Calculus I (MAT 221)

Mathematica Project 2

For this project, we will be looking at implicitly deﬁned curves. The goal is to be able to plot an implicit curve using Mathematica, ﬁnd the derivative of that implicit curve, and then plot tangent lines together with the original curve. We will be using the operation ContourPlot to graph these implicit functions. For example, to plot the unit circle x^2+y^2 = 1 on a window of −1.5 ≤ x ≤ 1.5 and −1.5 ≤ y ≤ 1.5, we would write the code as:

ContourPlot[x^2 + y^2 == 1 , {x,−1.5,1.5} , {y,−1.5,1.5}]

1) Let’s start by looking at the the basic hyperbola, x^2−y^2 = 1.

a) Plot this hyperbola.

b) In Mathematica write a detailed step by step process on how to ﬁnd the derivative, dy/dx, by using implicit diﬀerentiation.

c) Using the Derivative operation in Matheamtica, conﬁrm your derivative from the previous step.

d) Using this, ﬁnd the derivative at the point (−2,√3). (-2, radical (3))

e) Graph the tangent line on the same plot as the hyperbola.

2) A famous implicitly deﬁned curve is the Lemniscate of Bernoulli. This curve is deﬁned by: (x^2 + y^2)^2 = 4(x^2 −y^2).

a) Plot the Lemniscate of Bernoulli.

b) In Mathematica write a detailed step by step process on how to ﬁnd the derivative, dy/dx, by using implicit diﬀerentiation.

c) Using the Derivative operation in Matheamtica, conﬁrm your derivative from the previous step.

d) Find the derivatives (yes plural) when x = 1. Look at the function, how many points have an x-coordinate of 1?

e) Graph the tangent lines at these points on the same set of axes as the Lemniscate.

3) Now let’s look at another famous implicitly deﬁned curve, the folium of Descartes. This curve is deﬁned by: x^3 + y^3 = 6xy.

a) Plot the folium of Descartes.

b) In Mathematica write a detailed step by step process on how to ﬁnd the derivative, dy/dx, by using implicit diﬀerentiation.

c) Using the Derivative operation in Matheamtica, conﬁrm your derivative from the previous step.

d) Find all points where dy/dx = 0 (horizontal tangent lines). You should use Mathematica and the Solve or Reduce options.

e) Plot the horizontal tangent lines on the same plot as the folium.

f) We could also ﬁnd the vertical tangent lines by looking where dy/dx is undeﬁned. But, this function is its own inverse so if you reﬂex your horizontal tangent lines over the line y = x, you will ﬁnd your vertical tangent lines. Try this to get the vertical tangent lines and plot them to conﬁrm they are the vertical tangent lines.

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