert verified
The value of \( g(e, 0) \) is 3.

Step by step solution

01

Substitute the Values into the Function

First, substitute the values of \( x = e \) and \( y = 0 \) into the function \( g(x, y) = \ln(x^3 - y^2) \). This gives us \( g(e, 0) = \ln(e^3 - 0^2) \).
02

Simplify the Expression

Now simplify the expression inside the natural logarithm. Compute \( e^3 \) and \( 0^2 \) which results in \( g(e, 0) = \ln(e^3) \).
03

Apply the Natural Logarithm Property

Recall the property of logarithms that \( \ln(a^b) = b \ln(a) \). Applying this property here gives us \( g(e, 0) = 3 \ln(e) \).
04

Evaluate the Natural Logarithm of e

Since \( \ln(e) = 1 \), the expression simplifies to \( g(e, 0) = 3 \cdot 1 = 3 \).

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a special logarithm where the base is the number \( e \). The number \( e \) is approximately 2.71828, and it is a fundamental constant in mathematics, similar to \( \pi \). Logarithms help us transform multiplicative relationships into additive ones, making complex calculations easier. In simpler terms:
  • The natural logarithm \( \ln(a) \) answers the question: "To what power must \( e \) be raised to yield \( a \)?"
  • For example, \( \ln(e) = 1 \) because \( e^1 = e \).
  • Similarly, \( \ln(1) = 0 \) since any number to the power of zero is 1.
Understanding the natural logarithm is crucial because of its applications in various fields like calculus, finance, and biology.
Functions of Two Variables
Functions of two variables like \( g(x, y) = \ln(x^3 - y^2) \) depend on two different inputs or variables - in this case, \( x \) and \( y \). These variables work together to produce a single output. Such functions are often visualized as surfaces in three-dimensional space.
  • In our example function, the two variables \( x \) and \( y \) are used to calculate \( x^3 - y^2 \).
  • The result of this calculation is then passed through the natural logarithm \( \ln \), transforming it further into the final output.
  • Functions of two variables are widely used in economic models, physics for representing surfaces, and in optimization problems where multiple factors influence outcomes.
This type of function demonstrates how single outcomes can be derived from the interaction of more than one input, which is powerful for modeling real-world phenomena.
Logarithmic Properties
Logarithmic properties simplify the handling of complex expressions. One key property used in the step-by-step solution is \( \ln(a^b) = b \ln(a) \). This power rule helps break down expressions involving exponents in a logarithmic function:
  • For our problem, when finding \( \ln(e^3) \), we applied this rule to simplify it to \( 3 \ln(e) \).
  • Another important property is \( \ln(ab) = \ln(a) + \ln(b) \), which converts products into sums.
  • Additionally, \( \ln(a/b) = \ln(a) - \ln(b) \) is useful for simplifying fractions inside a logarithm.
Understanding these properties allows us to easily manipulate and evaluate logarithmic expressions like \( g(x, y) \). They are indispensable in calculus for solving logarithmic equations and integrating functions.

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Most popular questions from this chapter

A farmer's wheat yield (bushels per acre) depends on the amount of fertilizer (hundreds of pounds per acre) according to the following table. Find the least squares line. Then use the line to predict the yield using 3 hundred pounds of fertilizer per acre. $$ \begin{array}{lrrrr} \hline \text { Fertilizer } & 1.0 & 1.5 & 2.0 & 2.5 \\ \text { Yield } & 30 & 35 & 38 & 40 \\ \hline \end{array} $$

True or False: Every function \(f(x, y)\) of two variables can be written as the sum of two functions of one variable, \(g(x)+h(y)\).

The following table shows the relationship between the sulfur dust content of the air (in micrograms per cubic meter) and the number of female absentees in industry. (Only absences of at least seven days were counted.) Find the least squares line for these data. Use your answer to predict absences in a city with a sulfur dust content of \(25 .\) $$ \begin{array}{lcc} \hline & & \text { Absences per } \\ & \text { Sulfur } & \text { 1000 Employees } \\ & 7 & 19 \\ \text { Cincinnati } & 7 & 44 \\ \text { Indianapolis } & 13 & 43 \\ \text { Woodbridge } & 14 & 53 \\ \text { Camden } & 17 & 61 \\ \text { Harrison } & 20 & 88 \\ \hline \end{array} $$

An automobile manufacturer sells cars in America, Europe, and Asia, charging a different price in each of the three markets. The price function for cars sold in America is \(p=20-0.2 x\) (for \(0 \leq x \leq 100\) ), the price function for cars sold in Europe is \(q=16-0.1 y \quad(\) for \(0 \leq y \leq 160),\) and the price function for cars sold in Asia is \(r=12-0.1 z\) (for \(0 \leq z \leq 120\) ), all in thousands of dollars, where \(x, y,\) and \(z\) are the numbers of cars sold in America, Europe, and Asia, respectively. The company's cost function is \(C=22+4(x+y+z)\) thousand dollars. a. Find the company's profit function \(P(x, y, z)\). [Hint: The profit will be revenue from America plus revenue from Europe plus revenue from Asia minus costs, where each revenue is price times quantity. b. Find how many cars should be sold in each market to maximize profit. [Hint: Set the three partials \(P_{x}, P_{y}\), and \(P_{z}\) equal to zero and solve. Assuming that the maximum exists, it must occur at this point.]

Suppose that the least squares line for a set of data points is \(y=a x+b\). If you doubled each \(y\) -value, what would be the new least squares line?
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