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Chapter 3: Applications of Differentiation

Section 3.7: What Derivatives Reveal about Graphs

Example 3.7.5

Graph $f (x) = \sin (4 x + 1 / 2) \cos (x / 2)$ , then use the tools of the calculus to analyze the significant features of this graph.

Solution

Initialize

•	Tools≻Load Package: Student Calculus 1

Loading Student:-Calculus1

•	Control-drag $f (x) = \dots$

•	Context Panel: Assign Function

$f (x) = \sin (4 x + 1 / 2) \cos (x / 2)$ $\overset{assign as function}{\to}$ $f$

Preliminary Analysis

•	The function is defined for all real numbers.

•

The function is a product of two trigonometric terms: $\sin (4 x + x / 2)$ , whose period is $π / 2$ ; and $\cos (x / 2)$ , whose period is $4 π$ . The more-rapid oscillations of the sine term will be "enveloped" by the more slowly varying cosine term. Since $f (x + 4 π) - f (x)$ = $0$ , the function has period $4 π$ .

•	Because each term in the product has range $[- 1, 1]$ , their product will take on values in $[- 1, 1]$ . Whether the range is $[- 1, 1]$ or a subset of $[- 1, 1]$ will shortly be discovered.

•	The function is not odd because $f (- x) \neq - f (x)$ , that is, because

$f (- x) + f (x)$ = $- \sin (4 x - \frac{1}{2}) \cos (\frac{x}{2}) + \sin (4 x + \frac{1}{2}) \cos (\frac{x}{2})$ $\neq 0$

•	The function is not even because $f (- x) \neq f (x)$ , that is, because

$f (- x) - f (x)$ = $- \sin (4 x - \frac{1}{2}) \cos (\frac{x}{2}) - \sin (4 x + \frac{1}{2}) \cos (\frac{x}{2})$ $\neq 0$

•	The $y$ -intercept occurs at $f (0)$ = $\sin (\frac{1}{2})$ $≐$ $0.47943$ .

•	The $x$ -intercepts occur at the zeros of $\sin (4 x + 1 / 2)$ and at the zeros of $\cos (x / 2)$ . These zeros are given exactly in Table 3.7.5(a).

Zeros of $\sin (4 x + 1 / 2)$	$4 x + 1 / 2 = n π$ ⇒ $x = (2 n π + 1) / 8, n = 1, \dots, 16$
Zeros of $\cos (x / 2)$	$x / 2 = (2 n + 1) π / 2$ ⇒ $x = (2 n + 1) π, n = 0, 1$
Table 3.7.5(a) Zeros of $f (x)$

The Context Panel option Solve≻Solve (general solution) returns

$\{x = 2 π_Z3 + π\}, \{x = - \frac{1}{8} + \frac{π_Z4}{4}\}$

where $_Z1$ and $_Z2$ represent arbitrary integers. Hence, the analytic solutions in Table 3.7.5(a) are contained in Maple's general solution of the equation $f (x) = 0$ .

Graphs

Figure 3.7.5(a) shows $f (x)$ in black, and the enveloping curves $\pm \cos (x / 2)$ , in red and green, respectively. Although $f$ is periodic with period $4 π$ , Figure 3.7.5(a), drawn on a larger domain, best illustrates the periodicity.

Graph of $f (x)$ (in black), $\pm \cos (x / 2)$ (in red and green, respectively)

Figure 3.7.5(b) shows $f$ graphed for one period on the interval $[0, 4 π]$ , an interval in which $f$ is to have 18 $x$ -intercepts. At first count it might seem that there are only 16 such intercepts, but careful inspection near $x = π$ and $x = 3 π$ reveals that in each location there are two very closely-spaced zeros.

On the interval $[0, 4 π]$ , graph of one period of $f$

Analysis

Figure 3.7.5(c) is produced by the FunctionChart (a.k.a FunctionPlot) command, and is equivalent to the graph produced by the tutor. Black portions of the curve indicate where $f$ is decreasing; red, increasing. Small "plus signs" in green indicate points of concavity; small circles, zeros; and small diamonds, relative extrema. Yellow shading indicates where the curve is concave downward; gray, concave upward.

