Download Mathable Troubleshooting

Student Response
Mathematica Troubleshooting
Student Response
1.
Clear[x, y, z, t];
{x[t], y[t], z[t]} = {t, t ^ 2, t ^ 3}
ParametricPlot3D[{x[t], y[t], z[t]}, {t, 0, 1}]
t, t2 , t3 
0.5
1.0
0.0
-0.5
-1.0
1.0
0.5
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
Student Response
This error is caused by not properly defining the parametric equation. Make sure to use an underscore after
each of the variables when defining the equation.
Clear[x, y, z, t];
{x[t_], y[t_], z[t_]} = {t, t ^ 2, t ^ 3}
ParametricPlot3D[{x[t], y[t], z[t]}, {t, 0, 1}]
t, t2 , t3 
1.0
0.5
0.0
1.0
0.5
0.0
0.0
0.5
1.0
Student Response
2.
Solve[x + 1 = 2, x]
Set::write: Tag Plus in 1 + x is Protected.
Solve::naqs: 2 is not a quantified system of equations and inequalities.
Solve[2, x]
Student Response
This error was caused by not including a double equal sign in the solve function.
Solve[x + 1 == 2, x]
{{x → 1}}
Student Response
3.
Clear[x, y, z]
Solve[x ^ 2 + y ^ 2 + z ^ 2 + 3 ^ 2 == 4 ^ 2, z]
top = Plot3D[Sqrt[7 - x ^ 2 - y ^ 2], {x, -Sqrt[7], Sqrt[7]}, {y, -Sqrt[7], Sqrt[7]}];
bottom = Plot3D[-Sqrt[7 - x ^ 2 - y ^ 2], {x, -Sqrt[7], Sqrt[7]}, {y, -Sqrt[7], Sqrt[7]}];
Show[top, bottom, PlotRange -> All]
z → -
7 - x2 - y2 , z →
7 - x2 - y2 
Student Response
Although the skin was plotted, the porportions are jank. Use BoxRatios -> Automatic to fix it.
Clear[x, y, z]
Solve[x ^ 2 + y ^ 2 + z ^ 2 + 3 ^ 2 == 4 ^ 2, z]
top = Plot3D[Sqrt[7 - x ^ 2 - y ^ 2], {x, -Sqrt[7], Sqrt[7]}, {y, -Sqrt[7], Sqrt[7]}];
bottom = Plot3D[-Sqrt[7 - x ^ 2 - y ^ 2], {x, -Sqrt[7], Sqrt[7]}, {y, -Sqrt[7], Sqrt[7]}];
Show[top, bottom, PlotRange -> All, BoxRatios -> Automatic]
z → -
7 - x2 - y2 , z →
7 - x2 - y2 
Student Response
4.
f[x_] = cos[x]
Plot[f[x], {x, 0, 2 Pi}]
cos[x]
1.0
0.5
1
2
3
4
5
6
-0.5
-1.0
Student Response
All of the Mathematica functions are capitalized. Courtesy of Lukas Janavicius
f[x_] = Cos[x]
Plot[f[x], {x, 0, 2 Pi}]
Cos[x]
1.0
0.5
1
2
3
4
5
6
-0.5
-1.0
Student Response
5.
Clear[x, y, z, m1, n1, p1, m2, n2, p2, Field1, Field2];
Field1[x_, y_, z_] = E ^ ((-x ^ 2 - y ^ 2) / 10) {x, y, z};
{m1[x_, y_, z_], n1[x_, y_, z_], p1[x_, y_, z_]} = Field1[x, y, z]
Clear[t, P];
{x[t_], y[t_], z[t_]} = {3 Cos[t], 4 Sin[t], -2 Sin[2 t] + 1};
P[t_] = {x[t], y[t], z[t]};
Integrate[Field1[x[t], y[t], z[t]].P '[t], {t, 0, 2 Pi}]
ⅇ1/10 -x
2 -y2 
x, ⅇ1/10 -x
2 -y2 
y, ⅇ1/10 -x
2 -y2 
z
Hilbert::timeconst:
This evaluation was aborted for exceeding time constraint:
Integrate[Field1[x[t],y[t],z[t]].P'[t],{t, 0, 2 Pi}]
$Aborted
Student Response
When trying to use exact values, sometimes Mathematica won't chooch. This will result in an abortion. Either
use NIntegrate or put a "." after one of the numbers.
