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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 49 "Exploring Derivatives for \+
Trigonometric Functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
f:=proc(x) x+cos(x) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"df:=proc(x) diff(f(x),x) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "df(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Note that since
  sin(x) < = 1  for all x, we see that df(x) is always greater than or
 equal to 0. (which means that  f  is always increasing.) Let's verify
 this by graph." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(df(
x),x=-2*Pi..Pi, y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "ddf:=proc(x) diff(df(x),x) end;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "ddf(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "You s
hould know the graph of  f''(x) or ddf(x). Let's see if you can explai
n the relationships between the following functions:" }}{PARA 0 "" 0 "
" {TEXT -1 18 "(1) f '  and f '';" }}{PARA 0 "" 0 "" {TEXT -1 18 "(2) \+
f   and  f '';" }}{PARA 0 "" 0 "" {TEXT -1 17 "(3) f   and  f '." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "plot(\{df(x),ddf(x)\},x=-2*Pi..2*Pi, y=-2..2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{f(x),ddf(x)\},x=-4*Pi..4*Pi, y=-10
..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{f(x),df(x)
\},x=-4*Pi..4*Pi, y=-10..10);" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 24 "
Exploring the Chain Rule" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "limit((f(g(x+h))-f(g
(x)))/h,h=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->(x+1
)/(3*x-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g:=x->x^3;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "limit((f(g(x+h))-f(g(x)))/
h,h=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "df:=x->diff(f(x)
,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(df(x));" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ans:=simplify(subs(x=g(x),
df(x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ans*diff(g(x),x
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "page 233 number 10" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=t->(1+tan(t))^(1/3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(t),t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "number 22:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "u:=t->(t^3+1)/(t^3-1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "diff(u(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "s:=u->u^(1/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d
iff(s(u),u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(s(u(t)
),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "32" 0 
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