Graph of $f$ produced by the FunctionChart command

With respect to the function $f,$ the first column in Table 3.7.5(b) lists the zeros $x_{k}$ ; the second, the critical numbers $c_{k}$ ; and the fourth, the candidates for inflection $p_{k}$ . The remaining two columns in the table give the function values at either $c_{k}$ or $p_{k}$ . This information could be culled (with great pains) from the Curve Analysis tutor, but behind the table cells there is code that generates and displays these results more efficiently.

$[\begin{array}{c} x_{k} & c_{k} & f (c_{k}) & p_{k} & f (p_{k}) \\ 0.6603981634 & 0.2635573723 & 0.9911937098 & 0.6400483812 & 0.07718101926 \\ 1.445796327 & 1.035331386 & −0.8667834369 & 1.394941916 & −0.1548402893 \\ 2.231194490 & 1.799457792 & 0.6142567631 & 2.126970400 & 0.1967240243 \\ 3.016592654 & 2.529688118 & −0.2800687070 & 2.782673221 & −0.1436911244 \\ 3.141592654 & 3.078455941 & 0.007730926441 & 3.321488736 & −0.08434345203 \\ 3.801990817 & 3.546139921 & −0.1715429240 & 3.934830080 & 0.1957405455 \\ 4.587388980 & 4.244795395 & 0.5135616414 & 4.651367720 & −0.1734468489 \\ 5.372787144 & 5.003285807 & −0.7986750251 & 5.401685258 & 0.1043128480 \\ 6.158185307 & 5.773622150 & 0.9672059621 & 6.161920823 & −0.01491404920 \\ 6.943583471 & 6.546742679 & −0.9911937097 & 6.923233688 & −0.07718102279 \\ 7.728981634 & 7.318516694 & 0.8667834372 & 7.678127223 & 0.1548402912 \\ 8.514379797 & 8.082643099 & −0.6142567631 & 8.410155708 & −0.1967240236 \\ 9.299777961 & 8.812873425 & 0.2800687076 & 9.065858528 & 0.1436911240 \\ 9.424777961 & 9.361641248 & −0.007730926382 & 9.604674043 & 0.08434345245 \\ 10.08517612 & 9.829325228 & 0.1715429242 & 10.21801539 & −0.1957405499 \\ 10.87057429 & 10.52798070 & −0.5135616410 & 10.93455303 & 0.1734468567 \\ 11.65597245 & 11.28647111 & 0.7986750249 & 11.68487057 & −0.1043128672 \\ 12.44137061 & 12.05680746 & −0.9672059621 & 12.44510613 & 0.01491405048 \end{array}]$

Table 3.7.5(b) Zeros, critical numbers, candidates for inflection

The left-hand column in Table 3.7.5(c) lists the local minima; the right-hand, the local maxima. Again, this information could be culled (with great pains) from the Curve Analysis tutor, but behind the table cells there is code that generates and displays these results more efficiently.

Local Minima

Local Maxima

interface(rtablesize=50):
use Student[Calculus1] in
amertxe:=module()
local f,g,h,P,k,L,LL,s,ss,C;
export MIN,MAX;
f:=x->sin(4*x+1/2)*cos(x/2);g:=D(f);
h:=(D@@2)(f);
C:=Roots(g(x)=0,x=0..4*Pi,numeric);
s:=0;
ss:=0;
for k from 1 to nops(C)-1 do
P:=[C[k],f(C[k])];
if h(C[k])>0 then s:=s+1; L||s:=[P]; else ss:=ss+1;LL||ss:=[P];end if;
end do;
MIN:=Matrix([[[0,f(0)]],L||(1..s)]);
MAX:=Matrix([LL||(1..ss),[[4*Pi,f(4*Pi)]]]);
end module;
end use:

>	amertxe:-MIN;

$[\begin{array}{c} [0, \sin (\frac{1}{2})] \\ [1.035331386, −0.8667834369] \\ [2.529688118, −0.2800687070] \\ [3.546139921, −0.1715429240] \\ [5.003285807, −0.7986750251] \\ [6.546742679, −0.9911937097] \\ [8.082643099, −0.6142567631] \\ [9.361641248, −0.007730926382] \\ [10.52798070, −0.5135616410] \end{array}]$

>	amertxe:-MAX;

$[\begin{array}{c} [0.2635573723, 0.9911937098] \\ [1.799457792, 0.6142567631] \\ [3.078455941, 0.007730926441] \\ [4.244795395, 0.5135616414] \\ [5.773622150, 0.9672059621] \\ [7.318516694, 0.8667834372] \\ [8.812873425, 0.2800687076] \\ [9.829325228, 0.1715429242] \\ [11.28647111, 0.7986750249] \\ [4 π, \sin (\frac{1}{2})] \end{array}]$

