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Implicit Graphs
Introduction
Implicit graphs are a special type of graph and not used as regularly as many others. Because of this, we need to load in a package before Mathematica can graph them. Evaluate the command line which begins Needs below.
Needs["Graphics`ImplicitPlot`"]
Be sure that the "In[1]:=" appears in front of the command above before continuing. Equations below are entered for you. Pay careful attention to spaces between the x and y to indicate multiplication and also the double equals sign (==), used to indicate an equation. Try these examples. You will see some graphs.
Now see what happens to the next command.
You should receive error messages because the equation is non-algebraic. It turns out that Mathematica can still graph the relation, but it does need a little more help. Therefore give values of y as well, and we also add an option called PlotPoints to make it smoother.
Try a couple more graphs if you wish or continue to the next section.
Problem 1
We will begin with an ellipse. Consider  . Evaluate the next command to give the equation a name: equat1.
Evaluate the next cell to see the graph.
ImplicitPlot[equat1, {x, -4, 4}]
To differentiate the same equation, we use the command Dt[equat1, x], which refers to the total derivative. This takes the derivative of everything with respect to x. We will call it deriveq.
Then solve that result for the derivative, Dt[y, x], which is the same thing as  .
Horizontal tangents
What must happen if the derivative is to be horizontal? We see from the derivative that  . Substitute this into equat1 using the /. notation.
equat1 /. y -> 4x
Solve this resulting equation for x.
Solve[%, x]
Substitute these values of x into the equation of the line to find the corresponding y-values. Or, if you really want, just do it by hand!
y == 4x /. %
Do these points seem to agree with your graph above? To check, click on the graph so that a box appears around the graph. Hold the command key ( ) (Mac) or Control key (Windows) down while you move the cursor around the graph. Don't click! The coordinates of the point at the cursor are shown at the lower left. Note that this does not trace, but you can get quite good approximations.
Vertical tangents
What must happen for vertical tangent lines? Use steps similar to the ones above to find both coordinates of the points where the tangent lines are vertical. Be sure to check that your answers are reasonable by looking at the graph.
When finished with this problem, be sure to clear your equation name.
Clear[equat1]
Problem 2
This time, look at  . Execute the cells below to name and graph the equation.
We follow the same routine as in the first example.
We wish to find both coordinates of all points with horizontal tangents. Create new cells below and use the steps from Problem 1 or get out your pencil, some scratch paper, and your calculator. Do these points seem to agree with the graph above?
What must happen for the tangent line to be vertical? Again, find both coordinates (exact or approximate) of all points where this happens. (You might want to get a help from Mathematica or your calculator.)
What happens to f '(x) at (0, 0)? What happens to the graph at (0, 0)?
Problem 3
This time, consider  . We begin as before.
Now it is time for you to do more typing. Create new cells to do each of the following. Find the derivative using Dt[equat3, x].
Solve for Dt[y, x].
Find both coordinates of any points where the tangent line is horizontal. (You will definitely want to use Mathematica.)
Find both coordinates of any points where the tangent line is vertical.
Problem 4
Our fourth equation is  . We name the equation.
Plot the graph, using  .
Find the derivative.
Find the coordinates of the points with horizontal tangent lines. (You will want the NSolve command for this problem.)
Find the coordinates of the points with vertical tangent lines.
    
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