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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 40 "Slopes of Secant Lines a
nd Tangent Lines" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 53 "Average and \+
Instantaneous Velocity and Rate of Change" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 256 13 "The function " }{XPPEDIT 18 0 "f(t) = -16*t^2+128*
t;" "6#/-%\"fG6#%\"tG,&*&\"#;\"\"\"*$)F'\"\"#F+F+!\"\"*&\"$G\"F+F'F+F+
" }{TEXT -1 1 " " }{TEXT 257 36 "gives the height of an object after \+
" }{TEXT 258 2 "t " }{TEXT 259 86 "seconds if it was projected from gr
ound level with an initial velocity of 128 ft./sec." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 261 129 "Input the following and use them to fi
nd the average velocity for the period between t = 2 and t = 3, t = 2.
5, t = 2.1, t = 2.01." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=
t->-16*t^2+128*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g:=h->
(f(2+h)-f(2))/h;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 111 "The above values \+
can also be thought of as the slope of the secant line between t = 2 a
nd the various t values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 
51 "To find the instantaneous velocity at t = 2, input:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(g(h),h=0);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 91 "This can also be thought of as \+
the slope of the tangent line to the graph of f(t) at t = 2." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 264 30 "Exercise 1:  For the function " }{XPPEDIT 18 0 "f(x) = 6
*x-x^2;" "6#/-%\"fG6#%\"xG,&*&\"\"'\"\"\"F'\"\"\"F,*$)F'\"\"#F,!\"\"" 
}{TEXT -1 2 ", " }{TEXT 265 82 "find the slopes of the secant lines be
tween x = 2 and x = 1 + h for the following " }}{PARA 0 "" 0 "" {TEXT 
266 139 "                  values of h:  -1, -0.8, -0.6, -0.4, -0.2, 0
.2, 0.4, 0.6, 0.8, 1.0 and observe that they are approaching a limit a
s h gets" }}{PARA 0 "" 0 "" {TEXT 268 95 "                  closer to \+
zero.  Then find the instantaneous rate of change of f(x) at x = 2." }
{TEXT -1 0 "" }{TEXT 269 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Slopes of Secant Lines
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 116 "For a function, f(x)
, the slope of the secant line between the points (x, f(x)) and (x+h, \+
f(x+h)) can be written as " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "(f(x+h)-f
(x))/h;" "6#*&,&-%\"fG6#,&%\"xG\"\"\"%\"hGF*F*-F&6#F)!\"\"F*F+F." }
{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 19 "Inpu
t the function " }{XPPEDIT 18 0 "f(x) = x^3-3*x^2+4*x-2;" "6#/-%\"fG6#
%\"xG,**$)F'\"\"$\"\"\"F,*&\"\"$F,*$)F'\"\"#F,F,!\"\"*&\"\"%F,F'F,F,\"
\"#F2" }{TEXT -1 2 ", " }{TEXT 272 135 "find a general expression for \+
the slope of the secant line between x = 2 and x = 2+h, simplify it an
d find the limit as h approaches 0." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "f:=x->x^3-3*x^2+4*x-2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "s:=(f(2+h)-f(2))/h;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "s1:=simplify(s);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "limit(s1,h=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 273 41 "Exercise 2:  Do as above for the function" }{XPPEDIT 18 
0 "f(x) = -2*x^2-3*x+4;" "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\"*$)F'\"\"#F+F
+!\"\"*&\"\"$F+F'F+F/\"\"%F+" }{TEXT -1 1 " " }{TEXT 274 8 "at x = 1" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 134 "To plo
t a function along with its tangent line at a given point, first input
 the student package and then use the showtangent command." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "showtangent(f(x),x=1,x=-2..2,y=-5..5,thickne
ss=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 276 32 "Exercise 3:  Input the function " }{XPPEDIT 18 0 "f(
x) = 1/(x-1);" "6#/-%\"fG6#%\"xG*&\"\"\"\"\"\",&%\"xGF*\"\"\"!\"\"F." 
}{TEXT -1 1 " " }{TEXT 277 210 ", find the slopes of secant lines for \+
x values above and below 0, find a general                  expression
 for the slope of the secant line at x = 0, and find the limit of this
 expression as x approaches 0.  " }}{PARA 0 "" 0 "" {TEXT -1 21 "     \+
                " }{TEXT 278 94 "Then graph the function along with it
s tangent line at x = 0 in an appropriate viewing window." }}{PARA 0 "
" 0 "" {TEXT -1 22 "                      " }{TEXT 279 70 "The option \+
discont=true will get rid of the vertical line in the plot." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }