Solving Equations in Maple

Section 1.5 Solving Equations in Maple

Subsection 1.5.1 Recommended Tutorials

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

Equation Solvers

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Subsection 1.5.2 Introduction

In this activity, you will investigate the two main solvers that Maple uses, solve() and fsolve(). In some cases, you may find one is much more helpful than the other. Both commands provide unique benefits in specific circumstances.

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Exercises 1.5.3 Exercises

1.

Suppose we want to find the \(x\)-intercepts of the function

\begin{equation*} f(x) = x^5+x^4-4x^3-3x^2+3x+1\text{.} \end{equation*}

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Aside

(a)

Assign the function to \(f(x)\) using the assignment operator :=.

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(b)

Plot \(f(x)\text{,}\) choosing ranges for \(x\) and \(y\) so that you can clearly see all five \(x\)-intercepts on the graph (you may need to adjust each a few times).

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(c)

Try factoring \(f(x)\) to see if it is factorable. Does this method work for finding the \(x\)-intercepts?

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(d)

Try solving \(f(x)=0\) using the solve() command. Maple will attempt to give exact values of all five roots. Note that for challenging equations such as this one, the output may be more symbolic than you may prefer.

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(e)

Finally, try solving \(f(x)=0\) using the fsolve() command. Maple will only attempt to give numerical approximations of all five roots. The number of digits given will depend on the current value assigned to Digits.

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2.

Let’s look at how these two solvers behave differently for the quadratic equation \(x^2+4x+6=0\text{.}\)

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Aside

(a)

First, start off by solving the quadratic without Maple using the formula

\begin{equation*} x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \end{equation*}

to see if the equation has two real roots, one repeated root, or no real roots.

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(b)

Now, solve the quadratic using the solve() command. This solver allows for the display of imaginary roots using \(I = \sqrt{-1}\text{.}\)

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(c)

Finally, solve the quadratic using the fsolve() command. Since this solver uses numerical approximation, it will only display real roots (if it finds any).

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3.

In this exercise, you will be finding the intersection point (x,y) of the curves \(y=x^2\) and \(y=\frac{3}{x}\text{.}\)

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(a)

Plot \(x^2\) and \(\frac{3}{x}\) using a single plot() command and adjust the command as necessary to view both curves and their intersection point.

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Hint.

When multiple curves are plotted on the same set of axes, it is a good practice to specify the colour of each one. You can do this after specifying the ranges for \(x\) and \(y\text{.}\)

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(b)

Use either solver (solve() or fsolve()) to find the \(x\)- and \(y\)-values of the point of intersection.

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Hint.

There are multiple ways that you can solve for both! You may start by solving \(x=\frac{3}{x}\) to find the \(x\)-coordinate first and then use subs() to find \(y\text{.}\) Instead, you may want to solve a system of equations in one command. An example is provided in Section 10.4.

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