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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 38 "Derivative of a function \+
at one point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "f:=t->40*t-16*t^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "f(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(t),t=-10
..10,y=-50..40,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "showtangent(f,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
Q:=proc(a,h) (f(a+h)-f(a))/h end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
56 "Evaluate the function  Q  numerically to estimate f'(a)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for n from 1 to 5 do evalf(Q
(2,10^(-n))) od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for n from 1 to 5 do evalf(Q(2,-10^
(-n))) od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(limit(Q
(2,h),h=0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Therefore, the s
lope of the tangent line of  f  at x=2 is  -24. Let's check this with \+
Maple by computing f'(2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f1:=proc(t) diff(f(t),t) end;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f1(t);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "subs(t=2,f1(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
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