Mathematica Constructions Steps

                  Part II: Mathematica

Construction on Mathematica

           

Q.No.1: Plot the parabola from –3 to 3.

Construction Steps:

1.      Use Plot command for plotting the parabola .

2.      Give the range for x, e.g. {x -3, 3}.

3.      Use command PlotStyle to show the color of the graph. e.g. PlotStyle Red.

4.      To show the name of the graph, use the command PlotLegends. e. g. PlotLegends

5.      For example:

Plot[ , {x, -3, 3}, PlotStyle {Red}, PlotLegends { }]

[Note: Use the command Shift + Enter for the result]

Q.No.2: Plot the parabola and from –3 to 3.

Construction Steps:

1.      Input the two functions  and

2.      Use Plot command for plotting the two parabolas' f(x) and g(x).

3.      Give the range for the x i. e. {x, -3, 3}.

4.   Use command PlotStyle to show the color of the graph. e. g. .

For example.

f(x_) : =  

g(x_) : = 9-

Plot [{f[x], g[x], {x, -3, 3}, ]

Q.No.3: Plot a function whose axes origin is (1, 6).

Construction Steps:

1.      Use plot the command for plotting the function .

2.      Give the range for x, i. e. {x, 1, 2}.

3.   Use the AxesOrigin {1, 6}. To change the origin of axes.

4.      For example:

Plot [ , {x, 1, 2}, AxesOrigin {1, 6}]

Q.No.4: Plot the graph of from x = 0 to x = 5.

Construction Steps:

1.   Use the Plot command for plotting the function

2.      Use palettes and go to Basic Math Assistant for the exponential function [ ].

3.      Give the range for x. from 0 to 5.

For example:

Plot[ -x, {x, 0, 5}]

Q.No.5: Plot the graph of on the interval [-3, 3].

Construction Steps:

1.      Use the Plot command for plotting the function y = Abs[1-Abs[x]].

2.      Give the range for x, -3 to 3.

For example:

Plot[Abs[1-Abs[x]], {x, -3, 3}.

Q.No.6: Plot the graph of standard normal curve form x = -3, to x = 3.

Construction Steps:

1.      Use the Plot command for plotting the function for standard normal curve eq.

2.      Use Palettes and go to Basic Math assistant for standard normal equation.

3.      Give the range for x. -3 to 3.

For example:

Plot[

Q.No.7: Plot the graph of , - and Sin[10x] on the interval [-2 Pi, 2 Pi].

Construction Steps:

1.      Use the Plot command for three functions. , - and Sin[10 x].

2.   Give the range for x. i. e. { }.

Example:

Plot[{ , - 2, Sin[10 x]}, { }].

Q.No.8: Plot a function ranged from (-3,  3) in 3D.

Construction Steps:

1.    Use 3D Plot command for plotting the graph

Note: This function is associated with three variable so, we need 3D Plot command.

2.      Give the different range for x and y. i.e. {x, -3, 3} and {y,  -3,  3}.

Example:

Plot3D [Sin [x-y], {x, -3, 3}, {y, -3, 3}]

Q.No.9: Plot a parametric curve the value of t ranged from (0 to 13) in 3D.

Construction Steps:

1.      Since, t is a parameter. Use parametric Plot 3D for plotting the function (Sin (t), Cos (t), t/3).

Note: This function is associated with three co-ordinates so, use the 3D plot command.

2.      Give the range for t. i.e. {t, 0, 13}.

Example:

ParametricPlot3D [{Sin[t], Cos[t], t/3}, {t, 0, 13}]

Q.No.10: Sketch the surface of revolution generated when the curve   from x=, 0 to x = 4, is rotated about the z-axis.

Construction Steps:

For, plotting generating function.

1.    Use plot command for the function .

2.      Give the range for x. i.e. { x,  0,  4}.

3.   Use AspectRatio 1.

Note: We use AspectRatio for ratio between height and wide.

4.   Use the AxesLabel command to show the axes name. i. e. AxesLabel  {"x", "z"}

For, Plotting Revolution function 3D.

1.    Use the RevolutionPlot3D command for generating the revolution solid. i.e. .

2.      Give the range for x. i.e. {x, 0, 4}

3.   Use the command BoxRatio 1.

4.   Use the command ViewPoint {1, -5, 1}.

5.   Use the command AxesLabel {"x", "y", "z"}. To show the name of axes.

Example:

Plot[√x, {x, 0, 4}, AspectRatio 1, AxeLable {"x", "z"}]

RevolutionPlot3D[√x, {x, 0, 4}, BoxRatios 1, ViewPoint {1, -5, 1}, AxesLabel {"x", "y", "z"}]

Q.No.11: Use Manipulate to control the graph of , 0 ≤ x < 2 π, with controls for a, b, and c varying between 1 and 10. Move the sliders and observe the affect upon the graph.

Construction Steps:

1.      Use the Manipulate command to control the, b, c.

2.      Use Plot command under manipulation for the plotting function f(x) = a Sin (b x+ c).

3.   Give the range x. i.e. {x, 0, 2 }

4.      Give the range of manipulation for the a, b and c. i.e. {a, 1, 10}, {b, 1, 10} and {c, 1, 10}

Example:

Manipulate[Plot[a Sin[b x+ c], {x, 0, 2 }], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}]

Q.No.12: Show that the function satisfies Rolle's Theorem on the interval [0, 1] and find the value of c referred to in the theorem.

Construction Steps:

1.      Input the function f[x_] = ( +2 +15x +2)Sin[π x] Check the value for the initial point of the interval. i. e. f [0] and final point of the interval. i. e. f [1].

2.      Check, whether f[0] = f[1].

3.      Give the command FindRoot for f'[c] = = 0. Where, C Î (0, 1).

4.      Note: We use double equal to sign (= =) for the command finding the root.

5.      It gives the value of c.

6.      Use the plot command for plotting the function f[x] and f [0.640241]. To show the Rolle's theorem.  

7.      Give the range for x. i.e. {x, 0, 1}.

Example:

f[x_] = ( +2 +15x +2)Sin[π x] Press Shift+ Enter.

f[0] Press Shift+ Enter.

f[1] Press Shift+ Enter.

FindRoot[f'[c] = = 0, {x, 0, 1} ] Press Shift+ Enter

Plot[{f[x], f[0.640241]}, {x, 0, 1}].

Q.No.13: Sketch the graph of f and its derivative, on the set of axes, for -10<x<10.

Construction Steps:

1.    Input ;  press Shift+ Enter

2.      Give the range of x i.e. {x, -10, 10},

3.   Decorate the plot by using

Example:

           

Use Manipulate to show that the tangent line at various points of the curve  for 0<x<2pi.

Construction Steps:

1.    Input ;  press Shift+ Enter

2.      Give the input t[x_, u_] := f[u] + f'[u] (x - u) /; u - 0.5 < x < u + 0.5 for a the tangent at a point u

3.      Use the command manipulate for manipulation of tangent

4.      Use plot command to plot graph and its tangent

5.      Give the range of x i.e. {x, 0, 6},

6.      Decorate the plot by using PlotStyle and Graphic command

Example :

Manipulate[Show[Plot[{f[x], t[x, u]}, {x, 0, 6}, PlotStyle → {Red, {Green, Thickness[0.01]}}],Graphics[{PointSize[0.02], Point[{u, f[u]}]}]], {u, 0, 6}]

Q.No.14: Area enclosed by two curve.

Construction Steps:

1.       In put the function f[x_] = 1-  and g[x_] = -3

2.       Use Plot the command for plotting the function f[x] and g[x].

3.       Give the range for x. i.e. {x, -2, 2}.

4.    Use the PlotStyle Command for coloring curve line. i. e. PlotStyle  {Red, Blue}.

5.    Use the Filling command for filling the intersection of two curves. i.e. Filling  {1  {{2}, {None, Yellow}}}

Example:

f[x_]= 1-  Press Shift+ Enter

g[x_] =   -3 Press Shift + Enter

Plot[{f[x], g[x]}, {x, -2, 2}, PlotStyle  {Red, Blue}, Filling{1   {{2}, {None, Yellow}}}] Press Shift + Enter.

