Problem 92 - Calculus Single Variable

Question

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\frac{\ln (x)}{1+x} \quad I=[1, e] $$

Step-by-Step Solution

Answer

Find the average value of \( f \) over \( [1, e] \), evaluate the integral, and solve \( f(c) = f_{\text{avg}} \) to find \( c \).

Step 1: Understand the Formula for Average Value of a Function

The average value of a continuous function \( f \) over the interval \( [a, b] \) is given by the formula \( f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\, dx \). In this problem, \( f(x) = \frac{\ln(x)}{1+x} \) and the interval \( I = [1, e] \). So, \( a = 1 \) and \( b = e \).

Step 2: Set Up the Integral

Substitute \( f(x) = \frac{\ln(x)}{1+x} \) and the interval limits \( a = 1 \) and \( b = e \) into the formula: \[ f_{\text{avg}} = \frac{1}{e-1} \int_1^e \frac{\ln(x)}{1+x}\, dx \].

Step 3: Evaluate the Integral

To find the integral \( \int_1^e \frac{\ln(x)}{1+x}\, dx \), apply integration techniques (e.g., substitution if necessary). Calculators or symbolic software can be used, as this problem might require complex steps. Calculate it to three decimal places.

Step 4: Calculate the Average Value

After evaluating the integral, substitute back into the formula \( f_{\text{avg}} = \frac{1}{e-1} \times \text{[integral value]} \). The result should be given to three decimal places.

Step 5: Find the Value of \( c \)

Find \( c \) within the interval \( [1, e] \) where \( f(c) = f_{\text{avg}} \). You solve \( \frac{\ln(c)}{1+c} = f_{\text{avg}} \) numerically, as algebraic solutions may not be feasible. This can be done via graphing or using numerical solvers.

Step 6: Solution Verification and Rounding

After finding \( c \), verify the solution to ensure \( f(c) \approx f_{\text{avg}} \) matches accurately. Round \( c \) to three decimal places as well.

Key Concepts

Understanding Integration

Integration is like finding the area under a curve. In mathematics, it is a core concept for solving problems involving averages, areas, and accumulated quantities.
To find the average value of a function over an interval, we use integration. We integrate the function over the interval and then divide by the length of the interval.
For this exercise, our function is \( f(x) = \frac{\ln(x)}{1+x} \) and the interval is \([1, e]\). First, we set up the integral of the function over this interval:

• Calculate \( \int_1^e \frac{\ln(x)}{1+x} \, dx \): this represents summing up tiny slices (areas) under the curve between 1 and \(e\).
• Find the result using proper techniques or tools, since this integral might be complex to evaluate manually.

Integration gives us the accumulation of the function's values over the interval, a crucial step to calculate the average.

Characteristics of Continuous Functions

Continuous functions are smooth without any breaks, jumps, or holes. Imagine a smooth line that can be drawn without lifting your pen. These functions have valuable properties in integration and calculus.
In the exercise, \( f(x) = \frac{\ln(x)}{1+x} \) is continuous on the interval \([1, e]\) because the natural logarithm, \( \ln(x) \), is defined and continuous for all \(x > 0\), and \(1 + x\) is never zero over that interval.

• Continuity ensures the average value calculation is valid, as it guarantees no sudden changes occur that could disrupt the process.
• It also allows us to apply calculus rules more smoothly when solving integrals.

Continuous functions make averaging values using integration possible, offering more accurate answers.

Exploring the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithmic function with the constant \(e\) (approximately 2.71828) as its base. It is extensively used in mathematics and science.
In the provided exercise, the logarithm is part of the function \( f(x) = \frac{\ln(x)}{1+x} \).

• The natural logarithm has vital properties which make it suitable for integration, such as differentiability and continuity for \(x > 0\).
• It increases slower as \(x\) increases, which can affect the shape and slope of the graph, impacting the integral outcome.

This makes it essential to understand how \( \ln(x) \) behaves, as it influences the integral's solution and how we calculate the average value.

Numerical Solving Techniques

After analytically finding a function's average, numerical solving often aids in locating exact values. Not every problem has a straightforward algebraic solution, especially when dealing with equations that involve transcendental functions like the natural logarithm.
In our exercise, after finding the average \( f_{\text{avg}} \), we need \( c \) such that \( f(c) = f_{\text{avg}} \). Finding \(c\) might require numerical solutions:

• Graphical methods can help visualize where \( f(c) \) intersects \( f_{\text{avg}} \).
• Tools like numerical solvers or iteration methods (e.g., the Newton-Raphson method) find precise \(c\) when algebraic approaches are impractical.

Numerical solving bridges the gap when exact solutions are challenging to derive analytically, ensuring precise and practical answers.