Wolfram mathematica software project calculus i (mat


wolfram mathematica software device

Calculus I (MAT 221)

Mathematica Device 2

For this device, we conquer be observeing at impliedly defined deflexions. The intent is to be conducive to conspire an implied deflexion using Mathematica, find the derivative of that implied deflexion, and then conspire tangent rows contemporaneously after a while the former deflexion. We conquer be using the agency ContourPlot to graph these implied discharges. For stance, to conspire the individual dispersion x^2+y^2 = 1 on a window of −1.5 ≤ x ≤ 1.5 and −1.5 ≤ y ≤ 1.5, we would transcribe the principle as:

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

1) Let’s set-on-foot by observeing at the the basic hyperbola, x^2−y^2 = 1.

a) Conspire this hyperbola.

b) In Mathematica transcribe a specific stalk by stalk arrangement on how to find the derivative, dy/dx, by using implied differentiation.

c) Using the Derivative agency in Matheamtica, confirm your derivative from the anterior stalk.

d) Using this, find the derivative at the sharp-end (−2,√3). (-2, extreme (3))

e) Graph the tangent row on the selfselfselfcorresponding conspire as the hyperbola.

2) A celebrated impliedly defined deflexion is the Lemniscate of Bernoulli. This deflexion is defined by: (x^2 + y^2)^2 = 4(x^2 −y^2).

a) Conspire the Lemniscate of Bernoulli.

b) In Mathematica transcribe a specific stalk by stalk arrangement on how to find the derivative, dy/dx, by using implied differentiation.

c) Using the Derivative agency in Matheamtica, confirm your derivative from the anterior stalk.

d) Find the derivatives (yes plural) when x = 1. Observe at the discharge, how divers sharp-ends own an x-coordinate of 1?

e) Graph the tangent rows at these sharp-ends on the selfselfselfcorresponding set of axes as the Lemniscate.

3) Now let’s observe at another celebrated impliedly defined deflexion, the folium of Descartes. This deflexion is defined by: x^3 + y^3 = 6xy.

a) Conspire the folium of Descartes.

b) In Mathematica transcribe a specific stalk by stalk arrangement on how to find the derivative, dy/dx, by using implied differentiation.

c) Using the Derivative agency in Matheamtica, confirm your derivative from the anterior stalk.

d) Find all sharp-ends where dy/dx = 0 (downright tangent rows). You should use Mathematica and the Solve or Reduce options.

e) Conspire the downright tangent rows on the selfselfselfcorresponding conspire as the folium.

f) We could to-boot find the upright tangent rows by observeing where dy/dx is undefined. But, this discharge is its own inverse so if you reflex your downright tangent rows aggravate the row y = x, you conquer find your upright tangent rows. Try this to get the upright tangent rows and conspire them to confirm they are the upright tangent rows.