For Area of enclosed by two curve i.e. Yellow area.

1.      Input point=Solve[f[x]= =g[x]] Press Shift + Enter.

2.      Input {a, b, c, d}=x/.point [Press Shift+ Enter]

3.   Use palettes and go to basic math assistant and input the symbol //N [Press Shift + Enter]

Q.No.15: Binomial Probability Distribution Template.

Construction Steps:

1.      Define the binomial parameters n, p, q for example: n = 2; p = 0.5; q = 1 - p;

2.   Define  = ProbabilityDistribution[Binomial[n, x] p^x q^(n - x), {x, 0, n, 1}] then, Shift+ Enter

3.   Give the input PDF[ , x] Press Shift+ Enter

4.   Give the input Mean[ ] for finding mean of the distribution

5.   Give the input Variance[ ] for finding variance of the distribution

6.   Write a command TableForm[{Table[PDF[ , x], {x, 0, n, 1}]}, TableHeadings →{{"f(x)"}, Table[x, {x, 0, n, 1}]}] to show distribution in table form.

7.   Give  DiscretePlot[PDF[ , x], {x, -1, n + 1}, PlotStyle →Black, FillingStyle→Directive[Opacity[1], Red],AxesOrigin → {-1, 0}, ExtentSize → 0.2, AxesLabel → {"Value of X", "Probability f(x)"}] command for ploting the graph of the probability distribution.

Q.No.16: Plot the straight line in Mathematica using Contour plot. Show Axis and AxesLable. Take the range from - 2 to 2.

Construction Steps:

1.      Use ContourPlot command for the Polting the Straightline 2x+3y == 1.

Note: Use = = instead of = for ContourPlot command.

2.      Give the range of x and y i.e. {x , -2, 2}, {y,-2, 2}.

3.   Use Axes True to show Axes and AxesLabel {"X","Y"} to give the name of the Axes.

Example:

ContourPlot[2 x + 3 y == 1, {x, -2, 2}, {y, -2, 2},Axes True, AxesLabel  {X - Axis, Y - Axis}]

Q.No.17: Solve and Plot two linear equation x+ y=7 and x-y=1 in Mathematica using solve and contourplot command. Show Axis and intersection point. Take the range from -6 to 6.

Construction Steps:

1.      Give the input for solving Solve[ x + y == 7 && x - y == 1, {x, y}].

2.      Use the show command to show the CountourPlot {x + y == 7, x - y == 1}.

3.      Give the range of x and y i.e. {x, -6, 6}, {y, - 6, 6}.

4.   Use Axes  True and AxesLabel  {"X","Y"} to show and name for the Axes.

5.   Use AxesStyle   Directive[Orange, 12] for coluring and Width of the Axes.

6.      Use Graphics [{PointSize[0.03], Point[{4, 3}]}] for point size and intersection point.

Example:

Solve[ x + y == 7 && x - y == 1, {x, y}] press Shift + enter.

Show[ContourPlot[{x + y == 7, x - y == 1}, {x, -6, 6}, {y, -6, 6}, Axes   True,

AxesLabel   {X - Axis, Y - Axis}, AxesStyle   Directive[Orange, 12]],

Graphics[{PointSize[0.03], Point[{4, 3}]}]].

Q.No.18: Plot the hyperboloid using Contourplot3D. Take the range of plot from -2 to 2. ShowAxesLable and PlotLabel.

Construction Steps:

1.    Use the ContourPlot3D command for Plotting hyperboloid .

2.      [Note: This equation is related with three variables so we use 3D Plot 

3.      Use = = instead of = for ContourPlot.]

4.      Give the range of x, y and z. i.e, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}.

5.   Give the command AxesLabel l  {x,y,z} to Labled the Axes.

6.   Give the command PlotLabel  l   x^2 + y ^ 2 - z^ 2 == 1 to name the graph.

Example:

ContourPlot3D[ , {x, -2, 2}, {y, -2, 2}, {z, -2, 2},

AxesLabel l   {x, y, z}, PlotLabel l   ].

Q.No.19: Plot a helix  using ParametricPlot3D.  And Manipulate the parameter a from 1 to 5.

