Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the average value of the function on the given interval. $$ f(x)=e^{-x} ; \quad[0,2] $$

Step-by-Step Solution

Answer

The average value is \( \frac{e^2 - 1}{2e^2} \).

Step 1: Formula for Average Value of a Function

The formula for the average value of a continuous function \( f(x) \) on an interval \([a, b]\) is given by:\[\frac{1}{b-a} \int_a^b f(x) \, dx\]For this problem, \(a = 0\) and \(b = 2\). We need to find \(\frac{1}{2-0} \int_0^2 e^{-x} \, dx\).

Step 2: Compute the Definite Integral

Calculate the definite integral of \( f(x) = e^{-x} \) from \(0\) to \(2\):\[\int_0^2 e^{-x} \, dx\]The integral of \( e^{-x} \) is \( -e^{-x} \). So,\[- e^{-x} \Big|_0^2\]Evaluate at the limits:\[[-e^{-2}] - [-e^{0}] = -e^{-2} + 1\]

Step 3: Evaluate the Integral

Substituting the exponential values, the evaluation yields:\[-\frac{1}{e^2} + 1 = 1 - \frac{1}{e^2}\]

Step 4: Calculate the Average Value

Now we divide the result of the definite integral by the length of the interval \([0, 2]\):\[\frac{1}{2} (1 - \frac{1}{e^2})\]Simplifying further:\[\frac{1}{2} - \frac{1}{2e^2}\]

Step 5: Simplify the Expression

Add fractions to get a single expression:\[\frac{e^2 - 1}{2e^2}\]This represents the average value of the function on the interval \([0, 2]\).

Key Concepts

Integral Calculation

Integral calculation is a fundamental process in calculus used to find the area under a curve. For a given function \(f(x)\), the integral represents a summation of values of \(f(x)\) over a specified interval. Think of it like summing up tiny slices of a curve to find the total area beneath it.

When we calculate an integral, we often deal with a definite integral, which specifies the bounds of the area we want to find. In our example, this involves calculating \(\int_0^2 e^{-x} \, dx\). Learning to compute integrals often requires practicing different techniques, such as substitution or integration by parts.

• The notation \(\int\) signifies the integral.
• \( e^{-x} \) is our function to integrate.
• The limits \(0\) and \(2\) indicate the interval.

Being comfortable with these concepts is critical for effectively solving calculus problems.

Continuous Function

A continuous function is a mathematical concept where a function does not have any gaps, jumps, or abrupt changes in value across its domain. For the function \( f(x) = e^{-x} \), it is continuous over any real interval. This continuity ensures smooth behavior without interruptions, which is crucial for integrating such a function.

Continuous functions allow us to apply the Fundamental Theorem of Calculus. This theorem connects differentiation and integration, showing how they are inverse processes.

• Continuity means no sudden breaks in the graph of the function.
• \( e^{-x} \) is defined and smooth for all real numbers.
• Being continuous across the interval \([0,2]\) makes integration straightforward.

Understanding continuity helps simplify many mathematical procedures, making solving integrals more intuitive.

Definite Integral Evaluation

Evaluating a definite integral involves calculating the exact value of the integral over a given range. This process takes the form \(\int_a^b f(x) \, dx\), where \(a\) and \(b\) define the interval. For \(f(x) = e^{-x}\), we evaluated the definite integral by:

• Finding the antiderivative, which was \(-e^{-x}\).
• Substituting the limits into the antiderivative.
• Calculating \(-e^{-2} + 1\).

This yielded the definite value of \(1 - \frac{1}{e^2}\). Calculating definite integrals is vital for finding precise areas under curves and understanding accumulated quantities.

Exponential Functions

Exponential functions are represented by \(f(x) = a^x\) where \(a\) is a positive constant. In our case, \(e^{-x}\) is an exponential function with base \(e\), a mathematical constant approximately equal to 2.71828. Exponential functions are powerful tools in mathematics and are widely used due to their rapid rates of increase or decrease.

Key properties of \(e^x\) and its inverse, \(e^{-x}\), include:

• \( e^{-x} \) yields decreasing values as \(x\) increases.
• The derivative of \(e^{-x}\) is \(-e^{-x}\), demonstrating the function's inherent simplicity.
• Exponential decay, as seen in \(e^{-x}\), models natural phenomena like radioactive decay and cooling processes.

Understanding these properties is crucial for solving calculus problems involving exponential functions effectively.