{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {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 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }