Mathematica implicit differentiation

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 x2+y2 = 1 on a window of −1.5 ≤ x ≤ 1.5 and −1.5 ≤ y ≤ 1.5, we would write the code as:

ContourPlot[x2 + y2 == 1 , {x,−1.5, 1.5} , {y,−1.5, 1.5}]

1) Let’s start by looking at the the basic hyperbola, x2 − y2 = 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. e) Using this, find the derivative at the point (−2,

√ 3).

f) 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: (x2 + y2)2 = 4(x2 − y2). 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: x3 + y3 = 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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