Clear[x, y, z, m1, n1, p1, m2, n2, p2, Field1, Field2];
Field1[x_, y_, z_] = E ^ ((-x ^ 2 - y ^ 2) / 10) {x, y, z};
{m1[x_, y_, z_], n1[x_, y_, z_], p1[x_, y_, z_]} = Field1[x, y, z]
Clear[t, P];
{x[t_], y[t_], z[t_]} = {3 Cos[t], 4 Sin[t], -2 Sin[2 t] + 1};
P[t_] = {x[t], y[t], z[t]};
NIntegrate[Field1[x[t], y[t], z[t]].P '[t], {t, 0, 2 Pi}]
Integrate[Field1[x[t], y[t], z[t]].P '[t], {t, 0, 2. Pi}]
ⅇ1/10 -x
2 -y2 
x, ⅇ1/10 -x
2 -y2 
y, ⅇ1/10 -x
2 -y2 
z
-1.27951
-1.27951
Student Response
6.
Show[Vector[{-10, 3}], Vector[{3 / 10, 1}], Axes -> True, AxesLabel -> {"x", "y"}, AspectRatio -> 1]
{-10, 3}.{3 / 10, 1}
y
3.0
2.5
2.0
1.5
1.0
0.5
-10
-8
-6
-4
-2
x
0
Student Response
Sometimes perpendicular vectors do not appear to be perpendicular when plotted. Use AspectRatio>Automatic to solve this.
Show[Vector[{-10, 3}], Vector[{3 / 10, 1}], Axes -> True, AxesLabel -> {"x", "y"}, AspectRatio -> Automatic]
y
3.0
2.5
2.0
1.5
1.0
0.5
-10
-8
-6
-4
-2
x
Student Response
7.
Solve[x + xy == 3, {x}]
{{x → -xy}}
Student Response
Good bait. When multiplying two variables, use parenthesis or astericks or Mathematica will count is as one
variable.
Solve[x + x * y == 3, {x}]
3
x →

1+y
Student Response
8.
Clear[ElectricField1, ElectricFieldF2, CombinedElectricField, x, y];
q2 = 20;
{a2, b2} = {0, 0};
ElectricField2[x_, y_] = (q2 {x - a2, y - b2}) / ((x - a2) ^ 2 + (y - b2) ^ 2);
Show[Graphics[{Red, PointSize[0.03], Point[{a1, b1}]}],
Table[Vector[ElectricField2[x, y] / (Sqrt[ElectricField2[x, y].ElectricField2[x, y]]), Tail -> {x, y}],
{x, -2, 2, .5}, {y, -2, 2, .5}], Axes -> Automatic]
1
Power::infy: Infinite expression --encountered.
0.
Infinity::indet:Indeterminateexpression 0. ComplexInfinity encountered.
1
Power::infy: Infinite expression --encountered.
0.
3
2
1
-3
-2
1
-1
2
3
-1
-2
-3
Make sure that Mathematica is not dividing by zero in vector fields. To fix this, simply move the field vectors
in the table to avoid singularities.
Student Response
Make sure that Mathematica is not dividing by zero in vector fields. To fix this, simply move the field vectors
in the table to avoid singularities.
Clear[ElectricField1, ElectricFieldF2, CombinedElectricField, x, y];
q2 = 20;
{a2, b2} = {0, 0};
ElectricField2[x_, y_] = (q2 {x - a2, y - b2}) / ((x - a2) ^ 2 + (y - b2) ^ 2);
Show[Graphics[{Red, PointSize[0.03], Point[{a1, b1}]}],
Table[Vector[ElectricField2[x, y] / (Sqrt[ElectricField2[x, y].ElectricField2[x, y]]), Tail -> {x, y}],
{x, -2.1, 2.1, .5}, {y, -2.1, 2.1, .5}], Axes -> Automatic]
3
2
1
-3
-2
1
-1
2
3
-1
-2
-3
Student Response
9.
E[t] = {Cos[t], Sin[t]}
Set::write: Tag E in E[t] is Protected.
{Cos[t], Sin[t]}
Student Response
None of the inherent Mathematica functions or numbers can be saved to other functions.
f[t] = {Cos[t], Sin[t]}
{Cos[t], Sin[t]}
Student Response
10.
Log[10.]
2.30259
Student Response
In Mathematica, Log is actually ln. For logarithms: Log[argument, base]
Log[E]
Log[10, 10]
1
1
Student Response
11.
Clear[x, y, z, r, s, t];
x[s_, t_] = .1 Sin[s] Cos[t]
y[s_, t_] = .1 Sin[s] Sin[t] + 1
z[s_, t_] = .1 Cos[s]
Field2[x_, y_, z_] = (3 ({x, y, z} - {0, 1, 0})) / ((x - 0) ^ 2 + (y - 1) ^ 2 + (z - 0) ^ 2) ^ (3 / 2)
inner = Integrate[Field2[x[s, t], y[s, t], z[s, t]].Cross[{D[x[s, t], s], D[y[s, t], s], D[z[s, t], s]},
{D[x[s, t], t], D[y[s, t], t], D[z[s, t], t]}], {s, 0, Pi}, {t, 0, 2. Pi}]
0.1 Cos[t] Sin[s]
1 + 0.1 Sin[s] Sin[t]
0.1 Cos[s]
{z, 4 x, -3 y}
3x

x2 + (-1 + y)2 +
3/2
z2 
3 (-1 + y)
,
x2 + (-1 + y)2 +
3/2
z2 
3z
,

x2 + (-1 + y)2 + z2 
3/2
37.6991 + 0. ⅈ
Student Response
Sometimes, Mathematica will include a very small complex number in your answer. Use Chop to get rid of
this.
Clear[x, y, z, r, s, t];
x[s_, t_] = .1 Sin[s] Cos[t]
y[s_, t_] = .1 Sin[s] Sin[t] + 1
z[s_, t_] = .1 Cos[s]
Field2[x_, y_, z_] = (3 ({x, y, z} - {0, 1, 0})) / ((x - 0) ^ 2 + (y - 1) ^ 2 + (z - 0) ^ 2) ^ (3 / 2)
inner =
Chop[Integrate[Field2[x[s, t], y[s, t], z[s, t]].Cross[{D[x[s, t], s], D[y[s, t], s], D[z[s, t], s]},
{D[x[s, t], t], D[y[s, t], t], D[z[s, t], t]}], {s, 0, Pi}, {t, 0, 2. Pi}]]
0.1 Cos[t] Sin[s]
1 + 0.1 Sin[s] Sin[t]
0.1 Cos[s]
3x

x2
+ (-1 +
37.6991
12.
y)2
+
3/2
z2 
3 (-1 + y)
,
x2
+ (-1 +
y)2
+
3/2
z2 
3z
,

x2
+ (-1 + y)2 + z2 
3/2
Student Response
12.
Clear [x, y, f, misterkush, mrkush, wandering]
{x[t_], y[t_]} = (t / 2) {Cos[t], Sin[t]};
f[x_, y_] = 2 E ^ (-x ^ 2 - y ^ 2);
mrkush = ParametricPlot3D[{x, y, f[x, y]}, {x, -3 Pi, 3 Pi},
{y, -3 Pi, 3 Pi}, BoxRatios -> Automatic, AxesLabel -> {"x", "y", "z"}];
misterkush[w_] = ParametricPlot3D[{x[t], y[t], f[x[t], y[t]]},
{t, 0, w}, PlotStyle -> {Red, Thickness[0.02]}];
Manipulate[Show[mrkush, misterkush[w], Graphics3D[
{Red, Cone[{{x[w], y[w], f[x[w], y[w]]}, {x[w], y[w], f[x[w], y[w]] + 1}}, .3]}],
PlotRange -> {{-3 Pi, 3 Pi}, {-3 Pi, 3 Pi}, {0, 5}}, AxesLabel -> {"x", "y", "z"}], {w, 0.05, 6 Pi}]
ParametricPlot3D::plln:
Limiting value w in {t, 0, w} is not a machine-sizedreal number.
ParametricPlot3D::plln:
Limiting value w in {t, 0, w} is not a machine-sizedreal number.
w
Student Response
When using Manipulates with parametric equations that have variables is the bounds, Mathematica tries but
fails to evaluate the equation. The plain = tells Mathematica to evaluate the equation immediatley while the :=
calls for a delayed response. This delayed response means that the equation is not used until it is called by the
Manipulate thereby plugging a number into the bounds and dodging the error.
Clear [x, y, f, misterkush, mrkush, wandering]
{x[t_], y[t_]} = (t / 2) {Cos[t], Sin[t]};
f[x_, y_] = 2 E ^ (-x ^ 2 - y ^ 2);
mrkush = ParametricPlot3D[{x, y, f[x, y]}, {x, -3 Pi, 3 Pi},
{y, -3 Pi, 3 Pi}, BoxRatios -> Automatic, AxesLabel -> {"x", "y", "z"}];
misterkush[w_] := ParametricPlot3D[{x[t], y[t], f[x[t], y[t]]},
{t, 0, w}, PlotStyle -> {Red, Thickness[0.02]}];
Manipulate[Show[mrkush, misterkush[w], Graphics3D[
{Red, Cone[{{x[w], y[w], f[x[w], y[w]]}, {x[w], y[w], f[x[w], y[w]] + 1}}, .3]}],
PlotRange -> {{-3 Pi, 3 Pi}, {-3 Pi, 3 Pi}, {0, 5}}, AxesLabel -> {"x", "y", "z"}], {w, 0.05, 6 Pi}]
w
Student Response
13.
Solve[Sin[x] == 0, x]
Solve::ifun: Inverse functions are being used by Solve, so some solutions may
not be found; use Reduce for complete solution information.
{{x → 0}}
Student Response
When using Solve with trig functions, not all of the answers will be shown. Instead, use Reduce.
Reduce[Sin[x] == 0, x]
C[1] ∈ Integers && (x ⩵ 2 π C[1] || x ⩵ π + 2 π C[1])
Student Response
14.
{x[t_], y[t_], z[t_]} = {3 Cos[t], 4 Sin[t], -2 Sin[2 t] + 1};
P[t_] = {x[t], y[t], z[t]} 0;
ParametricPlot[P[t], {t, 0, 1}]
h[t_] = {t, t ^ 2}
ParametricPlot3D[h[t], {t, 0, 1}]
1.0
0.5
-1.0
0.5
-0.5
-0.5
-1.0
t, t2 
1.0
0.5
1.0
0.0
-0.5
-1.0
1.0
0.5
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
Student Response
Make sure to use the proper dimension when plotting.
Student Response
15.
Solve[Sin[x] == 0, x]
Solve::ifun: Inverse functions are being used by Solve, so some solutions may
not be found; use Reduce for complete solution information.
{{x → 0}}
Student Response
You can set bounds for the Solve function if you only need a result within a specific range of values. For
Solve: Solve[{equation, bounds}, variable]
Solve[{Sin[x] == 0, -1 <= x <= 1}, x]
{{x → 0}}
Student Response
16.
Clear[Field, m, n, p, x, y, z, q, a, b, c];
point = {a, b, c};
Field[x_, y_, z_] = (q ({x, y, z} - point)) / ((x - a) ^ 2 + (y - b) ^ 2 + (z - c) ^ 2) ^ (3 / 2);
{m[x_, y_, z_], n[x_, y_, z_], p[x_, y_, z_]} = Field[x, y, z];
Integrate[m[x, y, z], x]
Integrate[n[x, y, z], y]
Integrate[p[x, y, z], z]
f[x_, y_, z_] = -(q / Sqrt[(a - x) ^ 2 + (b - y) ^ 2 + (c - z) ^ 2])
gradf[x, y, z] = {D[f[x, y, z], x], D[f[x, y, z], y], D[f[x, y, z], z]}
gradf[x, y, z] == Field[x, y, z]
q
(a - x)2 + (b - y)2 + (c - z)2
q
(a - x)2 + (b - y)2 + (c - z)2
q
(a - x)2 + (b - y)2 + (c - z)2
q
(a - x)2 + (b - y)2 + (c - z)2
q (a - x)
(a - x)2 + (b - y)2 + (c -
3/2
z)2 
q (a - x)
(a - x)2 + (b - y)2 + (c -
3/2
z)2 
(a - x)2 + (b - y)2 + (c -
q (-a + x)
3/2
z)2 
3/2
z)2 
q (b - y)
, -

(-a + x)2 + (-b + y)2 + (-c +
q (b - y)
, -
(a - x)2 + (b - y)2 + (c ,
3/2
z)2 
q (c - z)
, -

(a - x)2 + (b - y)2 + (c - z)2 
q (c - z)
, -
⩵
(a - x)2 + (b - y)2 + (c - z)2 
q (-b + y)
(-a + x)2 + (-b + y)2 + (-c +
3/2
3/2
z)2 
,
3/2
q (-c + z)

(-a + x)2 + (-b + y)2 + (-c + z)2 
3/2
Student Response
Mathematica has trouble determining if two expressions are equal. Use Simplify to remedy this.
Simplify[gradf[x, y, z]] == Simplify[Field[x, y, z]]
True
Student Response
17.
Clear[x, y, t, a, f];
{x[t_], y[t_]} = {(4 t) / (1 + t ^ 3), (2 t ^ 2) / (1 + t ^ 3)};
f[a_] = {x[t], y[t]};
plot = ParametricPlot[{f[a]}, {t, 1, 10}, PlotStyle -> {{Green, Thickness[0.015]}},
AspectRatio -> Automatic, PlotRange -> {{-4, 4}, {-4, 4}}];
datapoint = Graphics[{Maroon, PointSize[0.03], Point[{x[1.25992], y[1.25992]]}];
Show[plot, datapoint]
Hilbert::syn: There is a syntax error starting somewhere within this part of
the expression: datapoint = Graphics[{Maroon,PointSize[0.03],
Point[{x[1.25992],
y[1.25992]]
$Failed
Student Response
When the code returns a syntax error check to make sure that you have all of the commas, brackets, and
parenthesis.
Clear[x, y, t, a, f];
{x[t_], y[t_]} = {(4 t) / (1 + t ^ 3), (2 t ^ 2) / (1 + t ^ 3)};
f[a_] = {x[t], y[t]};
plot = ParametricPlot[{f[a]}, {t, 1, 10}, PlotStyle -> {{Green, Thickness[0.015]}},
AspectRatio -> Automatic, PlotRange -> {{-4, 4}, {-4, 4}}];
datapoint = Graphics[{Maroon, PointSize[0.03], Point[{x[1.25992], y[1.25992]}]}];
Show[plot, datapoint]
4
2
-4
2
-2
4
-2
-4
Student Response
18.
Clear[Field, x, y, z, m, n, p, divField1];
Field[x_, y_, z_] =
{z, 4 x, -3 y} + (5 ({x, y, z} - {0, 1, 0})) / ((x - 0) ^ 2 + (y - 1) ^ 2 + (z - 0) ^ 2) ^ (3 / 2);
{m[x_, y_, z_], n[x_, y_, z_], p[x_, y_, z_]} = Field[x, y, z]
divField[x_, y_, z_] = D[m[x, y, z], x] + D[n[x, y, z], y] + D[p[x, y, z], z]
5x
z +
x2
+ (-1 +
y)2
+
3/2
z2 
5 (-1 + y)
, 4x+
15 x2
x2
+ (-1 +
y)2
15 (-1 + y)2
-
x2
+ (-1 +
y)2
+
3/2
z2 
+
5/2
z2 
5z
, -3 y +

x2
+ (-1 + y)2 + z2 
15 z2
15
x2
+ (-1 +
y)2
+
5/2
z2 
3/2
+
x2
+ (-1 +
y)2
+
5/2
z2 
x2
+ (-1 + y)2 + z2 
3/2
Student Response
Mathematica will not always combine long expressions automatically. Use Together to force it to.
divField[x_, y_, z_] = Together[D[m[x, y, z], x] + D[n[x, y, z], y] + D[p[x, y, z], z]]
0
Student Response
19.
FindMinimum[{1 / (x !), 0 <= x <= 200}, x]
1 / (200 !) < 1 / (103.98 !)
1.06787 × 10-166 , {x → 103.98}
True
Student Response
Mathematica has trouble with very small numbers. Use common sense to determine the correct answer is zero
or, if the answer is not expected to be zero, check your code to make sure it is entered correctly.
Student Response
20.
FindMaximum[x !, x]
General::ovfl: Overflow occurred in computation.
General::ovfl: Overflow occurred in computation.
General::ovfl: Overflow occurred in computation.
4.112840277612305 × 10278 919 263 , x → 3.89743 × 107 
Student Response
Mathematica has a largest number. Use common sense to determine the correct answer is infinity or, if the
answer is not expected to be infinity, check your code to make sure it is entered correctly.
Student Response
21.
{3, 3} + {4, 5, 6}
Thread::tdlen:Objects of unequal length in {3, 3} + {4, 5, 6}
cannot be combined.
{3, 3} + {4, 5, 6}
Student Response
The "objects of unequal length" error most commonly is encountered with vectors or parametric equations.
Make sure that everything in the problem has the same dimension.
{3, 3, 0} + {4, 5, 6}
{7, 8, 6}
Student Response
22.
Clear [x, y, f, misterkush, mrkush, wandering]
{x[t_], y[t_]} = (t / 2) {Cos[t], Sin[t]};
f[x_, y_] = 2 E ^ (-x ^ 2 - y ^ 2);
mrkush = ParametricPlot3D[{x, y, f[x, y]}, {x, -3 Pi, 3 Pi},
{y, -3 Pi, 3 Pi}, BoxRatios -> Automatic, AxesLabel -> {"x", "y", "z"}];
misterkush[w_] := ParametricPlot3D[{x[t], y[t], f[x[t], y[t]]},
{t, -0.1, w}, PlotStyle -> {Red, Thickness[0.02]}];
Manipulate[Show[mrkush, misterkush[w], Graphics3D[
{Red, Cone[{{x[w], y[w], f[x[w], y[w]]}, {x[w], y[w], f[x[w], y[w]] + 1}}, .3]}],
PlotRange -> {{-3 Pi, 3 Pi}, {-3 Pi, 3 Pi}, {0, 5}}, AxesLabel -> {"x", "y", "z"}], {w, 0, 6 Pi}]
In[44]:=
w
Out[49]=
Student Response
You cannot have a range for an equation start and end at the same number. This is commonly found with
Manipulates. Make sure the bounds always have a different starting and ending point.
Student Response
You cannot have a range for an equation start and end at the same number. This is commonly found with
Manipulates. Make sure the bounds always have a different starting and ending point.
Clear [x, y, f, misterkush, mrkush, wandering]
{x[t_], y[t_]} = (t / 2) {Cos[t], Sin[t]};
f[x_, y_] = 2 E ^ (-x ^ 2 - y ^ 2);
mrkush = ParametricPlot3D[{x, y, f[x, y]}, {x, -3 Pi, 3 Pi},
{y, -3 Pi, 3 Pi}, BoxRatios -> Automatic, AxesLabel -> {"x", "y", "z"}];
misterkush[w_] := ParametricPlot3D[{x[t], y[t], f[x[t], y[t]]},
{t, 0, w}, PlotStyle -> {Red, Thickness[0.02]}];
Manipulate[Show[mrkush, misterkush[w], Graphics3D[
{Red, Cone[{{x[w], y[w], f[x[w], y[w]]}, {x[w], y[w], f[x[w], y[w]] + 1}}, .3]}],
PlotRange -> {{-3 Pi, 3 Pi}, {-3 Pi, 3 Pi}, {0, 5}}, AxesLabel -> {"x", "y", "z"}], {w, 0.05, 6 Pi}]
w
Student Response
23.
RegionPlot3D[-Sqrt[x] <= z <= Sqrt[x] && 0 <= x <= 9 && -Sqrt[- z ^ 2 + x] <= y <= Sqrt[-z ^ 2 + x],
{x, -2, 12}, {y, -5, 5}, {z, -5, 5}, BoxRatios -> Automatic,
Mesh -> False, ColorFunction -> "BlueGreenYellow"]
LessEqual::nord: Invalid comparison with 0. - 1.41421 I attempted.
LessEqual::nord: Invalid comparison with 0. + 1.41421 I attempted.
LessEqual::nord: Invalid comparison with 0. - 1.41421 I attempted.
Student Response
When using RegionPlot and RegionPlot3D, make sure that there are no complex numbers being chooched by
Mathematica. Above, this is the result of having the bounds of x start at a negative number and the first
inequality has a square root of x. Usually reordering the inequalities is enough.
When using RegionPlot and RegionPlot3D, make sure that there are no complex numbers being chooched by
Mathematica. Above, this is the result of having the bounds of x start at a negative number and the first
inequality has a square root of x. Usually reordering the inequalities is enough.
RegionPlot3D[0 <= x <= 9 && -Sqrt[x] <= z <= Sqrt[x] && -Sqrt[- z ^ 2 + x] <= y <= Sqrt[-z ^ 2 + x],
{x, -2, 12}, {y, -5, 5}, {z, -5, 5}, BoxRatios -> Automatic,
Mesh -> False, ColorFunction -> "BlueGreenYellow"]
Student Response
24. If you encounter Internal Server Error 500, it usually means that you lost connection to the server. Try
evaluating again. If you get Cell Evaluation Error 0, you do not have internet connection. Check your connection, then try again.
Student Response
25.
Student Response
If at anytime you do not know what a specific function does, use ?[Function] to find out.
? Cos
? Plot
Cos[z] gives the cosine of z.
Plot[f, {x, x , x }] generates a plot of f
min max
as a function of x from x
min
to x . Plot[{f , f , ...},{x, x , x }]
max
1 2
min max
plots several functions f .
i
Student Response
By Zachary "Coop" Cooper, Class of 2016