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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