Problem 7 - Applied Calculus

Question

Find the average value of each function over the given interval. \(f(x)=2 x+1\) on [0,4]

Step-by-Step Solution

Answer

The average value of the function is 5.

Step 1: Understand the Problem

To find the average value of a continuous function \(f(x)\) over a specific interval \([a, b]\), we calculate the integral of the function over the interval and divide by the interval's length.

Step 2: Write the Formula

The formula to find the average value of a function \(f(x)\) on the interval \([a, b]\) is given by:\[\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx\]In this case, \(a = 0\), \(b = 4\), and \(f(x) = 2x + 1\).

Step 3: Set Up the Integral

Substitute the given function and limits into the integral:\[\int_{0}^{4} (2x + 1) \, dx\]

Step 4: Calculate the Integral

To evaluate \(\int_{0}^{4} (2x + 1) \, dx\), find the antiderivative:\[ \int (2x + 1) \, dx = x^2 + x + C \]Evaluate from 0 to 4:\[\left[ x^2 + x \right]_{0}^{4} = (4^2 + 4) - (0^2 + 0) = 16 + 4 = 20\]

Step 5: Divide by the Interval Length

The length of the interval \([0, 4]\) is \(4 - 0 = 4\). Now, divide the integral result by this interval length:\[\text{Average value} = \frac{20}{4} = 5\]

Key Concepts

calculus integral

In calculus, the integral is a fundamental concept, representing the area under a curve of a function. The integral calculates the total accumulation of values, which can relate to many real-world phenomena like distance, area, or total income over time.
For a given function, when you integrate over a specific interval [a, b], what you do is essentially summing up every infinitesimally small area under the curve from point a to point b.

• The process of integration makes use of the integral sign: \[ \int \]
• It uses limits of integration, such as \( a \) and \( b \) in our example, and expresses the function: \( f(x) \).

The result of this operation often yields another function (often called the anti-derivative) which, when evaluated over the given limits, provides a numerical representation of the area.

antiderivative

The antiderivative, or indefinite integral, is an essential tool that we use to "reverse" differentiate a function. Imagine if you are given the derivative of a function and asked to find the original function—that's what finding an antiderivative involves.

• An antiderivative for a function \( f(x) \) is a function \( F(x) \) whose derivative is \( f(x) \).
• The notation for the antiderivative includes a constant of integration, \( C \), represented as \[ \int f(x) \, dx = F(x) + C \]

In our example involving the function \( f(x) = 2x + 1 \), calculating the antiderivative by integration gives us \( x^2 + x + C \). Evaluating this antiderivative at the endpoints of the interval helps in finding the definite integral, which we further use to find the average value.

continuous function

A continuous function is one where small changes in input yield small changes in output. You can visualize it as a smooth curve without breaks or holes.

• Mathematically, a function \( f(x) \) is continuous over an interval if at every point \( c \) in the interval, \( \lim_{x \to c} f(x) = f(c) \).

This attribute of functions is crucial because only continuous functions can be integrated conveniently across an interval.
For example, our function \( f(x) = 2x + 1 \) is continuous over the interval [0, 4], which makes it suitable for applying the integral to find its average value over this interval.

interval analysis

Interval analysis involves breaking down the domain of a function to examine its behavior over a range. The interval [a, b] acts as the window through which we observe the function's properties.

• In the context of finding average values, this interval determines the limits of integration.
• The formulation \([a, b]\) signifies we only focus on those x-values between a and b inclusively.

Applying interval analysis helps in seamlessly defining where to start and stop gathering data from a function. For instance, in the exercise, we examine \( f(x) = 2x + 1 \) over the interval [0, 4] to compute its average value. We calculate the integral of \( f(x) \) over this interval and then divide by the interval’s length (4-0) to get the average. This process provides insight into the function's behavior across the specified domain.