Problem 53
Question
Find the average value of the function on the given interval. \(f(x)=x^{2}-1,[1,3]\)
Step-by-Step Solution
Verified Answer
The average value of the function \(f(x)=x^{2}-1\) on the interval \([1,3]\) is 3.5.
1Step 1: Understand the Average Value of a Function
The formula for the average value of a function \(f(x)\) over an interval \([a,b]\) is: \[\frac{1}{b-a} \int_{a}^{b} f(x) \, dx\]In this case, \(f(x) = x^{2}-1\), \(a = 1\) and \(b = 3\).
2Step 2: Substitute Values into the Formula
Substitute \(f(x), a,\) and \(b\) into the formula:\[\frac{1}{3 - 1} \int_{1}^{3} (x^{2}-1) \, dx\]This simplifies to: \[\frac{1}{2} \int_{1}^{3} (x^{2}-1) \, dx\]
3Step 3: Compute the Integral
Calculate the integral: \[\frac{1}{2} [ \frac{x^{3}}{3} - x] \Biggr|_1^3\]This evaluates to:\[\frac{1}{2} [( \frac{3^{3}}{3} - 3) - ( \frac{1^{3}}{3} - 1)]\]which simplifies to: \[\frac{1}{2} [(9 - 2)]\]
4Step 4: Calculate the Average Value
Simplify to get the average value: \[\frac{1}{2} \cdot 7 = 3.5\]
Key Concepts
Definite IntegralIntegral CalculusFunction Interval Analysis
Definite Integral
The concept of the definite integral is fundamental in understanding how to find the overall accumulation of a quantity, which could be area under a curve, work done by a force, or other similar cumulative effects. The definite integral is expressed as a numerical value representing the sum of infinitesimal parts under a curve within a specific interval.
For instance, when you see \[\int_{a}^{b} f(x) \, dx\], it means you’re looking at the total accumulation of the function f(x) as x goes from a to b. This is a powerful tool because it transitions beyond simply finding the total accumulation and into applications such as finding the average value of a function over an interval.
In our exercise, the integration bounds are 1 and 3 for the function \(f(x) = x^2 - 1\), indicating we’re interested in the cumulative effect of this function between those two points. This task is accomplished by evaluating the antiderivative (the function whose derivative is \(f(x)\)) at the upper limit and subtracting its value when evaluated at the lower limit.
For instance, when you see \[\int_{a}^{b} f(x) \, dx\], it means you’re looking at the total accumulation of the function f(x) as x goes from a to b. This is a powerful tool because it transitions beyond simply finding the total accumulation and into applications such as finding the average value of a function over an interval.
In our exercise, the integration bounds are 1 and 3 for the function \(f(x) = x^2 - 1\), indicating we’re interested in the cumulative effect of this function between those two points. This task is accomplished by evaluating the antiderivative (the function whose derivative is \(f(x)\)) at the upper limit and subtracting its value when evaluated at the lower limit.
Integral Calculus
Integral calculus is a branch of mathematics that deals with understanding and calculating integrals. While differential calculus focuses on the rate of change, integral calculus is all about the accumulation of quantities.
In real-world terms, think of integral calculus as a method to calculate total growth, area, volume, or other accumulative measures. It’s like knowing the speed at which a car travels and then figuring out the total distance covered over a period of time. In the context of our exercise, we use integral calculus to determine the area under the curve of \(f(x) = x^{2}-1\) from \(x = 1\) to \(x = 3\).
The calculation process involves finding the antiderivative, substituting the boundary points into this antiderivative, and then finding the difference. The capability to calculate definite integrals enables the evaluation of many real-life quantities, something that is central to the fields of physics, engineering, economics, and beyond.
In real-world terms, think of integral calculus as a method to calculate total growth, area, volume, or other accumulative measures. It’s like knowing the speed at which a car travels and then figuring out the total distance covered over a period of time. In the context of our exercise, we use integral calculus to determine the area under the curve of \(f(x) = x^{2}-1\) from \(x = 1\) to \(x = 3\).
The calculation process involves finding the antiderivative, substituting the boundary points into this antiderivative, and then finding the difference. The capability to calculate definite integrals enables the evaluation of many real-life quantities, something that is central to the fields of physics, engineering, economics, and beyond.
Function Interval Analysis
Function interval analysis is the investigation of a function's behavior within a certain range, or interval, of its domain. By analyzing a function on an interval, we can discover a great deal about the function's properties, such as its overall increase or decrease, concavity, and the average value of the function on that interval.
In the context of our problem, we have the function \(f(x) = x^{2}-1\) and the interval \[1,3\]. Analyzing \(f(x)\) on this interval, we determine the average value by integrating the function over the interval and then dividing by the width of the interval. This average value represents an equilibrium point where the function could be flattened out over the interval and it would balance evenly.
Therefore, function interval analysis not only helps us understand the nuances of the function's behavior but also equips us with a meaningful statistical measure which, in practical terms, could represent an average rate, an average temperature over a time period, or any other averaged quantity.
In the context of our problem, we have the function \(f(x) = x^{2}-1\) and the interval \[1,3\]. Analyzing \(f(x)\) on this interval, we determine the average value by integrating the function over the interval and then dividing by the width of the interval. This average value represents an equilibrium point where the function could be flattened out over the interval and it would balance evenly.
Therefore, function interval analysis not only helps us understand the nuances of the function's behavior but also equips us with a meaningful statistical measure which, in practical terms, could represent an average rate, an average temperature over a time period, or any other averaged quantity.
Other exercises in this chapter
Problem 52
Suppose that a car can come to rest from 60 mph in 3 seconds. Assuming a constant (negative) acceleration, find the acceleration (in miles per second squared) o
View solution Problem 53
Generalize exercise 51 to \(I=\int_{0}^{d} \frac{f(x)}{f(x)+f(a-x)} d x\) for any positive, continuous function \(f\) and then quickly evaluate \(\int_{0}^{\pi
View solution Problem 53
Suppose that the average value of a function \(f(x)\) over the interval [0,2] is 5 and the average value of \(f(x)\) over the interval [2,6] is \(11 .\) Find th
View solution Problem 54
Generalize the result of exercise 53 to \(\int_{2}^{4} \frac{f(9-x)}{f(9-x)+f(x+3)} d x\) for any positive, continuous function \(f\) on [2,4]
View solution