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Magazine Chapter 3: Problem 33
Give an example of: An interval where the Mean Value Theorem applies when \(f(x)=\ln x\)
Short Answer
Expert verified
An example interval is \([1, 2]\).
Step by step solution
01
Understanding the Function
The function given is the natural logarithm, denoted as \( f(x) = \ln x \). This function is defined for all \( x > 0 \). The graph of \( \ln x \) is continuous and differentiable everywhere on its domain.
02
Applying Mean Value Theorem Conditions
The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists some \( c \) in \( (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). We must choose an interval \([a, b]\) where \( f(x) = \ln x \) is both continuous and differentiable.
03
Choosing an Appropriate Interval
The function \( \ln x \) is continuous and differentiable for any interval where \( 0 < a < b \). Therefore, any such interval can be an example. A simple choice is \([1, 2]\), which lies entirely in the domain of \( \ln x \).
04
Conclusion: The Example Interval
Thus, the interval \([1, 2]\) is an interval where the Mean Value Theorem applies, because \( f(x) = \ln x \) is continuous on \([1, 2]\) and differentiable on \((1, 2)\).
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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 by \( f(x) = \ln x \), is a key mathematical function. This function is defined for all positive values of \( x \) (\( x > 0 \)). It is used frequently in calculus and many scientific fields because it has intriguing properties and behavorial attributes.
Key Characteristics of Natural Logarithm:
Understanding \( \ln x \) helps in grasping other mathematical concepts, including calculus theorems like the Mean Value Theorem.
Key Characteristics of Natural Logarithm:
- **Logs and Exponents:** The natural logarithm is the inverse function of the exponential function \( e^x \). This means if \( y = e^x \), then \( x = \ln y \).
- **Base \( e \):** The natural logarithm is based on Euler's number \( e \), which is approximately 2.71828. This number is irrational and transcendental, which means it cannot be represented as a simple fraction and does not satisfy any non-zero polynomial equation with rational coefficients.
- **Behavior:** As \( x \) increases, \( \ln x \) also increases, but at a decreasing rate. It approaches negative infinity as \( x \) approaches 0 from the positive side, and it crosses zero at \( x = 1 \).
Understanding \( \ln x \) helps in grasping other mathematical concepts, including calculus theorems like the Mean Value Theorem.
Continuous Function
A continuous function is a type of function that, informally, has no interruptions or 'jumps' in its graph. For a function to be continuous at a point \( x = c \), the left-hand and right-hand limits as \( x \) approaches \( c \) must exist and be equal to the function's value at that point, \( f(c) \).
Characteristics of Continuous Functions:
The continuity of \( \ln x \) on any interval like \([1, 2]\) assures its application under the Mean Value Theorem, since this theorem heavily relies on continuity over a set interval.
Characteristics of Continuous Functions:
- **No Gaps:** The graph of the function does not have any breaks or holes.
- **Limits and Values:** The value of the function at any point is exactly the value that the function approaches from both sides of that point.
- **Example with \( \ln x \):** The function \( \ln x \) is continuous for all \( x > 0 \). This means there are no values of \( x \) in this domain where the function suddenly 'jumps' or 'drops'.
The continuity of \( \ln x \) on any interval like \([1, 2]\) assures its application under the Mean Value Theorem, since this theorem heavily relies on continuity over a set interval.
Differentiable Function
A differentiable function is one that has a derivative at each point along its domain. This means you can find the slope of the tangent line to the function's graph at every point. Differentiation is a fundamental component of calculus, enabling us to understand and analyze the rate of change of a function.
Traits of Differentiable Functions:
The differentiability of \( \ln x \) within an interval like \((1, 2)\), means that the conditions for applying the Mean Value Theorem, which require differentiability over \((a, b)\), are met.
Traits of Differentiable Functions:
- **Slope of Tangents:** At any point \( x \), the derivative \( f'(x) \) exists, providing the slope of the tangent at the point.
- **Smooth Curves:** The graph of a differentiable function will appear smooth, without sharp corners or cusps, within its domain.
- **Example with \( \ln x \):** \( \ln x \) is differentiable for any \( x > 0 \). Its derivative is \( \frac{1}{x} \), showing it changes at a slowing rate as \( x \) increases.
The differentiability of \( \ln x \) within an interval like \((1, 2)\), means that the conditions for applying the Mean Value Theorem, which require differentiability over \((a, b)\), are met.
