Problem 36 - Calculus and its applications

Question

Find the average function value over the given interval. $$ y=e^{x} ; \quad[0,1] $$

Step-by-Step Solution

Answer

The average function value is \( e - 1 \).

Step 1: Understand the Problem

We need to find the average function value of the given function over the interval [0,1]. The function is \( y = e^x \). The average value of a function \( f(x) \) on an interval \([a, b]\) can be found using the formula: \[ \text{Average} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]

Step 2: Set Up the Integral

Substitute the function \( f(x) = e^x \), and the interval values \( a = 0 \) and \( b = 1 \) into the formula: \[ \text{Average} = \frac{1}{1-0} \int_{0}^{1} e^x \, dx \] This simplifies to: \[ \text{Average} = \int_{0}^{1} e^x \, dx \]

Step 3: Evaluate the Integral

The integral of \( e^x \) with respect to \( x \) is \( e^x \). Apply the limits from 0 to 1: \[ \int_{0}^{1} e^x \, dx = \left[ e^x \right]_{0}^{1} = e^1 - e^0 \] Simplifying further: \[ e^1 - e^0 = e - 1 \]

Step 4: Calculate the Average

Now calculate the average value using the result from the integral: \[ \text{Average} = e - 1 \] Thus, the average value of the function on the interval [0, 1] is \( e - 1 \).

Key Concepts

Integral Calculus

Integral calculus is a significant part of calculus focused on the concept of accumulation. It is commonly used to compute areas under curves, the total amount accumulated, or changes in quantities. In the exercise, integral calculus plays a role in determining the average value of the exponential function over a specified interval.

• The process of integrating involves finding the antiderivative of the function.
• In our case, the function is an exponential function, namely, \( y = e^x \).
• To find the average value of a function across an interval, integration is a key tool.

The integral from Step 2 is used to evaluate the accumulated area under the graph of the function from point \( x=0 \) to \( x=1 \). It provides insight into how much area is covered by the function, which reflects how the function accumulates over the interval.

Exponential Functions

Exponential functions have the general form \( f(x) = a \, e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. These functions are crucial in models related to growth and decay processes.

• In our problem, the specific function considered is \( y = e^x \).
• This form represents one of the simplest types of exponential growth, where the rate of increase is directly proportional to the current value.

This function does not intersect the x-axis; it is always greater than zero for any real \( x \), which affects the average value calculation. Studying these functions helps in comprehending phenomena like population growth or money compounding, where growth rates change continuously over time.

Definite Integrals

Definite integrals provide a means to compute the exact accumulation over an interval, which is essential for determining average values in calculus problems.

• The notation \( \int_{a}^{b} f(x) \, dx \) represents a definite integral from \( a \) to \( b \).
• The limits of integration \( a \) and \( b \) define the start and end points, respectively.

By evaluating the definite integral of \( y = e^x \) over the interval [0, 1], we determine how the function's value accumulates between these points. Performing the integration gives the result \( e^1 - e^0 \), or \( e - 1 \), elucidating the sum of the infinite small slices under the curve. This calculation enables us to derive an average that represents the function's behavior over the specified range.