Question

A function \(f(x)\) and interval \([a, b]\) are given. Check if the Mean Value Theorem can be applied to \(f\) on \([a, b] ;\) if so, find a value \(c\) in \([a, b]\) guaranteed by the Mean Value Theorem. \(f(x)=\ln x\) on [1,5]

Short answer

The Mean Value Theorem applies; \(c = \frac{4}{\ln 5}\).

Step-by-step solution

Check Continuity

The Mean Value Theorem (MVT) applies if the function is continuous on the closed interval \([a, b]\). Determine if \(f(x) = \ln x\) is continuous on \([1, 5]\). The natural logarithm function is continuous for \(x > 0\), and since the interval \([1, 5]\) is within the domain of \(f(x)\), \(f(x)\) is continuous on \([1, 5]\).

Check Differentiability

The MVT also requires that the function be differentiable on the open interval \((a, b)\). Determine if \(f(x) = \ln x\) is differentiable on \((1, 5)\). The derivative of \(f(x)\) is \(f'(x) = \frac{1}{x}\), which exists and is defined for \(x > 0\). So, \(f(x)\) is differentiable on \((1, 5)\).

Apply the Mean Value Theorem

Since \(f(x) = \ln x\) is continuous on \([1, 5]\) and differentiable on \((1, 5)\), the MVT applies. The theorem states that there exists at least one \(c\) in \((1, 5)\) such that \[ f'(c) = \frac{f(b) - f(a)}{b-a}\]Calculate \(f(5) = \ln 5\) and \(f(1) = \ln 1 = 0\).

Calculate the MVT Expression

Substitute the values found in Step 3 into the MVT expression: \[ \frac{\ln 5 - \ln 1}{5 - 1} = \frac{\ln 5}{4} \]Since \(f'(c) = \frac{1}{c}\), set this equal to \(\frac{\ln 5}{4}\) and solve for \(c\).

Solve for c

Solve \( \frac{1}{c} = \frac{\ln 5}{4} \).This implies that \[ c = \frac{4}{\ln 5} \]Calculate the numerical value of \(c\) using this expression.

Key concepts

Continuous Functions

A continuous function is a fundamental concept in calculus and mathematical analysis. If you've ever drawn a curve without lifting your pencil off the paper, you've essentially visualized a continuous function. For a function to be continuous at a point, say \(x = a\), the limit of the function as it approaches \(a\) from both sides must equal the value of the function at \(a\). In simpler terms, continuous functions have no breaks, holes, or jumps within the interval they're considered.
The Mean Value Theorem (MVT) requires continuity on a closed interval because it's essential for ensuring the function behaves predictably across that interval. Consider the function \(f(x) = \ln x\) defined on the interval \[1, 5\]. Since the natural logarithm is continuous for any \(x > 0\), it is continuous over our interval. This means there are no abrupt "leaps" in values when moving from \(x = 1\) to \(x = 5\).

• Continuity ensures smoothness within a given range.
• Necessary for MVT application: Guarantees no interruptions.

Differentiable Functions

Differentiability means a function has a derivative, or in other words, it has a defined slope at each point within its interval. Visualize a curve that always has a tangent; that's how you can identify differentiability. For the Mean Value Theorem (MVT), a function must be differentiable on an open interval. This means the function’s rate of change doesn’t explode into infinity or suddenly become undefined.Our function, \(f(x) = \ln x\), is differentiable as its derivative, \(f'(x) = \frac{1}{x}\), exists for all positive \(x\). In the interval \( (1, 5) \), \(f'(x)\) is well-defined, showing that the function's slope is always computable and the curve is smoothly changing.

• Smoothness & consistent change: A hallmark of differentiable functions.
• Critical for MVT: Identifies points where tangent represents average slope.

Natural Logarithm

The natural logarithm, symbolized by \(\ln x\), is a special mathematical function. It is the inverse of the exponential function \(e^x\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm answers the question: to what power must \(e\) be raised to obtain \(x\)?
The natural logarithm is crucial in many areas such as calculus, financial mathematics, and even biological processes. In our example, the function \(f(x) = \ln x\) plays a key role in proving the Mean Value Theorem on a specific interval. When dealing with \(\ln x\) over some interval like \[1, 5\], we're comfortably within its domain since \(\ln x\) is only defined for \(x > 0\).

• Inverse relation: Reverses the exponential \(e^x\).
• Key characteristics: Defined for positive \(x\).