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{SECT 0 {EXCHG {PARA 223 "" 0 "" {TEXT 236 43 "GENERAL SOLUTION TO PRO
BLEM 60 ON PAGE 106:" }{TEXT 236 0 "" }}{PARA 223 "" 0 "" {TEXT 236 0 
"" }}{PARA 224 "" 0 "" {TEXT 237 60 "Find the equation of the tangent \+
line(s) through the point (" }{TEXT 238 1 "d" }{TEXT 237 2 ", " }
{TEXT 238 1 "e" }{TEXT 237 35 ") that are tangent to the parabola " }
{TEXT 238 2 "f " }{TEXT 237 1 "(" }{TEXT 238 1 "x" }{TEXT 237 4 ") = "
 }{TEXT 238 2 "ax" }{TEXT 237 5 "^2 + " }{TEXT 238 2 "bx" }{TEXT 237 
3 " + " }{TEXT 238 1 "c" }{TEXT 237 1 "." }{TEXT 239 0 "" }}{PARA 225 
"" 0 "" {TEXT 240 0 "" }}}{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 8 
"restart:" }{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "" 0 "" {TEXT 236 
0 "" }}{PARA 227 "" 0 "" {TEXT 242 10 "Store the " }{TEXT 243 1 "x" }
{TEXT 242 50 "-coordinate of the given point to d and store the " }
{TEXT 243 1 "y" }{TEXT 242 36 "-coordinate of the given point to e:" }
{TEXT 242 0 "" }}}{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 16 "d := 2
; e := -3;" }{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "" 0 "" {TEXT 
236 0 "" }}{PARA 227 "" 0 "" {TEXT 242 19 "Store the standard " }
{TEXT 243 1 "a" }{TEXT 242 2 ", " }{TEXT 243 1 "b" }{TEXT 242 6 ", and
 " }{TEXT 243 1 "c" }{TEXT 242 55 " of the quadratic function to a, b,
 and c respectively:" }{TEXT 242 0 "" }}}{EXCHG {PARA 226 "> " 0 "" 
{MPLTEXT 1 241 24 "a := 1 ; b := 1; c := 0;" }{MPLTEXT 1 241 0 "" }}}
{EXCHG {PARA 223 "" 0 "" {TEXT 236 0 "" }}{PARA 223 "" 0 "" {TEXT 236 
34 "Store the quadratic function to f:" }{TEXT 236 0 "" }}}{EXCHG 
{PARA 226 "> " 0 "" {MPLTEXT 1 241 28 "f := x -> a*x^2+b*x+c; f(x);" }
{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "" 0 "" {TEXT 236 0 "" }}
{PARA 227 "" 0 "" {TEXT 242 46 "Find the derivative of the quadratic f
unction " }{TEXT 243 1 "f" }{TEXT 242 2 " :" }{TEXT 242 0 "" }}}
{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 8 "D(f)(x);" }{MPLTEXT 1 
241 0 "" }}}{EXCHG {PARA 223 "" 0 "" {TEXT 236 0 "" }}{PARA 227 "" 0 "
" {TEXT 242 82 "Using algebra to find the slope of the tangent line gi
ven that the two points of (" }{TEXT 243 1 "d" }{TEXT 242 2 ", " }
{TEXT 243 1 "e" }{TEXT 242 7 ") and (" }{TEXT 243 1 "x" }{TEXT 242 2 "
, " }{TEXT 243 2 "f " }{TEXT 242 1 "(" }{TEXT 243 1 "x" }{TEXT 242 19 
")) are on the line:" }{TEXT 242 0 "" }}}{EXCHG {PARA 226 "> " 0 "" 
{MPLTEXT 1 241 19 "(f(x) - e)/(x - d);" }{MPLTEXT 1 241 0 "" }}}
{EXCHG {PARA 223 "" 0 "" {TEXT 236 0 "" }}{PARA 227 "" 0 "" {TEXT 242 
133 "Since this slope is the same slope that is given by the derivativ
e of the function, then have Maple solve the following equation for " 
}{TEXT 243 1 "x" }{TEXT 242 40 ".  We will store these solutions to so
l:" }{TEXT 242 0 "" }}}{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 134 "
(f(x) - e)/(x - d) = D(f)(x); f(x) - e = (x - d)*D(f)(x); f(x) - e = e
xpand((x - d)*D(f)(x)); expand((x - d)*D(f)(x)) - f(x) + e = 0; " }
{MPLTEXT 1 241 7 "solve( " }{MPLTEXT 1 241 43 "(f(x) - e)/(x - d) = D(
f)(x), x); sol := %;" }{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "" 0 ""
 {TEXT 236 0 "" }}{PARA 223 "" 0 "" {TEXT 236 53 "Have Maple write the
 equation of the tangent line(s):" }{TEXT 236 0 "" }}}{EXCHG {PARA 
226 "> " 0 "" {MPLTEXT 1 241 43 "y = D(f)(sol[1])*(x - sol[1]) + f(sol
[1]); " }{MPLTEXT 1 241 42 "y = D(f)(sol[2])*(x - sol[2]) + f(sol[2]);
" }{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "" 0 "" {TEXT 236 0 "" }}
{PARA 223 "" 0 "" {TEXT 236 68 "Now, have Maple plot the quadratic fun
ction and the tangent line(s):" }{TEXT 236 0 "" }}}{EXCHG {PARA 226 ">
 " 0 "" {MPLTEXT 1 241 13 "plot( [f(x), " }{MPLTEXT 1 241 109 "D(f)(so
l[1])*(x - sol[1]) + f(sol[1]), D(f)(sol[2])*(x - sol[2]) + f(sol[2])]
 ,x, color=[red, green, blue] );" }{MPLTEXT 1 241 0 "" }}}{EXCHG 
{PARA 226 "> " 0 "" {MPLTEXT 1 241 231 "plot( [f(x), D(f)(sol[1])*(x -
 sol[1]) + f(sol[1]), D(f)(sol[2])*(x - sol[2]) + f(sol[2])] ,x=min(d,
sol[1],sol[2])-5..max(d,sol[1],sol[2])+5, y=min(e,f(sol[1]),f(sol[2]))
-5..max(e,f(sol[1]),f(sol[2]))+5, color=[red, green, blue] );" }
{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 229 0 "" }}
{PARA 229 "" 0 "" {TEXT 229 27 "Now, let's see what is the " }{TEXT 
244 1 "x" }{TEXT 229 45 "-coordinate of the point(s) on the graph of  \+
" }{TEXT 244 1 "f" }{TEXT -1 7 " (x) = " }{TEXT 244 2 "ax" }{TEXT -1 
5 "^2 + " }{TEXT 244 2 "bx" }{TEXT -1 3 " + " }{TEXT 244 1 "c" }{TEXT 
-1 31 " where the tangent line passes " }{TEXT 229 19 "through the poi
nt (" }{TEXT 244 1 "d" }{TEXT 229 2 ", " }{TEXT 244 1 "e" }{TEXT 229 
2 ")." }{TEXT -1 0 "" }}}{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 8 "
restart:" }{MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 
1 241 28 "f := x -> a*x^2+b*x+c; f(x);" }{MPLTEXT 1 241 0 "" }}}
{EXCHG {PARA 226 "> " 0 "" {MPLTEXT 1 241 174 "(f(x) - e)/(x - d) = D(
f)(x); f(x) - e = (x - d)*D(f)(x); f(x) - e = expand((x - d)*D(f)(x));
 expand((x - d)*D(f)(x)) - f(x) + e = 0; solve( (f(x) - e)/(x - d) = D
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