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Chapter 13: Problem 8

Find and simplify the function values. \(g(x, y)=\ln |x+y|\) (a) \((2,3)\) (b) \((5,6)\) (c) \((e, 0)\) (d) \((0,1)\) (e) \((2,-3)\) (f) \((e, e)\)

Short Answer

Expert verified
(a) \(\ln 5\), (b) \(\ln 11\), (c) 1, (d) 0, (e) 0, (f) \(\ln 2\)

Step by step solution

01

Evaluating at Point (2,3)

First, replace \(x\) and \(y\) in the function with 2 and 3 respectively, you get \(g(2, 3) = \ln |2 + 3|\). This will simplify to \(\ln |5|\) and hence further simplifies to \(\ln 5\)
02

Evaluating at Point (5,6)

Replace \(x\) and \(y\) in the function with 5 and 6 respectively, you get \(g(5, 6) = \ln |5 + 6|\). This will simplify to \(\ln |11|\) and hence further simplifies to \(\ln 11\)
03

Evaluating at Point (e,0)

Replace \(x\) and \(y\) in the function with \(e\) and 0 respectively, you get \(g(e, 0) = \ln |e + 0|\). This simplifies to \(\ln |e|\) and hence further simplifies to 1 because the natural logarithm of \(e\) equals 1.
04

Evaluating at Point (0,1)

Replace \(x\) and \(y\) in the function with 0 and 1 respectively, you get \(g(0, 1) = \ln |0 + 1|\). This simplifies to \(\ln |1|\) and hence further simplifies to 0 because the natural logarithm of \(1\) equals 0.
05

Evaluating at Point (2,-3)

Replace \(x\) and \(y\) in the function with 2 and -3 respectively, you get \(g(2, -3) = \ln |2 - 3|\). This simplifies to \(\ln |-1|\) and hence further simplifies to 0 because the natural logarithm of \(1\) equals 0.
06

Evaluating at Point (e, e)

Replace \(x\) and \(y\) in the function with \(e\) and \(e\) respectively, you get \(g(e, e) = \ln |e + e|\). This simplifies to \(\ln |2e|\) and hence further simplifies to \(\ln 2 + \ln e\), which is equal to 1. So, \(g(e, e) = \ln 2\).

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Key Concepts

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

Understanding Natural Logarithms
The natural logarithm is a special mathematical function denoted by \(\ln\). It is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Logarithms are the inverse of exponential functions. Simply put, they help us solve equations involving the exponent.

For example, if \(e^x = 5\), the natural logarithm allows us to express this as \(x = \ln 5\). The function \(\ln x\) is especially useful in calculus, particularly in integration and differentiation, due to its straightforward derivative and integral.
Function Evaluation Made Easy
Evaluating a function means finding its value at a given point. In multivariable calculus, it often involves substituting values into a given function with more than one variable.

For example, to find \(g(2, 3)\) of \(g(x, y) = \ln |x+y|\), you substitute 2 for \(x\) and 3 for \(y\), resulting in \(g(2,3) = \ln |2+3|\). Solving this gives \(\ln 5\).

Function evaluation requires attention to each variable and any operations performed on them, ensuring accuracy in your final conclusion.
The Role of Absolute Value
Absolute value, denoted by vertical bars \(|x|\), measures the distance a number is from zero on the number line, without considering direction. It transforms any negative input into a positive output.

In expressions involving addition or subtraction, like \(|x+y|\), the absolute value ensures that the result is always non-negative. For instance, in \(g(2, -3) = \ln |2-3|\), it becomes \(\ln |-1|\), which is the same as \(\ln 1\). Because the absolute value of \(-1\) is 1, the natural logarithm then simplifies accordingly.
Simplification in Calculus
Simplification is a fundamental skill in calculus, focusing on reducing expressions to their simplest form. It involves performing arithmetic operations, recognizing constants, and using identities or properties of mathematical functions.

For example, simplifying \(\ln |e|\) yields 1, since the natural logarithm of \(e\) is known to be 1. Meanwhile, expressions like \(\ln 2e\) can be broken down using logarithmic identities such as \(\ln ab = \ln a + \ln b\), transforming it to \(\ln 2 + \ln e\), simplifying further to \(\ln 2 + 1\).

Such steps leading to simplification are crucial in finding neat and interpretable answers.

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Most popular questions from this chapter

Consider the function \(f(x, y)=\left(x^{3}+y^{3}\right)^{1 / 3}\). (a) Show that \(f_{y}(0,0)=1\). (b) Determine the points (if any) at which \(f_{y}(x, y)\) fails to exist.

Investigation Consider the function \(f(x, y)=\frac{\sin y}{x}\) on the intervals \(-3 \leq x \leq 3\) and \(0 \leq y \leq 2 \pi\). (a) Find a set of parametric equations of the normal line and an equation of the tangent plane to the surface at the point \(\left(2, \frac{\pi}{2}, \frac{1}{2}\right)\) (b) Repeat part (a) for the point \(\left(-\frac{2}{3}, \frac{3 \pi}{2}, \frac{3}{2}\right)\). (c) Use a computer algebra system to graph the surface, the normal lines, and the tangent planes found in parts (a) and (b). (d) Use analytic and graphical analysis to write a brief description of the surface at the two indicated points.

The table shows the world populations \(y\) (in billions) for five different years. (Source: U.S. Bureau of the Census, International Data Base) $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 1994 & 1996 & 1998 & 2000 & 2002 \\ \hline \text { Population, } \boldsymbol{y} & 5.6 & 5.8 & 5.9 & 6.1 & 6.2 \\ \hline \end{array} $$ Let \(x=4\) represent the year 1994 . (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use the regression capabilities of a graphing utility to find the least squares regression quadratic for the data. (c) Use a graphing utility to plot the data and graph the models. (d) Use both models to forecast the world population for the year \(2010 .\) How do the two models differ as you extrapolate into the future?

The endpoints of the interval over which distinct vision is possible are called the near point and far point of the eye. With increasing age, these points normally change. The table shows the approximate near points \(y\) in inches for various ages \(x\) (in years). $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Age, } x & 16 & 32 & 44 & 50 & 60 \\ \hline \text { Near Point, } y & 3.0 & 4.7 & 9.8 & 19.7 & 39.4 \\ \hline \end{array} $$ (a) Find a rational model for the data by taking the reciprocal of the near points to generate the points \((x, 1 / y)\). Use the regression capabilities of a graphing utility to find a least squares regression line for the revised data. The resulting line has the form \(\frac{1}{y}=a x+b\) Solve for \(y\). (b) Use a graphing utility to plot the data and graph the model. (c) Do you think the model can be used to predict the near point for a person who is 70 years old? Explain.

A measure of what hot weather feels like to two average persons is the Apparent Temperature Index. A model for this index is \(A=0.885 t-22.4 h+1.20 t h-0.544\) where \(A\) is the apparent temperature in degrees Celsius, \(t\) is the air temperature, and \(h\) is the relative humidity in decimal form. (Source: The UMAP Journal, Fall 1984) (a) Find \(\partial A / \partial t\) and \(\partial A / \partial h\) when \(t=30^{\circ}\) and \(h=0.80\). (b) Which has a greater effect on \(A\), air temperature or humidity? Explain.
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