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{SECT 0 {PARA 259 "" 0 "" {TEXT -1 38 "Slopes, Tangent Lines, and Deri
vatives" }}{PARA 260 "" 0 "" {TEXT -1 18 "Jim Herod, Retired" }}{PARA
263 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 261 "" 0 ""
{TEXT -1 12 "Georgia Tech" }}{PARA 262 "" 0 "" {TEXT -1 21 "herod@math
.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 ""
{TEXT 256 207 "\n Calculus most often begins with derivatives. Der
ivatives are a most geometric notion. They begin with a picture, lead \+
to arithmetic of quotients and limits, develop a calculus, and return \+
to a picture." }}{PARA 256 "" 0 "" {TEXT 257 107 "\n\011 Here is th
e first picture. 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 point \{3/2, f(3/2) \} is a line that intersects the graph of f a
t the point \{3/2, f(3/2) \} and at another point \{a, f(a) \}. To dra
w 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
116 "\nWe are 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 dr
aw " }{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 that " }{TEXT 262 1 "a" }{TEXT 263 175 " approaches 3/2. This limit
ing line will be the definition of the tangent line. We make this visu
alization by using the animation properties of Maple. To do this, we n
eed the " }{TEXT 264 5 "plots" }{TEXT 265 157 " package. After the gra
ph is drawn, use the mouse to touch a point on the graph. On the menu \+
bar, icons 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=BLAC
K):\n od:\na:='a':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
display([seq(J||n,n=1..10)],insequence=true);" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 220 "This tangent line is co
mpletely determine by knowing its 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 found 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 "plo
t(\{[t,f(t),t=-1..2],[t,%*(t-3/2)+f(3/2),t=0..2]\});" }}{PARA 13 "" 0
"" {TEXT -1 0 "" }}}{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 an
y point on the graph of this f. We illustrate this by drawing an anima
tion 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(%));" }}}{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 "We end this worksheet by pointing out that Maple know
s 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 269 143 "Rather \+
than showing simply more examples, We point out that Maple knows the s
tandard 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 "diff
(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 "Exercis
e for the student" }}{PARA 0 "" 0 "" {TEXT -1 36 "On the same axis, dr
aw 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 point [2
, 4], and the line tangent to the graph of f at the point [- 3, 9]. Wh
ere do these two lines intersect?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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