Table 3.7.5(c) Local minima and maxima

The right-hand column in Table 3.7.5(c) (and Figure 3.7.5(c)) shows that $0.9911937098$ is the absolute maximum value of $f$ . Likewise, the left-hand column reveals that $- 0.9911937098$ is the absolute minimum. Hence, the range of $f$ is the interval $[- 0.9911937098, 0.9911937098]$ , which is a subinterval within $[- 1, 1]$ .

Table 3.7.5(d) lists the intervals where $f$ is either decreasing or increasing. Again, this information could be culled (with great pains) from the Curve Analysis tutor, but behind the table cells there is code that generates and displays these results more efficiently.

Intervals where $f$ is decreasing

Intervals where $f$ is increasing

>	interface(rtablesize=50):

use Student[Calculus1] in
IncDec:=module()
local f,g,temp,rp,m,P,k,L,LL,s,ss;
export M,MM;
f:=x->sin(4*x+1/2)*cos(x/2);g:=D(f);
temp:=Roots(g(x)=0,x=0..4*Pi,numeric);
rp:=[0,temp[],4*Pi];
s:=0;
ss:=0;
for k from 1 to nops(rp)-1 do
m:=(rp[k]+rp[k+1])/2;
P:=[rp[k],rp[k+1]];
if g(m)>0 then s:=s+1; L||s:=[P]; else ss:=ss+1;LL||ss:=[P];end if;
end do;
M:=Matrix([L||(1..s)]);
MM:=Matrix([LL||(1..ss)]);
end module;
end use:

>	IncDec:-M;

$[\begin{array}{c} [0, 0.2635573723] \\ [1.035331386, 1.799457792] \\ [2.529688118, 3.078455941] \\ [3.546139921, 4.244795395] \\ [5.003285807, 5.773622150] \\ [6.546742679, 7.318516694] \\ [8.082643099, 8.812873425] \\ [9.361641248, 9.829325228] \\ [10.52798070, 11.28647111] \\ [12.05680746, 4 π] \end{array}]$

$[\begin{array}{c} [0.2635573723, 1.035331386] \\ [1.799457792, 2.529688118] \\ [3.078455941, 3.546139921] \\ [4.244795395, 5.003285807] \\ [5.773622150, 6.546742679] \\ [7.318516694, 8.082643099] \\ [8.812873425, 9.361641248] \\ [9.829325228, 10.52798070] \\ [11.28647111, 12.05680746] \end{array}]$

Table 3.7.5(d) Intervals where $f$ is decreasing, and where $f$ is increasing

Table 3.7.5(e) lists the intervals where $f$ is either concave upward or concave downward. Again, this information could be culled (with great pains) from the Curve Analysis tutor, but behind the table cells there is code that generates and displays these results more efficiently.

Intervals where $f$ is concave upward

Intervals where $f$ is concave downward

>	interface(rtablesize=50):

use Student[Calculus1] in
ConcUpDn:=module()
local f,g,temp,rp,m,P,k,L,LL,s,ss;
export M,MM;
f:=x->sin(4*x+1/2)*cos(x/2);g:=(D@@2)(f);
temp:=Roots(g(x)=0,x=0..4*Pi,numeric);
rp:=[0,temp[],4*Pi];
s:=0;
ss:=0;
for k from 1 to nops(rp)-1 do
m:=(rp[k]+rp[k+1])/2;
P:=[rp[k],rp[k+1]];
if g(m)>0 then s:=s+1; L||s:=[P]; else ss:=ss+1;LL||ss:=[P];end if;
end do;
M:=Matrix([L||(1..s)]);
MM:=Matrix([LL||(1..ss)]);
end module;
end use:

>	ConcUpDn:-M;

$[\begin{array}{c} [0.6400483812, 1.394941916] \\ [2.126970400, 2.782673221] \\ [3.321488736, 3.934830080] \\ [4.651367720, 5.401685258] \\ [6.161920823, 6.923233688] \\ [7.678127223, 8.410155708] \\ [9.065858528, 9.604674043] \\ [10.21801539, 10.93455303] \\ [11.68487057, 12.44510613] \end{array}]$

