Problem 22 - Applied Calculus

Question

Find the average value of each function over the given interval. \(f(x)=e^{k x}\) on \([0,1],\) where \(k\) is a constant \((k \neq 0)\)

Step-by-Step Solution

Answer

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

Step 1: Write the formula for the average value of a function

The average value of a function \( f(x) \) over the interval \([a, b]\) is given by the formula:\[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \]In our case, \(a = 0\) and \(b = 1\). Thus, the formula becomes:\[ \text{Average value} = \int_0^1 f(x) \, dx \]

Step 2: Substitute the function into the formula

Substitute the given function \(f(x) = e^{kx}\) into the formula for the average value:\[ \text{Average value} = \frac{1}{1-0} \int_0^1 e^{kx} \, dx \]This simplifies to:\[ \text{Average value} = \int_0^1 e^{kx} \, dx \]

Step 3: Integrate the function over the interval

To integrate \(e^{kx}\), we recognize its integral as \( \frac{1}{k} e^{kx} + C \) (where \(C\) is the constant of integration, but not needed for a definite integral). Applying the limits 0 to 1, we get:\[ \int_0^1 e^{kx} \, dx = \left. \frac{1}{k} e^{kx} \right|_0^1 \]This evaluates to:\[ \frac{1}{k} e^{k} - \frac{1}{k} e^{0} = \frac{1}{k}(e^{k} - 1) \]

Step 4: Express the average value

Thus, the average value of \(f(x) = e^{kx}\) over the interval \([0, 1]\) is:\[ \frac{1}{k} (e^k - 1) \]

Key Concepts

Integration

Integration is a fundamental concept in calculus, representing the accumulation of quantities and the area under a curve. In this context, it helps us find the average value of a function over a specified interval. When practicing integration, it's essential to understand the basic operations and how they apply to various functions.

• The primary operation of integration is finding the antiderivative, or the inverse of differentiation. This process determines the original function given its derivative.
• In our problem, the function involves an exponential, but integration techniques apply to all sorts of functions, each with its rules.

For beginner learners, useful methods for integration include substitution, integration by parts, and recognizing standard forms. In this example, we identified the standard integral for exponential functions, aiding in calculating the average value.

Definite Integrals

Definite integrals are used to compute the total accumulation of a quantity, such as area, over a given interval. They are denoted by: \[\int_a^b f(x) \, dx\]where \(a\) and \(b\) are the limits of integration.

• The process involves finding the antiderivative of the function and evaluating it from \(a\) to \(b\).
• In distinct contrast to indefinite integrals, definite integrals result in a specific numerical value, representing the precise area under the curve for the specified interval.

In our exercise, the integration of \(e^{kx}\) from 0 to 1 gives the amount needed to compute the average value accurately.

Exponential Functions

Exponential functions are essential mathematical models that describe many phenomena in science and engineering. The function \(f(x) = e^{kx}\), where \(e\) is the base of natural logarithms and \(k\) is a constant, exhibits exponential growth or decay.

• These functions have unique properties: their rate of change is proportional to their current value, making them useful for modeling population growth, finance, and radioactive decay, among others.
• For integration purposes, exponential functions retain their form. Integrating \(e^{kx}\) gives the result \(\frac{1}{k} e^{kx} + C\), which is straightforward given their self-replicating derivative and integral forms.

In the given exercise, recognizing the form of the function and applying the correct integration formula is crucial for solving the average value efficiently.