Describing the Shapes of Integral Functions

Section 3.4 Describing the Shapes of Integral Functions

Subsection 3.4.1 Recommended Tutorials

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

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Subsection 3.4.2 Introduction

In this activity, we will examine two functions that are defined by integrals, in the form

\begin{equation*} f(x) = \int_0^x g(t) \, dt\text{.} \end{equation*}

We may view these integral functions as the accumulated area under a function \(g(t)\) over an interval from \(0\) to \(x\text{,}\) where \(x\) is the variable of the function \(f(x)\text{.}\) Integral functions frequently appear in analysis and in differential equations. Determining critical points and inflection points of integral functions is incredibly important in the analysis of these types of problems.

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Aside

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Exercises 3.4.3 Exercises

1.

The sine integral function

\begin{equation*} Si(x) = \begin{cases}\displaystyle\int_{0}^x \dfrac{\sin(t)}{t}\, dt \amp x \neq 0 \\ 1 \amp x = 0 \end{cases} \end{equation*}

is important in electrical engineering. Note that by defining \(Si(0)=1\) in the piecewise definition above, \(Si(x)\) is a continuous function.

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Aside

(a)

The sine integral function is already defined in Maple, so you don’t have to manually assign it. Plot the graph of \(Si(x)\) over the interval \([-15,15]\text{.}\)

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(b)

On the graph of \(Si(x)\text{,}\) you will notice that there are many local minimum and maximum values. Use the fsolve() command to find the critical numbers of \(Si(x)\) corresponding to the location of the absolute minimum and maximum values.

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Hint.

When you use the fsolve() command, you can specify an interval in which you wish to search for solutions. An example of this can be found in Solving an Equation of One Variable.

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(c)

What are the absolute minimum and maximum values of \(Si(x)\text{?}\)

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(d)

There is an inflection point just to the right of the absolute maximum value. Use the second derivative \(Si''(x)\) and the fsolve() command to find its location.

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(e)

Use the limit() command to find the horizontal asymptote(s) of \(Si(x)\text{.}\)

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Hint.

Recall that if

\begin{equation*} \lim_{x\rightarrow\infty} f(x) = L ~\text{ or } ~ \lim_{x\rightarrow-\infty} f(x) = L \end{equation*}

and \(L\) is finite, then \(y=L\) is a horizontal asymptote of \(f(x)\text{.}\)

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2.

Assign the integral function

\begin{equation*} f(x) = \displaystyle\int_{0}^x \dfrac{1}{1 + t + t^2} \, dt \end{equation*}

using the assignment operator, :=.

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(a)

Plot the integral function, \(f(x)\text{.}\) Try to specify a plot interval that gives you a good idea of the shape of \(f(x)\text{.}\)

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(b)

Use the second derivative, \(f''(x)\text{,}\) to determine where \(f(x)\) is concave up and where \(f(x)\) is concave down.

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Hint.

You may make use of the factor() or solve() commands to help determine where \(f''(x)\) equals zero or is undefined. From there, your graph may help determine where \(f''(x)\) is positive or negative.

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(c)

Determine the inflection points of \(f(x)\text{.}\)

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