8/10/2019 Implicit Differentiation and Tangent Lines Mathematica Lab

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Implicit Differentiation and Tangent LinesBy Bruce Bordwell

Objective: To graph implicitly defined functions and use implicit differentiation to find tangent lines.

Narrative: Implicitly defined functions offer a wide variety of creative graphs when plotted. This lab will explorethe graphs of some implicit functions and use implicit differentiation to find equations of tangent lines to the graphs of

these functions.

Part 1: Use implicit differentiation to find the equation of the tangent line to the Devils Curve defined as

( ) ( )2 2 2 24 4y y x x = .Task:

1. Type the command lines in the left-hand column below into Mathematica in the order in which they are listed. Theeffect of each command is described in the right-hand column for your reference.

In[1] :=(* Your Name Todays date *)

In[2] :=(* Ch 3 Computer Lab Implicit Differentiation *)

In[3] :=

8/10/2019 Implicit Differentiation and Tangent Lines Mathematica Lab

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Part 2: Use Mathematica to answer question #24 from Sec 3.4in your textbook that refers to the BouncingWagon curve implicitly defined by

3 2 5 4 3 22 2y y y x x x+ = +

2. Type the command lines in the left-hand column below into Mathematica in the order in which they are listed. The

effect of each command is described in the right-hand column for your reference.

In[8] := ImplicitPlot[2 y^3+y^2-y^5x^4-2 x^3+x^2,{x,-2,3}]

In[9] := D[2 y[x]^3+y[x]^2-y[x]^5x^4-2 x^3+x^2,x]

In[10] := Solve[%,y'[x]]

In[11] :=

SolveA2Hx 3x2+ 2x3L

y@xD H2 6y@xD + 5y@xD3L 0, xE

(Solves y[x] = 0, to findhorizontal tangents. Cut and pastethe expression you found for y[x]in the Solve function.)

At this time make a hard-copy of your input and Mathematicas responses.

3. On the graphic you created in Part 1, draw by hand the tangent line at the point (0, 2) and label the tangent line

with its equation.

4.On the graphic you created in Part 2, draw by hand the points on the graph where the tangent is horizontal and label

the points with theirx- coordinate.

If you have extra time, you can explore these other functions with the methods you learned from this Lab.

a)2 3 23y x x= + (Tschirnhausen Cubic) analyzed in #22, Sec 3.6in your textbook

b)

( )( ) ( )( )2

1 2 1 2y y y x x x = analyzed in #23, Sec 3.6in your textbook

c) ( ) ( )2

2 2 2 22 25x y x y+ = (Lemniscates) - analyzed in #19, Sec 3.6in your textbook

d)2 4 25y x x= (Kampyle of Eudoxus) - analyzed in #21, Sec 3.6in your textbook