DERIVE for Windows version 5.05 DfW file saved on 01 Oct 2002
Y   ANS(x, data, ddata, a, n, v):=PROG(v := SOLUTIONS(EQNS(data, ddata, a, n), a), IF(DIM(v) = 0, [], POLY(x, v™1, n)))
AUTONOMOUS(r, v):=r/v
AUTONOMOUS_CONSERVATIVE(q, x, y, x0, y0, v0):=x = x0 + IF(v0 = 0, 1, SIGN(v0))·INT((2·INT(q, y, y0, y) + v0^2)^(- 1/2), y, y0, y)
BISECT(u, x, v0, k):=IF(k > 0, ITERATES(BIS_AUX(u, x, v™1, v™2), v, v0, k), ?, BIS_AUX(u, x, v0™1, v0™2))
BIS_AUX(u, x, a, b):=IF(SUBST(u, x, a)·SUBST(u, x, (a + b)/2) < 0, [a, (a + b)/2], [(a + b)/2, b])
CURVEFIT(x, data, ddata):=IF(DIM(ddata) > 0, CURVEFIT2(x, data, ddata), CURVEFIT1(x, data), CURVEFIT1(x, data))
CURVEFIT1(x, data):=ANS(x, data, [], UNK(DIM(data)), DIM(data) - 1)
CURVEFIT2(x, data, ddata):=ANS(x, data, ddata, UNK(DIM(data) + DIM(ddata)), -1 + DIM(ddata) + DIM(data))
DE(p, q, x, y, x0, y0):=y = ê^(- INT(p, x, x0, x))·(y0 + INT(ê^INT(p, x, x0, x)·q, x, x0, x))
DE2(b, c, x, x0, y0, v0):=DSOLVE2_IV(b, c, 0, x, x0, y0, v0)
DF(r, x, x0, xm, m, y, y0, yn, n):=VECTOR(VECTOR(SEG(LIM(r, [x, y], [x0 + j·(xm - x0)/m, y0 + k·(yn - y0)/n]), x0 + j·(xm - x0)/m, y0 + k·(yn - y0)/n, (xm - x0)/(4·m), (yn - y0)/(4·n)), j, 0, m), k, 0, n)
DISC_AUX(p, q, x):=p^2/4 - q + DIF(p, x)/2
DPOLY(x, a, n):=¤(i·a™(i + 1)·x^(i - 1), i, 1, n)
DRAW_COMPLEX(v):=VECTOR([RE(z), IM(z)], z, v)
DRAW_LEFT(u, x, n, a, b):=VECTOR(D_BOX(t, 0, t + (b - a)/n, SUBST(u, x, t)), t, a, b - (b - a)/n, (b - a)/n)
DRAW_NEWT(u, x, x0, k):=VECTOR([v, 0; v, SUBST(u, x, v); NEWT(u, x, v), 0], v, ITERATES(NEWT(u, x, w), w, x0, k))
DRAW_RIGHT(u, x, n, a, b):=VECTOR(D_BOX(t, 0, t + (b - a)/n, SUBST(u, x, t + (b - a)/n)), t, a, b - (b - a)/n, (b - a)/n)
DRAW_TRAP(u, x, n, a, b):=VECTOR(D_TRAP(t, 0, t + (b - a)/n, 0, t + (b - a)/n, SUBST(u, x, t + (b - a)/n), t, SUBST(u, x, t)), t, a, b - (b - a)/n, (b - a)/n)
DSOLVE2(p, q, r, x, c1, c2):=LIN2_AUX(DISC_AUX(p, q, x), p, r, x, [c1, c2])
DSOLVE2_BV(p, q, r, x, x0, y0, x2, y2):=LIN2_BV_AUX(DISC_AUX(p, q, x), p, r, x, x0, y0, x2, y2)
DSOLVE2_IV(p, q, r, x, x0, y0, v0):=LIN2_IV_AUX(DISC_AUX(p, q, x), p, r, x, x0, y0, v0)
D_BOX(x1, y1, x2, y2):=[x1, y1; x2, y1; x2, y2; x1, y2; x1, y1]
D_TRAP(x1, y1, x2, y2, x3, y3, x4, y4):=[x1, y1; x2, y2; x3, y3; x4, y4; x1, y1]
EQNS(data, ddata, a, n):=APPEND(VECTOR(POLY(data™i™1, a, n) = data™i™2, i, 1, DIM(data), 1), VECTOR(DPOLY(ddata™i™1, a, n) = ddata™i™2, i, 1, DIM(ddata), 1))
