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Magazine Chapter 14: Problem 36
At what points \((x, y, z)\) in space are the functions in continuous? a. \(f(x, y, z)=\ln x y z\) b. \(f(x, y, z)=e^{x+y} \cos z\)
Short Answer
Expert verified
a. Continuous where \( xyz > 0 \).
b. Continuous for all real \( x, y, z \).
Step by step solution
01
Identify Domain of ln Function
The function given is \( f(x, y, z) = \ln(xyz) \). The natural logarithm \( \ln(u) \) is defined and continuous for \( u > 0 \). Therefore, \( xyz > 0 \) must hold true for the function to be continuous.
02
Determine Domain for \( \ln(xyz) \)
For \( xyz > 0 \), there are specific conditions for \( x, y, \) and \( z \): either all are positive, or an odd number of them must be negative. Hence, the function \( f(x, y, z) \) is continuous where \( xyz > 0 \).
03
Identify Domain of \( e^{x+y} \)
The function given is \( f(x, y, z) = e^{x+y} \cos z \). Notice that \( e^{x+y} \) is an exponential function which is continuous for all real values of \( x \) and \( y \). Thus, it does not place any restrictions on \( x \) and \( y \).
04
Identify Domain of \( \cos z \)
The \( \cos z \) function is continuous for all real numbers \( z \). Therefore, there are no restrictions for \( z \). The entire function \( f(x, y, z) = e^{x+y} \cos z \) is continuous in all of \( \mathbb{R}^3 \).
05
Combine Domain Conditions for Each Function
For \( f(x, y, z) = \ln(xyz) \), the function is continuous wherever \( xyz > 0 \). For \( f(x, y, z) = e^{x+y} \cos z \), the function is continuous for all \( x, y, \) and \( z \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Continuous Functions
In mathematics, a function is continuous if it does not have any abrupt changes in value. This means if you are moving through the domain of the function, the output doesn鈥檛 jump unexpectedly. A common example of a continuous function is a straight line.
- Continuity is crucial for many applications in calculus and real-world scenarios because it ensures predictability and smooth transitions.
- A function is continuous at a point if the limit of the function as it approaches that point is the same as the function value at the point.
- The function involving a logarithm, such as \( f(x, y, z) = \ln(xyz) \), requires \( xyz > 0 \) for continuity, as logarithmic functions break down when dealing with non-positive values.
- Exponential functions like \( f(x, y, z) = e^{x+y} \), along with trigonometric parts like \( \cos z \), are simpler since they remain continuous across their entire domains.
Domain and Range
When studying functions, the domain and range are paramount. The domain of a function is the set of all possible input values (usually \( x \), \( y \), and \( z \) for multivariable functions) for which the function is defined. The range is the set of all possible output values.
- For \( f(x, y, z) = \ln(xyz) \), the domain restriction is \( xyz > 0 \). This means only combinations of \( x \), \( y \), and \( z \) that result in a positive product are part of the domain.
- The range for \( \ln(u) \) is all real numbers, as the logarithmic function can produce any real value if given the right positive input.
- The exponential part \( e^{x+y} \) is defined for any real \( x \) and \( y \).
- \( \cos z \) is continuous for all real \( z \).
Exponential and Logarithmic Functions
Exponential and logarithmic functions are two fundamental types of functions in multivariable calculus. They often appear in models describing growth, decay, and many natural phenomena.
### Exponential FunctionsExponential functions are characterized by a constant base raised to a variable exponent. A key feature of these functions:
### Exponential FunctionsExponential functions are characterized by a constant base raised to a variable exponent. A key feature of these functions:
- The base, typically \( e \), leads to continuous, smooth curves.
- If \( f(x) = e^x \), then for multivariable functions like \( f(x, y, z) = e^{x+y} \), it implies continuity and smoothness for all \((x, y, z)\).
- Exponential functions grow very quickly and are never zero.
- \( \ln(u) \) is only defined for positive \( u \), hence the restriction \( xyz > 0 \).
- They describe phenomena that increase slowly and can handle huge ranges of input.
