3.4 Derivatives of Trigonometric Functions

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

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

Gallery of Some Trigonometric Functions and Their Derivatives

Stewart, Section 3.4

In[201]:=

<<Graphics`Colors`

In[202]:=

Clear[f, t]  f[t_] = Sin[t] ;   Plot[{f[t], f '[t]}, {t, 0, 2π},  ...          AxesLabel  {"t", None}] ;

[Graphics:HTMLFiles/3.4_derTrigFns_3.gif]

In[205]:=

Clear[f, t]  f[t_] = Cos[t] ;   Plot[{f[t], f '[t]}, {t, 0, 2π},  ...          AxesLabel  {"t", None}] ;

[Graphics:HTMLFiles/3.4_derTrigFns_5.gif]

In[208]:=

Clear[f, t]  f[t_] = Tan[t] ;   Plot[{f[t], f '[t]}, {t, 0, 2π},  ...          AxesLabel  {"t", None}] ;

[Graphics:HTMLFiles/3.4_derTrigFns_7.gif]

Finding Horizontal Tangents

Stewart, Section 3.4, Example 2

First, let's define and then differentiate a certain trigonometric function f.

In[211]:=

Clear[f, x]  f[x_] := Sec[x]/(1 + Tan[x])   Plot[{f[x], f '[x]}, {x, 0, 2π ...          AxesLabel  {"t", None}] ;

[Graphics:HTMLFiles/3.4_derTrigFns_9.gif]

Now, try to discover where the graph of f has a horizontal tangent.
Ans: The graph of f has a horizontal tangent when f'(x) == 0.

In[214]:=

f '[x]  Solve[f '[x]  0, x]

Out[214]=

-Sec[x]^3/(1 + Tan[x])^2 + (Sec[x] Tan[x])/(1 + Tan[x])

                                                 π Solve :: verif : Potential solution  { ...  hand. May require use of limits. More…                                                    4

                                                3 π Solve :: verif : Potential solution   ...  hand. May require use of limits. More…                                                    4

Solve :: ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.  More…

Out[215]=

{{x -(3 π)/4}, {xπ/4}}

We were warned by Mathematica to check the points 3π/4 and -π/4 by hand.
Should we retain or discard those points?
Should we retain or discard the points which Solve returned: -3π/4 and π/4 ?

Taking into account the periodicity of the trigonometric functions involved,
what is the complete list of ALL of the points where f'(x) == 0 ?


Created by Mathematica  (April 19, 2004)