EULER(r, x, y, x0, y0, xn, n):=ITERATES(v + (xn - x0)/n·[1, LIM(r, [x, y], v)], v, [x0, y0], n)
EXACT2(p, q, x, y, v, c):=IF((v·DIF(p, y, 2) + 2·DIF(DIF(p, x), y) - DIF(DIF(q, y), v))·v + DIF(p, x, 2) - DIF(DIF(q, x), v) + DIF(q, y) = 0  v·DIF(DIF(p, y), v) + DIF(DIF(p, x), v) + 2·DIF(p, y) - DIF(q, v, 2) = 0, EXACT2_AUX1(INT(p, v), q, x, y, v) = c, "inapplicable", "inapplicable")
EXACT2_AUX1(p, q, x, y, v):=p + EXACT2_AUX2(q - DIF(p, x) - v·DIF(p, y), x, y, v)
EXACT2_AUX2(r, x, y, v):=EXACT2_AUX3(INT(DIF(r, v), y), LIM(r, v, 0), x)
EXACT2_AUX3(r, u, x):=r + INT(u - DIF(r, x), x)
EXTRACT_2_COLUMNS(a, c1, c2):=EXTRACT_2_ELEMENTS(a`, c1, c2)`
EXTRACT_2_ELEMENTS(v, p, q):=[v™p, v™q]
LEFT(u, x, n, a, b):=(b - a)/n·¤(SUBST(u, x, a + k·(b - a)/n), k, 0, n - 1)
LIN2_AUX(d, p, r, x, c):=IF(DIF(d, x) = 0, LIN2_AUX2(ê^(INT(p, x)/(-2))·LIN2_HOM(d, x), r, x, c), "inapplicable", "inapplicable")
LIN2_AUX2(h, r, x, c):=c • h + LIN2_PARTIC(h, r, x)
LIN2_BV_AUX(d, p, r, x, x0, y0, x2, y2):=IF(DIF(d, x) = 0, LIN2_BV_AUX2(ê^(INT(p, x)/(-2))·LIN2_HOM(d, x), r, x, x0, y0, x2, y2), "inapplicable", "inapplicable")
LIN2_BV_AUX2(h, r, x, x0, y0, x2, y2):=LIN2_BV_AUX3(h, LIN2_PARTIC(h, r, x), x, x0, y0, x2, y2)
LIN2_BV_AUX3(h, p, x, x0, y0, x2, y2):=p + h • LIN_SOLVE([LIM(h, x, x0), LIM(h, x, x2)], [y0 - LIM(p, x, x0), y2 - LIM(p, x, x2)])
LIN2_HOM(d, x):=IF(d = 0, [1, x], LIN2_HOM_AUX(d, x), LIN2_HOM_AUX(d, x))
LIN2_HOM_AUX(d, x):=IF(d < 0, [COS(‹(-d)·x), SIN(‹(-d)·x)], [ê^(‹d·x), ê^(- ‹d·x)], [ê^(‹d·x), ê^(- ‹d·x)])
LIN2_IV_AUX(d, p, r, x, x0, y0, v0):=IF(DIF(d, x) = 0, LIN2_IV_AUX2(ê^(INT(p, x)/(-2))·LIN2_HOM(d, x), r, x, x0, y0, v0), "inapplicable", "inapplicable")
LIN2_IV_AUX2(h, r, x, x0, y0, v0):=LIN2_IV_AUX3(h, LIN2_PARTIC(h, r, x), x, x0, y0, v0)
LIN2_IV_AUX3(h, p, x, x0, y0, v0):=p + h • LIN_SOLVE(LIM([h, DIF(h, x)], x, x0), LIM([y0 - p, v0 - DIF(p, x)], x, x0))
LIN2_PARTIC(h, r, x):=h • INT([- ELEMENT(h, 2), ELEMENT(h, 1)]·r/DET([h, DIF(h, x)]), x)
LIN_SOLVE(a, b):=ELEMENT(ROW_REDUCE(a, b)`, 1 + DIM(a))
LIOUVILLE(p, q, x, y, c1, c2):=INT(ê^INT(q, y), y) - c1 + c2·INT(ê^INT(-p, x), x) = 0
MID(u, x, n, a, b):=(b - a)/n·¤(SUBST(u, x, a + (k + 1/2)·(b - a)/n), k, 0, n - 1)
NEWT(u, x, x0, k):=IF(k > 0, NEWT_ITERATES(u, x, x0, k), ?, SUBST(x - u/DIF(u, x), x, x0))
NEWT_ITERATES(u, x, x0, k):=ITERATES(x - u/DIF(u, x), x, x0, k)
