Riemann Sums for Monotonic Functions

Section 3.1 Riemann Sums for Monotonic Functions

Subsection 3.1.1 Recommended Tutorials

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

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Subsection 3.1.2 Introduction

A monotonic function is one that is strictly increasing or strictly decreasing. In this activity, you will use the ApproximateInt() command to visualize and calculate Riemann sums for the area below the function

\begin{equation*} f(x) = x - 2\ln(x)\text{.} \end{equation*}

When analyzing Riemann sums for this function, it is useful to know whether the function is monotonically increasing or decreasing on the specified interval. This is because a right-endpoint or left-endpoint method may reliably give either an overestimate or underestimate of the true area under the curve.

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You will need to load the Student[Calculus1] package to use the ApproximateInt() command. You can do with the command with(Student[Calculus1]): in your worksheet. It is typically a good idea to load any necessary packages at the start of your Maple worksheet. If you close and reopen your worksheet, you will need to run the command to load the packages again.

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Exercises 3.1.3 Exercises

1.

Start by assigning the function \(f(x) = x - 2\ln(x)\) using the assignment operator, :=.

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(a)

Use the ApproximateInt() command to estimate the area under \(f(x)\) on the interval \([2,10]\) with \(10\) partitions. Use the options method=left and output=plot.

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(b)

Use the ApproximateInt() command to estimate the area under \(f(x)\) on the interval \([2,10]\) with \(10\) partitions. Use the options method=right and output=plot.

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(c)

For a monotonically increasing function on a given interval such as this, which method will always be an underestimate? Which method always gives an overestimate? Provide an answer to these questions in your worksheet, using paragraph (text) mode.

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(d)

Now suppose a function is monotonically decreasing on a given interval. Which method will always be an underestimate? Which method always gives an overestimate? Provide an answer to these questions in your worksheet.

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2.

In this exercise, you will change the output of the ApproximateInt() command to be the numerical value only, without the plot. Evaluate each of the following to 15 digit accuracy.

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(a)

Use the ApproximateInt() command to estimate the area under \(f(x)\) on the interval \([2,10]\) with \(10\) partitions. Use the options method=left and output=value.

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Hint.

Don’t forget that you can convert an exact value to a decimal using the evalf() command. You can set the accuracy to 15 digits at the top of your worksheet using the command Digits := 15. Note that the first letter is always capitalized in Digits.

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(b)

Use the ApproximateInt() command to estimate the area under \(f(x)\) on the interval \([2,10]\) with \(10\) partitions. Use the options method=right and output=value.

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3.

In order to find the true area under \(f(x)\) over the interval \([2,10]\text{,}\) you may express the Riemann sum in terms of an arbitrary number of partitions, \(n\text{,}\) and take the limit as \(n \rightarrow \infty\text{.}\)

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(a)

Use the ApproximateInt() command to estimate the area under \(f(x)\) on the interval \([2,10]\) with \(n\) partitions. Use output=sum and either method=right or method=left.

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(b)

Evaluate the limit of the sum from part (a) as \(n \rightarrow \infty\text{.}\)

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4.

Evaluate the Riemann sum for \(f(x)\) on the interval \([0.5, 3]\) using \(10\) partitions and both left- and right-endpoint methods. Is \(f(x)\) monotonic on this interval? Is it obvious from a graph which method will result in an overestimate or underestimate of the true area?

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