3.6 Implicit Differentiation

Mathematica script by Chris Parrish,

cparrish@sewanee.edu

Problems from James Stewart,

"Calculus," Second Edition, Brooks/Cole, 2001

Acknowledgement: Thanks to Prof. Clay Ross for showing me how to use ContourPlot to produce these pictures.

Graphing Implicit Functions: Folium of Descartes

Stewart, Section 3.6

Let's plot the Folium of Descartes.

Start by graphing the surface f[x,y] =

Here are some level curves of that surface - a kind of topographic map showing the lay of the land..

And here is the Folium of Descartes -- the level curve at level 0.

Let's make a transect of theough this region, over hill and dale, always keeping y = 2.

Which points on this orange curve correspond to points on the Folium of Descartes?

Graphing Implicit Functions: Concoid of Nicomedes

Stewart, Exercise 3.6.18

Let's plot the Conchoid of Nicomedes.

Implicit Differentiation Using Mathematica

This exercise is based on the development shown in Green, Evans, Johnson, "Exploring Calculus with Mathematica, to Accompany Calculus, by Hughes-Hallett, Gleason, et. al.," pages 127-129

Define y = y(x) by an equation it must satisfy, and then find dy/dx by implicit differentiation.

Here is the curve we will study.

Now let's find dy/dx by implicit differentiation.

Find dy/dx when x= and y=-.

(1) First, verify that the point P(,-) lies on the curve.

(2) Now evaluate y' at the point P(,-).

Tangent Lines for Implicit Functions

Find the equation of the tangent line to the previous graph at the point P(,-).

Plot the tangent line on the same screen as the curve.

Use the tangent line to approximate the value of y when x = 1.5.

We calculated the tangent line at a = Sqrt[2], so let's first of all verify that 1.5 is pretty close to .

Now, use the tangent line to the curve at x = to approximate the value of y when x = 1.5.

Finally, use the original equation to evaluate the accuracy of this last approximation.

Calculate the relative accuracy of the tangent line approximation to y[1.5],

taking Mathematica's approximation as the best answer available at this stage.

The relative difference is less than 0.89%

Created by Mathematica (April 19, 2004)