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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 31 "Derivatives and Antideriv
atives" }}{PARA 0 "" 0 "" {TEXT -1 11 "page 101 #8" }}{PARA 0 "" 0 "" 
{TEXT -1 34 "(1) Can you find the f ' and f ''?" }}{PARA 0 "" 0 "" 
{TEXT -1 90 "(2) Can you plot f, f ' and f '' together and explain the
 relationship among their graphs?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "f:=proc(x) 4*x^(3/2) -5*x^(-3/2) end;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{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 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{f(x),df(x)\},x=-
10..10, y=-5..20);" }}}{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 should know th
e graph of  f''(x) or ddf(x). Let's see if you can explain the relatio
nships 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 41 "plot(\{d
f(x),ddf(x)\},x=-10..10, y=-5..20);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "plot(\{f(x),df(x),ddf(x)\},x=-10..10, y=-5..20,color=
black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Question: Do you know \+
which one is which? Do you think the function  f  looks like a straigh
t line?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(x),x=0..4
, y=-5..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(ddf(
x)=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Notice that this indica
tes the inflection point of f is at around  x =1.842015749" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 25 "Exercise: page 191, #4,5." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 30 "Quiz 4 Problem 1: p
age 191 #4:" }}{PARA 0 "" 0 "" {TEXT -1 39 "(1) Find the derivative of
  f  by hand." }}{PARA 0 "" 0 "" {TEXT -1 47 "(2) Use Maple to plot f,
 f ' and f '' together." }}{PARA 0 "" 0 "" {TEXT -1 90 "(3) Explain th
e relationships for (a)  f '  and f ''; (b) f   and  f ''; (c) f   and
  f '." }}{PARA 0 "" 0 "" {TEXT 256 15 "Antiderivatives" }}{PARA 0 "" 
0 "" {TEXT -1 65 "Example: Find the antiderivatives for the following \+
function  f  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=proc(x)
 4*x^(3/2)-5*x^(-3/2) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g:=proc(x) int(
f(x),x) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Notice that Maple only find one a
ntiderivative for  f  which satisfies g'(0)=0. In other words, the cor
rect answer for the antiderivative of  f  should be g(x)+C." }}{PARA 
0 "" 0 "" {TEXT -1 113 "Now, let's some possible antiderivatives for f
, say, g(x)+1, g(x)+2, g(x), g(x)-1 and g(x)-2 together with  f(x)." }
}{PARA 0 "" 0 "" {TEXT 257 8 "Question" }{TEXT -1 69 ": Can you tell t
he relationship between an antiderivative of f and f?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(\{f(x),g(x),g(x)+1,g(x)+2,g(x)
-1,g(x)-2\},x=-10..10, y=-5..20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
259 30 "Quiz 4 Problem 2: page 191#33." }}}}{MARK "16 1 0" 0 }
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