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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 37 "Derivative of a function \+
at one point" }}{PARA 19 "" 0 "" {TEXT -1 16 "Dr. Wei-Chi Yang" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wit
h(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=x->(x^2)*
cos(sqrt(1+2^x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(f(x),x=-10..10,y=-10
0..100,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sho
wtangent(f,4);" }}}{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 "Evalu
ate 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(4,10^(-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(4,-10^(-n))) od;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(limit(Q(4,h),h=0));" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Therefore, the slope of the tan
gent line of  f  at x=4 is  about 13.44. Let's check this with Maple b
y computing f'(4)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "f1:=proc(x) diff(f(x),x) end;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f1(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "evalf(subs(x=4,f1(x)));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "We will see how to find the derivative function by using \+
the definition." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->x^
3+3*x^2+4*x-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Q:=proc(a
,h) (f(a+h)-f(a))/h end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
Q(a,h);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(Q(a,h))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(Q(a,h),h=0);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(limit(Q(a,h),h=0));" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "16 0 0" 33 }
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