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.

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

We value our customers and so we ensure that what we do is 100% original..

With us you are guaranteed of quality work done by our qualified experts.Your information and everything that you do with us is kept completely confidential.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreThe Product ordered is guaranteed to be original. Orders are checked by the most advanced anti-plagiarism software in the market to assure that the Product is 100% original. The Company has a zero tolerance policy for plagiarism.

Read moreThe Free Revision policy is a courtesy service that the Company provides to help ensure Customer’s total satisfaction with the completed Order. To receive free revision the Company requires that the Customer provide the request within fourteen (14) days from the first completion date and within a period of thirty (30) days for dissertations.

Read moreThe Company is committed to protect the privacy of the Customer and it will never resell or share any of Customer’s personal information, including credit card data, with any third party. All the online transactions are processed through the secure and reliable online payment systems.

Read moreBy placing an order with us, you agree to the service we provide. We will endear to do all that it takes to deliver a comprehensive paper as per your requirements. We also count on your cooperation to ensure that we deliver on this mandate.

Read more
The price is based on these factors:

Academic level

Number of pages

Urgency