4.4 Graphing with Calculus and Calculators

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Problems from James Stewart,
  "Calculus: Concepts and Contexts,"
  Second Edition, Brooks/Cole, 2001

A Template for Drawing the Graphs of Functions and their Derivatives,
and Indicating Critical Points and Points of Inflection

Stewart, Section 4.4

Let's build on the template developed in Section 4.3.
It already images the graph of the function to be studied, together with its first and second derivatives.

In[54]:=

Needs["Graphics`Colors`"] Clear[f, x, derivativePlot]  f[x_] := x^3 - x^2 -  ...          AxesLabel  {"x", None}] ;

f[x] =  -5 - x - x^2 + x^3

f'[x] =  -1 - 2 x + 3 x^2

f''[x] =  -2 + 6 x

[Graphics:HTMLFiles/4.4_graphing_5.gif]

Now, calculate the critical points and points of inflection of this function.

In[37]:=

Solve[f '[x]  0, x]

Out[37]=

{{x -1/3}, {x1}}

In[38]:=

Solve[f''[x]  0, x]

Out[38]=

{{x1/3}}

As an "Epilog," add dots to the image to indicate critical points and points of inflexion.

In[61]:=

Needs["Graphics`Colors`"] Clear[f, x, a, b, c, criticalPoints, pointsOfInfection, do ... p;     , Orchid, ,, Map[Point, pointsOfInfection]}], }}]}]}], ]}]}], ;}]

[Graphics:HTMLFiles/4.4_graphing_11.gif]


Created by Mathematica  (October 19, 2003)