{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 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 14 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 "Heading 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 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Fo nt 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 128 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 35 "GRAPHICAL APPROACH TO THE PROOF OF " }}{PARA 18 "" 0 "" {TEXT -1 44 "THE SECOND FOUNDALMENTAL T HEOREM OF CALCULUS" }}{PARA 19 "" 0 "" {TEXT -1 12 "Wei-Chi Yang" }} {PARA 258 "" 0 "" {TEXT -1 40 "Department of Mathematics and Statistic s" }}{PARA 258 "" 0 "" {TEXT -1 37 "Radford University, Radford, VA 24 142" }}{PARA 258 "" 0 "" {TEXT -1 25 "E-mail: wyang@radford.edu" }} {PARA 258 "" 0 "" {TEXT -1 29 "http://www.radford.edu/~wyang" }}{PARA 3 "" 0 "" {TEXT 259 7 "Part I." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "In this note we shall use graphs of Riema nn sums to provide an intuitive" }}{PARA 0 "" 0 "" {TEXT -1 144 "appro ach of understanding the Second Fundamental Theorem of Calculus. We re call that the Second Fundamental Theorem of Calculus may be stated as " }}{PARA 0 "" 0 "" {TEXT -1 8 "follows:" }}{PARA 0 "" 0 "" {TEXT 256 7 "Theorem" }}{PARA 0 "" 0 "" {TEXT -1 55 "Let f be continuous on an open interval I, and let " }{TEXT 261 1 "a" }{TEXT -1 18 " be any \+ point in " }}{PARA 0 "" 0 "" {TEXT -1 23 "I. If F is defined by " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 5 "F(x)= " }{XPPEDIT 18 0 "int(f(t),t=a..x)" "6#-%$intG6$-%\"fG6#%\"tG/F);%\"aG %\"xG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " then F'(x) = f(x). For example, " }{XPPEDIT 18 0 "diff(int(cos(t),t=a. .x),x)" "6#-%%diffG6$-%$intG6$-%$cosG6#%\"tG/F,;%\"aG%\"xGF0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "cos(x) " "6#-%$cosG6#%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 15 "at each point " }{TEXT 262 1 "x" }{TEXT -1 20 " in t he interval I." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Note that if there is a closed form for " }{XPPEDIT 18 0 "int(f(t),t=a..x)" "6#-%$intG6$-%\"fG6#%\"tG/F);%\"aG%\"xG" }{TEXT -1 20 ", then it is easy to" }}{PARA 0 "" 0 "" {TEXT -1 32 "check if F'(x ) = f(x). However, " }{TEXT 257 33 "most of the time, we don't have a " }}{PARA 259 "" 0 "" {TEXT -1 16 "closed form for " }{XPPEDIT 18 0 "i nt(f(t),t=a..x) " "6#-%$intG6$-%\"fG6#%\"tG/F);%\"aG%\"xG" }{TEXT 263 2 ". " }{TEXT -1 53 "We hope the approach we take in this note can fil l in" }}{PARA 259 "" 0 "" {TEXT -1 9 "this gap." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "First, we define the func tion f. (Users can pick any complex functions on their own.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:=proc(x) abs(1-x*sin(x)) e nd;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We shall use a Riemann sum to approximate " }{XPPEDIT 18 0 "int(f(t),t=a..x" "6#-%$intG6$-%\"fG6#%\"tG/F);%\"aG %\"xG" }{TEXT -1 12 ". We use the" }}{PARA 0 "" 0 "" {TEXT -1 38 "midp oint rule to define the following " }{TEXT 260 12 "Riemann Sum," } {TEXT -1 9 " M, of f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "M:=proc(a,x,n) ((x-a)/n) * sum(f(a+(x-a)/(2*n) + i*(x-a)/n), i=0..n) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "M(a,x,n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Next, we define the derivative function, DM, of M(a,x,n) below:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DM:=proc(a,x,n) diff(M(a,x,n),x) end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "We now create the animation to visualize that when n goes to infinity, we shall obtain DM tends to f(x)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for k from 1 by 1 to 40 do " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P||k:= plot (\{f(x),DM(0,x,k)\}, x= -3..3, y=0..2,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plots[display]([seq(P||k, k=1..40)], insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT 258 11 "Conclusions" }}{PARA 0 "" 0 "" {TEXT -1 30 "We observe from the graphs of " }{XPPEDIT 18 0 "d iff(M[f],x)(0,x,10)" "6#--%%diffG6$&%\"MG6#%\"fG%\"xG6%\"\"!F+\"#5" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "diff(M[f],x)(0,x,20)" "6#--%%diffG6$&% \"MG6#%\"fG%\"xG6%\"\"!F+\"#?" }{TEXT -1 14 ", and f(x) to " }}{PARA 0 "" 0 "" {TEXT -1 52 "conjecture that indeed as n approaches to infin ity, " }{XPPEDIT 18 0 "diff(M[f],x)(x)" "6#--%%diffG6$&%\"MG6#%\"fG%\" xG6#F+" }}{PARA 0 "" 0 "" {TEXT -1 86 "tends to f(x), which makes the \+ second Fundamental Theorem of Calculus more believable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "Part II. How to use \+ the FTC" }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 1. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=x->x^3 -2*sin(x)-4*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g:=x->int (f(t),t=3..x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(g(x) ,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(g(x),x)-f(x); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Question 1. What is the meani ng of g(1)?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Question 2. Can you identify the f ollowing graphs, which one is f and which one is g?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{f(x),g(x)\},x=-5..5,y=-6..6);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Example 2." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "f:=x->sqrt(x)-2*sin(x)-4*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g:=x->int(f(t),t=sin(x)..cos(x));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Question 1. What is dg/dx?? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "h1:=x->int(f(t),t=sin(x)..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "h2:=x->int(f(t),t=0..cos(x));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "answer1:=diff(h1(x),x);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "answer2:=-f(sin(x))*cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(answer1-answer2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "answer3:=diff(h2(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "answer4:=f(cos(x))*(-sin(x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(answer3-answer4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Therefore, the answer to dg/dx \+ = answer 2 + answer 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a nswer2+answer4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(g(x ),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simplify(diff(g(x) ,x)-(answer2+answer4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{MARK "16 0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }