eQuate It CAS UDFs — Calculus (TI-Nspire CAS)
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1. avgroc(f, x, lo, hi) — Average rate of change
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Define avgroc(f, x, lo, hi) = Prgm
  Local result
  result := (f | x = hi) - (f | x = lo)) / (hi - lo)
  Disp "Average ROC on [", lo, ",", hi, "]: ", result
EndPrgm


2. avgval(f, x, lo, hi) — Average value of a function
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Define avgval(f, x, lo, hi) = Prgm
  Local result
  result := (1/(hi - lo)) * ∫(f, x, lo, hi)
  Disp "Average value on [", lo, ",", hi, "]: ", result
EndPrgm


3. boundarea(f1, f2, x) — Area between two curves
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Define boundarea(f1, f2, x) = Prgm
  Local pts, area, i, lo, hi
  pts := solve(f1 = f2, x)
  area := 0
  Disp "Intersection points: ", pts
  If dim(pts) >= 2 Then
    For i, 1, dim(pts) - 1
      lo := pts[i]
      hi := pts[i + 1]
      area := area + abs(∫(f1 - f2, x, lo, hi))
    EndFor
  EndIf
  Disp "Total bound area: ", area
EndPrgm


4. boundaread(f1, f2, x, lo, hi) — Area between curves in domain
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Define boundaread(f1, f2, x, lo, hi) = Prgm
  Local area
  area := abs(∫(f1 - f2, x, lo, hi))
  Disp "Bound area on [", lo, ",", hi, "]: ", area
EndPrgm


5. intguess — Integral multiple choice solver
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(Case 1: One integral given, find transformed integral)
Define intguess(known, trans, target) = Prgm
  Local l1, u1, v1, l2, u2, result
  l1 := known[1]
  u1 := known[2]
  v1 := known[3]
  l2 := target[1]
  u2 := target[2]
  Disp "Known: integral from ", l1, " to ", u1, " = ", v1
  Disp "Transformations: ", trans
  Disp "Target: integral from ", l2, " to ", u2
  Disp "Use substitution and transformation rules"
  Disp "to determine the target integral value."
EndPrgm


6. newtons(f, x, x0, n) — Newton's method
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Define newtons(f, x, x0, n) = Prgm
  Local xi, i, fp
  xi := x0
  fp := d(f, x)
  For i, 1, n
    xi := xi - (f | x = xi) / (fp | x = xi)
    Disp "Iteration ", i, ": x = ", approx(xi)
  EndFor
  Disp "Root approx: ", approx(xi)
EndPrgm


7. nroot(poly, x, k, n) — Parameter values for n roots
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Define nroot(poly, x, k, n) = Prgm
  Local disc, result
  Disp "Finding k for ", n, " roots"
  disc := discrim(poly, x)
  If n = 2 Then
    result := solve(disc > 0, k)
  ElseIf n = 1 Then
    result := solve(disc = 0, k)
  ElseIf n = 0 Then
    result := solve(disc < 0, k)
  EndIf
  Disp "k values: ", result
EndPrgm


8. nstp(poly, x, k, n) — Parameter values for n stationary points
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Define nstp(poly, x, k, n) = Prgm
  Local fp, disc, result
  fp := d(poly, x)
  Disp "f'(x) = ", fp
  disc := discrim(fp, x)
  If n = 2 Then
    result := solve(disc > 0, k)
  ElseIf n = 1 Then
    result := solve(disc = 0, k)
  ElseIf n = 0 Then
    result := solve(disc < 0, k)
  EndIf
  Disp "k for ", n, " stationary points: ", result
EndPrgm


9. npoi(poly, x, k, n) — Parameter values for n points of inflection
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Define npoi(poly, x, k, n) = Prgm
  Local fpp, disc, result
  fpp := d(d(poly, x), x)
  Disp "f''(x) = ", fpp
  disc := discrim(fpp, x)
  If n = 2 Then
    result := solve(disc > 0, k)
  ElseIf n = 1 Then
    result := solve(disc = 0, k)
  ElseIf n = 0 Then
    result := solve(disc < 0, k)
  EndIf
  Disp "k for ", n, " POI: ", result
EndPrgm


10. pois(f, x) — Points of inflection
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Define pois(f, x) = Prgm
  Local fpp, xvals, i, y
  fpp := d(d(f, x), x)
  xvals := solve(fpp = 0, x)
  Disp "f''(x) = ", fpp
  Disp "Points of inflection:"
  For i, 1, dim(xvals)
    y := f | x = xvals[i]
    Disp "(", xvals[i], ", ", y, ")"
  EndFor
EndPrgm


11. signtab(f, x) — Sign table for stationary points
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Define signtab(f, x) = Prgm
  Local fp, xvals, i, left, right, nature
  fp := d(f, x)
  xvals := solve(fp = 0, x)
  Disp "f'(x) = ", fp
  Disp "Stationary points at x = ", xvals
  For i, 1, dim(xvals)
    left := sign(fp | x = xvals[i] - 0.001)
    right := sign(fp | x = xvals[i] + 0.001)
    If left > 0 and right < 0 Then
      nature := "Local maximum"
    ElseIf left < 0 and right > 0 Then
      nature := "Local minimum"
    Else
      nature := "Stationary point of inflection"
    EndIf
    Disp "x = ", xvals[i], ": ", nature
  EndFor
EndPrgm


12. stps(f, x) — Stationary points
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Define stps(f, x) = Prgm
  Local fp, xvals, i, y
  fp := d(f, x)
  xvals := solve(fp = 0, x)
  Disp "f'(x) = ", fp
  Disp "Stationary points:"
  For i, 1, dim(xvals)
    y := f | x = xvals[i]
    Disp "(", xvals[i], ", ", y, ")"
  EndFor
EndPrgm


13. tangsolve(f, x, x0, y0) — Tangent lines through a point
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Define tangsolve(f, x, x0, y0) = Prgm
  Local fp, m, tang, eq, tvals
  fp := d(f, x)
  Disp "Finding tangent lines to f through (", x0, ",", y0, ")"
  tvals := solve(y0 - (f | x = t) = (fp | x = t) * (x0 - t), t)
  Disp "Tangent points at x = ", tvals
  For i, 1, dim(tvals)
    m := fp | x = tvals[i]
    tang := m * (x - tvals[i]) + (f | x = tvals[i])
    Disp "y = ", tang
  EndFor
EndPrgm


14. trapapprox(f, x, lo, hi, n) — Trapezoidal approximation
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Define trapapprox(f, x, lo, hi, n) = Prgm
  Local h, s, i, xi
  h := (hi - lo)/n
  s := (f | x = lo) + (f | x = hi)
  For i, 1, n - 1
    xi := lo + i*h
    s := s + 2*(f | x = xi)
  EndFor
  s := (h/2)*s
  Disp "h = ", h
  Disp "Trapezoid approx (", n, " strips): ", approx(s)
  Disp "Exact integral: ", approx(∫(f, x, lo, hi))
EndPrgm