Problem 29 - Calculus Early Transcendentals

Question

Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. \(f(x)=x^{n}\) on \([0,1],\) for any positive integer \(n\)

Step-by-Step Solution

Answer

Answer: The average value of the function \(f(x) = x^n\) on the interval \([0, 1]\) for any positive integer \(n\) is \(\frac{1}{n+1}\).

Step 1: Compute the integral of the function on the given interval

To find the average value of the function \(f(x)=x^n\) on \([0,1]\), we first need to compute the integral of the function on this interval. The integral will give us the area under the curve of the function, and when we divide by the length of the interval, we will get the average value. To compute the integral, we have: \(\int_{0}^{1} x^n dx\)

Step 2: Use the power rule for integration

We will now apply the power rule for integration to compute the integral of \(f(x)=x^n\). The power rule for integration states: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) Applying the power rule to our integral, we get: \(\int_{0}^{1} x^n dx = \left[ \frac{x^{n+1}}{n+1} \right]_0^1\)

Step 3: Evaluate the integral

Now, we will evaluate the integral by applying the limits of integration: \(\left[ \frac{x^{n+1}}{n+1} \right]_0^1 = \frac{1^{n+1}}{n+1} - \frac{0^{n+1}}{n+1} = \frac{1}{n+1}\)

Step 4: Find the average value

To find the average value of the function \(f(x)=x^n\) on the interval \([0,1]\), we divide the value of the integral by the length of the interval. The length of the interval is \(1-0=1\). Therefore, the average value is: \(\frac{1}{1} \cdot \frac{1}{n+1} = \frac{1}{n+1}\)

Step 5: Draw the graph and indicate the average value

To represent the average value graphically, you will draw the graph of the function \(f(x)=x^n\) for any positive integer \(n\). On the interval \([0,1]\), the graph will start at the point \((0, 0)\) and end at the point \((1, 1)\), where its shape will depend on the value of \(n\). To indicate the average value, draw a horizontal line at the height of the average value, which is \(\frac{1}{n+1}\). This line will represent the constant function \(y=\frac{1}{n+1}\) on the interval \([0,1]\). The area under this horizontal line and the interval \([0,1]\) will be equal to the area under the curve of the function \(f(x)=x^n\) on the interval \([0,1]\).

Key Concepts

Definite Integral

The term definite integral refers to the evaluation of an integral with specific boundaries; it represents the signed area under a curve defined by a function on a given interval. When calculating the average value of a function such as f(x) = x^n on an interval from 0 to 1, the definite integral \(\frac{1}{n+1}\) provides the total area under the curve between these two points.

Identifying the area under the curve is crucial because it offers valuable information about the quantity represented by the function within specified limits. In real-world scenarios, this could translate to the total distance traveled by a car when the function represents the car's speed over time, or the accumulated resources consumed when the function depicts the rate of consumption over a period.

Power Rule for Integration

One of the foundational tools in calculus is the power rule for integration, a technique used to integrate polynomials. According to this rule, to integrate a power of x, you increase the exponent by one and then divide by the new exponent, adding a constant of integration at the end. Mathematically, it is expressed as \(\frac{x^{n+1}}{n+1} + C\).

This simplification makes it straightforward to handle otherwise complex calculations involving polynomial functions. Notably, when you are dealing with a definite integral, the constant of integration is not required, as it cancels out when the boundaries are applied. This rule significantly reduces the effort required to determine areas under curves and solve various physical problems like finding the work done against a force.

Graphing Functions

Graphing functions is a crucial skill in calculus that helps visualize the behavior of mathematical equations. By plotting the function such as f(x) = x^n on a Cartesian plane, we gain insight into its characteristics, like continuity, slope, and symmetry. For different values of n, the graph will display distinct shapes, but it will always pass through the points (0,0) and (1,1) on the interval [0,1].

A function's graph can reveal the average value at a glance, as it is graphically represented by the horizontal line intersecting the y-axis at the average value \(\frac{1}{n+1}\). Understanding how to graph functions enhances comprehension of calculus concepts, as it converts abstract numerical data into a visual form, enabling easier interpretation and comparison.

Calculus

Calculus is an expansive field in mathematics, focusing on change and motion. It is divided into two branches: differential calculus, which deals with the rates at which quantities change, and integral calculus, the focus of our current exploration, which concerns the accumulation of quantities and the areas under and between curves. Understanding calculus is essential in many scientific and engineering fields because it enables the precise modeling of systems and prediction of behaviors.

Through concepts like the average value of a function, calculus allows us to distill complex and dynamic systems into more manageable figures that represent the 'average' state over an interval, offering a simplified yet powerful insight into the nature of the system being studied.