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2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 208 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{PSTYLE "_pstyle5" -1 204 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 36 "Slope of a Tangent Line at one point" }{TEXT 204 0 "" }}{PARA 201 "" 0 "" {TEXT 205 12 "Wei-C hi Yang" }{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 13 "f:=x->sin(x);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 14 "with(student):" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 55 " showtangent(f(x),x=Pi/2,x=-Pi..Pi,y=-1..1,thickness=2);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 12 "with(plots): " }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 9 "x0:=Pi/2;" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 15 "p0:=[x0,f(x0)];" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 19 "p1:=[x0+h,f(x0+h)];" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 16 "m:=slope(p 0,p1);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 206 49 "The following is the equation of the secant line." }{TEXT 206 0 "" }} {PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 20 "y-p0[2]=m*(x-p0[1]);" }{MPLTEXT 1 207 0 "" }}} {EXCHG {PARA 202 "" 0 "" {TEXT 206 65 "The isolate command turns the e quation into slope-intercept form." }{TEXT 206 0 "" }}{PARA 202 "" 0 " " {TEXT 206 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 13 "isol ate(%,y);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 206 45 "We need to turn the equation into a function." }{TEXT 206 0 "" }} {PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 26 "secant:=unapply(rhs(%),x);" }{MPLTEXT 1 207 0 "" }} }{EXCHG {PARA 202 "" 0 "" {TEXT 206 169 "We can now plot the secant an d the function itself together for different values of h. First, make a sequence of plots. We can use the slope of the secant as the title. " }{TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 29 "h_v alues:=[seq(1/i,i=1..50)];" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 " > " 0 "" {MPLTEXT 1 207 52 "### WARNING: semantics of type `string` ha ve changed" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 117 "\nS:=seq(plot([f (x),secant(x)],x=-Pi..Pi,thickness=2,view=[-Pi..Pi,-3..3],title=conver t(evalf(m),string)),h=h_values):" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 12 "with(plots):" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 46 "display(S,insequ ence=true,view=[0..Pi,-1..1]);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 30 "h_values:=[-seq(1/i,i=1..50)];" } {MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 52 "# ## WARNING: semantics of type `string` have changed" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 117 "\nS:=seq(plot([f(x),secant(x)],x=-Pi..Pi,thi ckness=2,view=[-Pi..Pi,-3..3],title=convert(evalf(m),string)),h=h_valu es):" }{MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 12 "with(plots):" }{MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 46 "display(S,insequence=true,view=[0..Pi,-1..1]);" }{MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }