{VERSION 6 0 "IBM INTEL NT" "6.0" }
{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 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 257 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Geneva"
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 1 10 0 0
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 1 10 0 0 0 1 0 0
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 10 0 0 0 1 1 0 0 0 0 0
0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 264 "Geneva" 1 10 0 0 0 1 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 265 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Geneva"
1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Geneva" 1 10 0 0
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Geneva" 1 10 0 0 0 1 0 0
0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1
10 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0
0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1
{CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0
0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0
0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 \+
Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0
0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257
1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1
-1 "Geneva" 1 10 0 0 0 1 1 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0
0 0 -1 0 }{PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0
0 0 0 0 0 -1 0 }{PSTYLE "" 5 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1
0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 259 "" 0 "" {TEXT -1 38 "Slopes, Tangent Lines, and Deri
vatives" }}{PARA 260 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 264 "" 0 "
" {TEXT -1 38 "P O Box 1038\nGrove Hill, Alabama 36451" }}{PARA 263 "
" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 21 "herod@math.gatec
h.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256
207 "\n Calculus most often begins with derivatives. Derivatives a
re a most geometric notion. They begin with a picture, lead to arithme
tic of quotients and limits, develop a calculus, and return to a pictu
re." }}{PARA 256 "" 0 "" {TEXT 257 107 "\n\011 Here is the first pi
cture. We make this picture with a cubic polynomial that is zero at -1
, 0, and 1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=t->(t+1)*t
*(t-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
5 " " }{TEXT 258 205 "A secant line for the graph of f at the poin
t \{3/2, f(3/2) \} is a line that intersects the graph of f at the poi
nt \{3/2, f(3/2) \} and at another point \{a, f(a) \}. To draw such a \+
line, we compute the slope." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
32 "slope:=a->(f(a)-f(3/2))/(a-3/2);" }}}{PARA 0 "" 0 "" {TEXT -1 5 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 259 120 "\nWe are no
w prepared to compute the equation for such a secant line. The secant \+
line will depend on two things: t and a." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 41 "secant:=(t,a)->slope(a)*(t-3/2) + f(3/2);" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We draw " }
{TEXT 260 52 "a graph of f and of the secant line in case a = 1/2." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(\{f(t),secant(t,1/2)\},
t=-1..2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT
261 63 "\nThe next idea is to visualize this secant line moving so tha
t " }{TEXT 262 1 "a" }{TEXT 263 175 " approaches 3/2. This limiting li
ne will be the definition of the tangent line. We make this visualizat
ion by using the animation properties of Maple. To do this, we need th
e " }{TEXT 264 5 "plots" }{TEXT 265 157 " package. After the graph is \+
drawn, use the mouse to touch a point on the graph. On the menu bar, i
cons for putting the animation in motion appear. Try them." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 121 "for n from 1 to 10 do\n a:=-1+5/2*n/11;
\n J||n:=plot(\{f(t),secant(t,a)\},t=-1..2,color=BLACK):\n \+
od:\na:='a':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display([se
q(J||n,n=1..10)],insequence=true);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT 266 220 "This tangent line is completely determine by knowing it
s slope and one point on the line. The one point is \{3/2, f(3/2) \}. \+
The notion of slope is the very center of the Calculus. The slope is f
ound by a limiting process." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
9 "slope(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(slope
(a),a=3/2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 52 "Thus, the tangent line is known. We draw a picture." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(\{[t,f(t),t=-1..2],[t,%*(t-3/2
)+f(3/2),t=0..2]\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT 267 252 "Of course, what we have done with 3/2 can be done at
any point on the graph of this f. We illustrate this by drawing an an
imation as the tangent line moves along the graph of f. This time, we \+
use the Calculus to compute the slopes of the tangent lines." }}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 28 "cubecurve:=[t,f(t),t=-2..2
];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(t),t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(t=a,simplify(%));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 31 "line:=[t,%*(t-a)+f(a),t=-2..2];" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 44 "animate(\{cubecurve,line\},a=-1..1,frames=32)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 101 "W
e end this worksheet by pointing out that Maple knows very well how to
do the differential calculus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 17 "diff(sin(x^2),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(sin(x)^2,x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sin(x)/x^2,x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(ln(x),x);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->exp(x^2); D(f)(ln(4));" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 144 "Rather \+
than simply showing more examples, we point out that Maple knows the \+
standard rules for taking derivatives -- the Differential Calculus." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT
1 0 18 "diff(J(x)*K(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
18 "diff(J(x)/K(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "d
iff(J(K(x)),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "E
xercise for the student" }}{PARA 0 "" 0 "" {TEXT -1 36 "On the same ax
is, draw the graph of " }{XPPEDIT 18 0 "f(x)=x^2" "6#/-%\"fG6#%\"xG*$F
'\"\"#" }{TEXT -1 156 ", the line tangent to the graph of f at the po
int [2, 4], and the line tangent to the graph of f at the point [- 3, \+
9]. Where do these two lines intersect?" }}}{PARA 0 "" 0 "" {TEXT -1
0 "" }}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS
0 1 2 33 1 1 }