Average Value of a Function on a Shrinking Interval

Section 3.6 Average Value of a Function on a Shrinking Interval

Subsection 3.6.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

🔗

Subsection 3.6.2 Introduction

The average value of a function \(f\) on the interval \([a,b]\) is defined as

\begin{equation*} f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx\text{.} \end{equation*}

In this activity, we will investigate the function

\begin{equation*} f(x) = \sqrt{1 + x^3} \end{equation*}

over a shrinking interval where \(a = 2\) is fixed and \(b\) approaches \(a\text{.}\) To do this, we can let \(b=2+h\) and take the limit as \(h \rightarrow 0\text{.}\) Specifically, we will determine the value of the integral

\begin{equation*} \displaystyle\lim_{h \to 0} \dfrac{1}{h} \displaystyle\int_{2}^{2+h} \sqrt{1 + x^3}\, dx\text{.} \end{equation*}

For convenience, we can view

\begin{equation} avg(h) = \frac{\displaystyle\int_{2}^{2+h} \sqrt{1 + x^3}\, dx}{h}\tag{3.5} \end{equation}

as a function of \(h\text{.}\) This function gives the average value of \(f(x) = \sqrt{1 + x^3}\) over the interval \([2,2+h]\text{.}\)

🔗

In this activity, you will ultimately need to determine the limit of \(avg(h)\) as \(h \to 0\text{.}\)

🔗

Exercises 3.6.3 Exercises

1.

In this exercise, you will use function defined in equation (3.5) to determine and visualize the average value of \(f(x) = \sqrt{1 + x^3}\) over the interval \([2,4]\text{.}\)

🔗

(a)

Start by assigning \(f(x) = \sqrt{1 + x^3}\) and the \(avg(h)\) function given in equation (3.5) using the assignment operator, :=.

🔗

Hint.

The \(avg(h)\) function can also be defined as

\begin{equation*} avg(h) = \frac{\displaystyle\int_{2}^{2+h} f(x)\, dx}{h}\text{,} \end{equation*}

assuming that you have assigned \(f(x)\) in Maple first. Note that you should use \(h\) as the independent variable of this function.

🔗

(b)

Use the \(avg(h)\) function to find the average value of \(f(x) = \sqrt{1+x^3}\) on the interval \([2,4]\text{.}\)

🔗

(c)

Plot \(avg(2)\) and \(f(x)\) on the same axes over the interval \([2,4]\text{.}\) Does \(avg(2)\) appear to be the average value of \(f(x) = \sqrt{1+x^3}\) on the interval \([2,4]\text{?}\)

🔗

2.

Try to determine the average value of \(f(x)\) as the interval width shrinks to zero (\(h \to 0\)) graphically. Plot \(avg(h)\) over the interval \(h\in[-1,1]\) and estimate the value

\begin{equation*} \displaystyle\lim_{h \to 0} \dfrac{\displaystyle\int_{2}^{2+h} \sqrt{1 + x^3}\, dx}{h} \end{equation*}

from your graph.

🔗

3.

In this exercise, you will determine the value the limit analytically, using Maple to assist with methods that you would otherwise use on paper.

🔗

(a)

Try evaluating the integral \(\displaystyle\int_{2}^{2+h} \sqrt{1 + x^3} \; dx\) using the int() command or Int() and value(%) commands. Notice how Maple evaluates the integral, but not in terms of elementary functions.

🔗

Aside

Hint.

There is no easy integration technique for this integral. You’ll notice that Maple evaluates the integral in terms of the Elliptic F function, a transcendental function.

🔗

(b)

Luckily, you should not need to evaluate this integral in order to evaluate the limit. Instead, you may apply l’Hôpital’s rule, so long as the limit is indeterminate of the form \(0/0\) or \(\infty/\infty\text{.}\) Which of these two indeterminate forms is applicable?

🔗

(c)

Applying l’Hôpital’s rule to the limit results in

\begin{equation*} \lim_{h \to 0} avg(h) = \lim_{h \to 0}\frac{\displaystyle\int_{2}^{2+h} f(x)\, dx}{h} = \lim_{h \to 0}\frac{\frac{d}{dh}\displaystyle\int_{2}^{2+h} f(x)\, dx}{\frac{d}{dh}h}\text{.} \end{equation*}

Obviously \(\frac{d}{dh}h=1\) in the denominator, but you will need to apply the Fundamental Theorem of Calculus for the numerator. Use the diff() command to evaluate

\begin{equation*} \dfrac{d}{dh}\displaystyle\int_{2}^{2+h} f(x)\, dx \end{equation*}

and see how the Fundamental Theorem of Calculus is applied.

🔗

(d)

Using this result, determine \(\displaystyle\lim_{h \to 0} avg(h)\) on your own.

🔗

4.

Finally, it’s time to let Maple evaluate this limit, letting the power of the limit() command and the Limit Methods tutor do all the work.

🔗

(a)

Check your answer to question 3.6.3.3 by using the limit() command to evaluate \(\displaystyle\lim_{h \to 0} avg(h)\text{.}\)

🔗

(b)

Use the Limit Methods tutor to evaluate \(\displaystyle\lim_{h \to 0} avg(h)\) to see how l’Hôpital’s Rule is applied, along with a variety of limit laws.

🔗

Hint.

After opening the Limit Methods tutor, you can simply type \(avg(h)\) for the function. Just don’t forget to change the variable to \(h\text{!}\)

🔗

Prev Top Next