POLY(x, a, n):=¤(a™(i + 1)·x^i, i, 0, n)
RIGHT(u, x, n, a, b):=(b - a)/n·¤(SUBST(u, x, a + k·(b - a)/n), k, 1, n)
RK(r, v, v0, h, n):=RK_AUX0(h·APPEND([1], r), v, v0, n)
RK_AUX0(p, v, v0, n):=ITERATES(u_ + RK_AUX1(p, v, u_, LIM(p, v, u_)), u_, v0, n)
RK_AUX1(p, v, u_, c1):=RK_AUX2(p, v, u_, c1, LIM(p, v, u_ + c1/2))
RK_AUX2(p, v, u_, c1, c2):=RK_AUX3(p, v, u_, c1, c2, LIM(p, v, u_ + c2/2))
RK_AUX3(p, v, u_, c1, c2, c3):=(2·(c2 + c3) + LIM(p, v, u_ + c3) + c1)/6
SECANT(u, x, a, h):=y = (SUBST(u, x, a + h) - SUBST(u, x, a))/h·(x - a) + SUBST(u, x, a)
SEG(rc, x, y, «x, «y):=IF(ABS(rc) > 1  «y < ABS(rc)·«x, [x - «y/rc, y - «y; x + «y/rc, y + «y], [x - «x, y - rc·«x; x + «x, y + rc·«x])
SIMP(u, x, n, a, b):=(b - a)/(6·n)·(4·¤(SUBST(u, x, a + (k + 1/2)·(b - a)/n), k, 0, n - 1) + 2·¤(SUBST(u, x, a + k·(b - a)/n), k, 1, n - 1) + SUBST(u, x, b) + SUBST(u, x, a))
SLOPE(data, m1):=ITERATES([v™1 + 1, 2·(data™(v™1 + 1)™2 - data™(v™1)™2)/(data™(v™1 + 1)™1 - data™(v™1)™1) - v™2], v, [1, m1], DIM(data) - 1)
SLOPE1(data, m1):=ITERATES([v™1 + 1, 2·(data™(v™1 + 1)™2 - data™(v™1)™2)/(data™(v™1 + 1)™1 - data™(v™1)™1) - v™2], v, [1, m1], DIM(data) - 1)
SPLINE(x, data, m1):=SPLINE_AUX(x, data, SLOPE1(data, m1))
SPLINE_AUX(x, data, m):=¤(CURVEFIT(x, [data™k, data™(k + 1)], [[data™k™1, m™k™2]])·CHI(data™k™1, x, data™(k + 1)™1), k, 1, DIM(data) - 1)
TANGENT(u, x, a):=y = SUBST(u, x, a) + SUBST(DIF(u, x), x, a)·(x - a)
TRAP(u, x, n, a, b):=(b - a)/(2·n)·(2·¤(SUBST(u, x, a + k·(b - a)/n), k, 1, n - 1) + SUBST(u, x, b) + SUBST(u, x, a))
UNK(n):=VECTOR((SOLUTIONS(½ = ½, ½))™1, i, 1, n, 1)
c1:=
c2:=
c3:=
data:=
ddata:=
m1:=
rc:=
u_:=
v0:=
x0:=
x1:=
x2:=
x3:=
x4:=
xm:=
xn:=
y0:=
y1:=
y2:=
y3:=
y4:=
yn:=
«x:=
«y:=
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\viewkind4\uc1\pard\cf1\b\f0\fs24 Name: 
\par Utility Functions for "Calculus Concepts Using Derive for Windows":
\par }
€   >   ô  P    ÿ,{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fmodern\fprq1 Arial;}}
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\viewkind4\uc1\pard\cf1\f0\fs24 Open add-head-explanation for explanations of these utility functions. (On the web you can go to http://www.math.hawaii.edu/lab/utilities.html)
\par }
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\viewkind4\uc1\pard\cf1\b\f0\fs24 Your file starts here:
\par }
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