$[\begin{array}{c} [0, 0.6400483812] \\ [1.394941916, 2.126970400] \\ [2.782673221, 3.321488736] \\ [3.934830080, 4.651367720] \\ [5.401685258, 6.161920823] \\ [6.923233688, 7.678127223] \\ [8.410155708, 9.065858528] \\ [9.604674043, 10.21801539] \\ [10.93455303, 11.68487057] \\ [12.44510613, 4 π] \end{array}]$

Table 3.7.5(e) Intervals where $f$ is concave upward, and where $f$ is concave downward

Some Useful Commands

Applicable Commands

$CriticalPoints (f (x), x = 0 .. 4 π, numeric)$

$[0.2635573723, 1.035331386, 1.799457792, 2.529688118, 3.078455941, 3.546139921, 4.244795395, 5.003285807, 5.773622150, 6.546742679, 7.318516694, 8.082643099, 8.812873425, 9.361641248, 9.829325228, 10.52798070, 11.28647111, 12.05680746]$

$ExtremePoints (f (x), x = 0 .. 4 π, numeric)$

$[0., 0.2635573723, 1.035331386, 1.799457792, 2.529688118, 3.078455941, 3.546139921, 4.244795395, 5.003285807, 5.773622150, 6.546742679, 7.318516694, 8.082643099, 8.812873425, 9.361641248, 9.829325228, 10.52798070, 11.28647111, 12.05680746, 12.56637061]$

$InflectionPoints (f (x), x = 0 .. 4 π, numeric)$

$[0.6400483812, 1.394941916, 2.126970400, 2.782673221, 3.321488736, 3.934830080, 4.651367720, 5.401685258, 6.161920823, 6.923233688, 7.678127223, 8.410155708, 9.065858528, 9.604674043, 10.21801539, 10.93455303, 11.68487057, 12.44510613]$

$Roots (f (x), x = 0 .. 4 π)$

$[- \frac{1}{8} + \frac{π}{4}, - \frac{1}{8} + \frac{π}{2}, - \frac{1}{8} + \frac{3 π}{4}, - \frac{1}{8} + π, π, - \frac{1}{8} + \frac{5 π}{4}, - \frac{1}{8} + \frac{3 π}{2}, - \frac{1}{8} + \frac{7 π}{4}, - \frac{1}{8} + 2 π, - \frac{1}{8} + \frac{9 π}{4}, - \frac{1}{8} + \frac{5 π}{2}, - \frac{1}{8} + \frac{11 π}{4}, - \frac{1}{8} + 3 π, 3 π, - \frac{1}{8} + \frac{13 π}{4}, - \frac{1}{8} + \frac{7 π}{2}, - \frac{1}{8} + \frac{15 π}{4}, - \frac{1}{8} + 4 π]$

$Roots (f (x), x = 0 .. 4 π, numeric)$

$[0.6603981634, 1.445796327, 2.231194490, 3.016592654, 3.141592654, 3.801990817, 4.587388980, 5.372787144, 6.158185307, 6.943583471, 7.728981634, 8.514379797, 9.299777961, 9.424777961, 10.08517612, 10.87057429, 11.65597245, 12.44137061]$

The return of the Roots command is consistent with the analysis in Table 3.7.5(a). Note that while the equation $f (x) = 0$ can be solve analytically, the equations $f' (x) = 0$ and $f ″ (x) = 0$ cannot, and must be solved numerically.

Conclusions

The typical calculus text still presents the unit on "Curve Sketching" as if modern computing devices had not yet been invented. The evidence of this lies in the belief that the graph of a function should be constructed from the information derived from it, and its first two derivatives. Example 3.7.5 has been included to illustrate the folly of that belief. Were Figure 3.7.5(b) to be generated by such a paradigm, that is, by obtaining the many points detailed in Tables 3.7.5(b-e), and plotting them on a sheet of graph paper, the task would have taken many hours of tedious labor, and would most likely have been rife with computational errors.

The author of these notes takes this opportunity to "climb onto a soapbox" to orate a message that often gets just lip service: Use technology wisely, and let it infuse and modify the curriculum appropriately. In other words, when it comes to sketching a curve, admit that with today's technology a graph is readily obtained. The value of the calculus is not in obtaining the graph, but in ascertaining that the graph drawn by a calculator or computer is correct.

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