Problem 45 - Essential Calculus

Question

Find the average value of the function over the given interval and all values of \(x\) in the interval for which the function equals its average value. $$ f(x)=2 e^{x}, \quad[-1,1] $$

Step-by-Step Solution

Answer

The average value of the function \(f(x)=2e^x\) over the interval [-1,1] is \(\frac{1}{2} (2e - 2/e)\). The x-values for which the function equals its average value within the interval [-1,1] are those values that satisfy the equation \(x = \ln((e - 1/e)/2)\),

Step 1: Calculating the average value of the function over the interval [-1,1]

The average value of the function is obtained using the formula \(\frac{1}{b-a} \int_{a}^{b} f(x) dx\). For this particular function \(f(x)=2e^x\) over the interval [-1,1], the average value is computed as: \(\frac{1}{1-(-1)} \int_{-1}^{1} 2e^x dx = \frac{1}{2} [2e^x]_{-1}^{1} = \frac{1}{2} (2e - 2/e)\)

Step 2: Finding all x-values in the interval for which the function equals its average value

Now, to find the x-values at which \(f(x)=2e^x\) equals the average value obtained in step 1, the function is set equal to the average value and then solved for x. The equation thus obtained is: \(2e^x = \frac{1}{2} (2e - 2/e)\). After simplifying, the equation becomes \(e^x = \frac{1}{2} (e - 1/e)\). Solving it will yield the x-values within the interval [-1,1] at which the function equals its average value.

Step 3: Solving the equation

To solve the equation \(e^x = \frac{1}{2} (e - 1/e)\), it's likely to use a logarithmic function. Taking the natural logarithm on both sides yields \(x = \ln((e - 1/e)/2)\). It's then necessary to check whether this x-value falls within the interval [-1,1].

Key Concepts

Exponential Function

An exponential function is a type of mathematical function where a constant base is raised to a variable exponent. The general form is given by \( f(x) = a \cdot b^x \), where

• \( a \) is a constant, which stretches or shrinks the graph vertically.
• \( b \) is the base of the exponential, determining the rate of growth or decay. If \( b > 1 \), the function grows as \( x \) increases; if \( 0 < b < 1 \), it decays.

In the given exercise, our function is \( f(x) = 2e^x \). Here, \( e \) is the base of the natural logarithm, approximately 2.718, and our function grows exponentially as \( x \) increases. Exponential functions, such as this one, model continuous growth processes.

They are extensively used in fields like economics for compound interest, biology for population models, and many areas of physics and engineering.

Definite Integral

The definite integral is a mathematical concept used to calculate the area under a curve between two points on the x-axis. It is not just a simple area, but a signed area, taking into account parts of the function above or below the x-axis. To find the definite integral of a function \( f(x) \) from \( a \) to \( b \), we use the notation \[\int_{a}^{b} f(x) \ dx.\]The solution provides the accumulated value from \( a \) to \( b \). In our exercise, the function \( 2e^x \) is integrated from -1 to 1, aiming to determine the average value of the function over this interval. In mathematical terms, this looks like the formula\[\frac{1}{b-a} \int_{a}^{b} f(x) \ dx,\]which gives the mean or "average" of the function's values over the specified interval.

Interval Notation

Interval notation is a method of describing a set of numbers along an interval on the number line. This notation is particularly useful in calculus when defining domains or specifying intervals over which integrals are evaluated. For an interval \([a,b]\) in interval notation,

• \([a,b]\) denotes a closed interval, including both endpoints \(a\) and \(b\).
• \((a,b)\) is an open interval, excluding both endpoints.
• \([a,b)\) or \((a,b]\) are half-open intervals, including one endpoint.

In the exercise, \([-1,1]\) is a closed interval, indicating that both -1 and 1 are included in the interval. This means we consider all x-values from -1 to 1, inclusive.

Solving Equations with Logarithms

Solving equations that involve exponential functions often requires the use of logarithms. This is because logarithms are the inverse operations of exponentials. If you have an equation of the form \( e^x = k \), you can solve for \( x \) by applying the natural logarithm, \( \ln \), to both sides, resulting in \( x = \ln(k) \).

In the given problem, the equation \( e^x = \frac{1}{2}(e - 1/e) \) offers a perfect opportunity to use logarithms. By taking the natural logarithm of both sides, we can isolate \( x \) to get\[x = \ln\left(\frac{e - 1/e}{2}\right).\]This approach simplifies the problem and allows us to find the specific x-values in the interval where the exponential function equals its average. It's important to check that the solution lies within the specified interval, ensuring it's a valid answer.