Graphs of Implicit Equations

  2. Calculus
  3. Implicit Differentiation
  4. Graphs of Implicit Equations
    5. 30'

Implicit Equations are often used to represent relations that cannot be expressed as functions (one to many relations or many to many relations). These often have really interesting graphs. This page will allow to explore some of these graphs and consider why we need to use implicit diffrentiation. If you are looking for help with the techniques of implicit differentiation, examples and exam-style questions then you should visit  Implicit Differentiation .

Essentials

Here are the graphs of some implicit relations

Circle

A circle is the locus of a point that is a fixed distance from a point

The equation can be written implicity like this \(x^2+y^2=a^2\) where a is the radius of the circle. Drag the point around the circle to consider the gradient of this relation.

We can write the eqution of this circle explicity \(y=\pm\sqrt{a^{2}-x^{2}}\)

This creates 2 derivatives: one for when y>0 and one for when y<0.

Let’s take the case when the radius if the circle is equal to 1. The graph of \(y=\pm\sqrt{1^{2}-x^{2}} \)for y>0 is plotted below.

Sketch the graph of the gradient function. You can check your answer by clicking in the box.

You can find the gradient function of this explicit function using the chain rule. Try it for yourself!

Answer

 

Ellipse

An ellipse is the locus of a point in which the sum of the distances to two fixed points is constant.

The equation can be written implicity like this \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \)

We can write it explicitly like this \(y=\pm b\sqrt{1-\frac{x^{2}}{a^{2}}}\)

A graph of an ellipse can be seen below. You can drag the blue point to check the property of the locus.

It is possible to find the gradient! Consider what the gradient function might look like.

 

 

Casini Oval

A Cassini Oval has a beautiful graph. It is defined as the locus of a point in which the product of the distances to two fixed points is constant.

The equation can be written implicity like this \(((x-a)^2+y^2)((x+a)^2+y^2)=b^4\)

Here is the graph. Adjust the parameters a and b using the sliders and explore the gradient of the graph.

It is rather difficult to write the function of this graph explicitly, but we can still describe the gradient!

 

 

Another Beautiful Implicit Relation 

 

\(sin(x+y) – cos(xy) – 1 = 0\) is another implicit relation with a beautiful graph. It is impossible to write explicitly, but we can still find its gradient.

Try plotting this implicit function by typing the equation in  the input bar below

 
 

 

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