Implicit Functions

Section 1.12 Implicit Functions

Subsection 1.12.1 Recommended Tutorials

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

🔗

Subsection 1.12.2 Introduction

In this activity, you will learn how to plot implicit functions and compute their derivatives. You will need to ensure that the plots package is loaded using the with() command. It is typically a good idea to include 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.

🔗

Exercises 1.12.3 Exercises

1.

The equation of a circle with radius \(r\) and centre \((h,k)\) is given by

\begin{equation*} (x-h)^2+(y-k)^2=r^2\text{.} \end{equation*}

🔗

Start by assigning the equation of a circle with radius \(5\) and centre at the origin to the name circle. In the following exercise, you will be determining the equations of two tangent lines to this circle and plotting them with the circle. A very similar example may be found in Example 13.2.

🔗

(a)

Start by plotting this circle using the implicitplot() command. Make sure that the entire circle is visible and appears smooth.

🔗

Hint.

When plotting an equation using the implicitplot() command, the graph may not always appear smooth. This was especially common in earlier editions of Maple. If this is ever the case, you may include either numpoints=30000 or grid=[250,250] as a parameter in the implicitplot() to increase the number of plot points.

🔗

(b)

There should be exactly two points on this circle where \(x=2\text{.}\) Substitute this value of \(x\) into circle and solve for the two \(y\)-coordinates. You may wish to assign these values for the next steps.

🔗

Hint.

If you are assigning the output of a solve() or fsolve() command to something like yCoords and there are multiple solutions, then you may use yCoords[1] and yCoords[2] as the first and second solution, respectively.

🔗

(c)

Use the implicitdiff() command to assign the derivative, \(\frac{dy}{dx}\text{,}\) of the circle to a name such as dydx. Then, evaluate this expression at the two points where \(x=2\) to obtain two slopes.

🔗

Hint.

You will need to substitute both an \(x\)-coordinate and a \(y\)-coordinate into the derivative for each point when using the subs() command.

🔗

(d)

Now, determine the equations of the two tangent lines at these points and assign them to line1 and line2. Make sure to include the y= in both equations, since the implicitplot() command requires each curve to be given as an equation.

🔗

Hint 1.

Recall that the tangent line equation to the curve at the point \((x_0,y_0)\) is

\begin{equation*} y=m\cdot(x-x_0)+y_0\text{,} \end{equation*}

where \(m=\frac{dy}{dx}\Bigr|_{(x_0,y_0)}\text{.}\)

🔗

Hint 2.

Do not forget to include multiplication between variables and brackets to indicate multiplication instead of function notation.

🔗

(e)

Plot the circle and the two tangent lines.

🔗

Hint 1.

The optional scaling=constrained parameter can be included to enforce \(1:1\) scaling. Alternatively, the optional scaling can be performed by clicking on the graph and then clicking on the \(1:1\) button in the plot toolbar at the top of the page.

🔗

Hint 2.

If you are plotting multiple graphs on the same set of axes, it is a good idea to specify plot colours.

🔗

2.

In this exercise, you will investigate the equations of tangent lines to the Folium of Descartes, given by the equation

\begin{equation*} x^3 + y^3 = 6xy\text{.} \end{equation*}

🔗

(a)

Assign the equation of this curve to descartes and plot it. You may need to adjust the range of your graph to get a good idea of the shape of this curve. You should be able to visibly see three points on the curve where \(x=1\text{.}\)

🔗

(b)

Compute the derivative, \(\frac{dy}{dx}\) and optionally, assign this to a name such as dydx2.

🔗

(c)

There are three points on the Foilum of Descartes where \(x=1\text{.}\) Determine the equations of the tangent lines at these three points. Then, plot the curve and all three lines on the same graph.

🔗

Hint 1.

It is a good idea to assign these equations to unused names such as line3, line4, and line5 so that they don’t get confused with your tangent lines in the previous exercise.

🔗

Hint 2.

Depending on the complexity of the equation of the curve, you may find that using fsolve() to get numerical values for the \(y\)-coordinates is a better option.

🔗

(d)

Looking at the graph of the Folium of Descartes, you should notice that there are multiple points where the tangent line is horizontal. Solve for these points.

🔗

Hint 1.

Recall that a curve has a horizontal tangent line when \(\frac{dy}{dx} = 0\text{.}\)

🔗

Hint 2.

In order to solve for both \(x\) and \(y\text{,}\) you will need to solve a system of two equations. One equation is that the derivative must equal zero, but the second equation is the curve itself.

🔗

An example of solving a system of equations can be found in Section 10.4.

🔗