Construction Steps:

1.      Since this function is determined by parameter, u and having 3 co-ordinates show we use parametricPlot3D.

2.      Use Manipulate command for manipulating a.

3.      Use ParametricPlot3D for [{a Sin[u], a Cos[u], u / 10}, {u, 0, 20}].

4.      Give the manipulation rang for a i. e {a, 1, 5}.

Example:

Manipulate[ParametricPlot3D[{a Sin[u], a Cos[u], u / 10}, {u, 0, 20}], {a, 1, 5}]

Q.No.20: Surface plot

Construction Steps:

1.      Use the command ParametricPlot3D[{u + v, u - v, u^2 + v^2}].

2.      Give the range for u and v i. e {u,-3, 3},{v,-3, 3}.

3.   Use the PlotStyle Yellow command for coloring the surface.

4.   Use the AxesLabel {X, Y, Z} to name the Axes.

5.   Use the command ImageSize Large for enlarging the surface.

Example:

ParametricPlot3D[{u + v, u - v, u^2 + v^2}, {u, -3, 3},{v, -3, 3}, BoxRatios  {1, 1, 1}, PlotStyle  Yellow, AxesLabel  {X - Axis, Y - Axis, Z - Axis}, ImageSize  Large].

Q.No.21: Sphere plot

Construction Steps:

1.      Use the ParametricPlot3D[{Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]}]

2.      Give the range for u and v i. e {u, 0, Pi}, {v,-Pi, Pi + 3 Pi / 4}.

3.   Use the PlotStyle Opacity[0.5]

Example:

ParametricPlot3D[{Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]},{u, 0, Pi}, {v, -Pi, Pi + 3 Pi / 4}, PlotStyle  Opacity[0.5]].

Q.No.22: Cylinder plot.

Construction Steps:

1.      Use the ContourPlot3D[(x - 1)^2 + y^2 == 1].

2.      Give the range for x, y and z i. e, {x,-2,2},{y,-2, 2},{z,-2, 2}.

3.   Use AxesLabel {X, Y, Z} for name the axes for naming for the Axes.

4.   Use PlotRnge Full for complet appear of cylinder inside the axes.

Example:

ContourPlot3D[(x - 1)^2 + y^2 == 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},BoxRatios  {1, 1, 1}, AxesLabel  {X - Axis, Y - Axis, Z - Axis}, PlotRange Full].

Q.No.23: Tangent Plane plot

Construction Steps:

1.      Use Manipulate command to Manipulate to Tangent Plane.

2.      Use ParametricPlot3D[{u + v, u - v, u^2 + v^2}] under Show command.

3.      Give BoxRatios, PlotStyle, AxesLabel, ImageSize as you’re interested.

For example:

BoxRatios {1, 1, 1},PlotStyle Yellow, AxesLabel  {X -Axis, Y-Axis, Z-Axis}, ImageSize  Large.

Use Graphics3D command for decorating the tangent Plane.

Example:

Manipulate[Show[ParametricPlot3D[{u + v, u - v, u^2 + v^2},{u, -3, 3}, {v, -3, 3}, BoxRatios  {1, 1, 1}, PlotStyle  Yellow,AxesLabel  {X - Axis, Y - Axis, Z - Axis}, ImageSize  Large],Graphics3D[{Red, PointSize[0.02], Point[{u + v, u - v, u^2 + v^2}]} /. {u  p, v q}],Graphics3D[Arrow[{{u + v, u - v, u² + v²}, {1 + u + v, 1 + u - v, 2 u + u² + v²}} /.{u  p, v  q}]], Graphics3D[Arrow[{{u + v, u - v, u² + v²}, {1 + u + v, -1 + u - v, u² + 2 v + v²}}/. {u  p, v ¦ q}]], ParametricPlot3D[({u + v, u - v, u^2 + v^2} + a {1, 1, 2 u} + b {1, -1, 2 v}) /.{u  p, v  q}, {a, -2, 2}, {b, -2, 2}, PlotStyle  Blue]], {p, -3, 3}, {q, -3, 3}].

Q.No.24: Surface plot

Construction Steps:

1.      Use ParametericPlot3D command for Ploting {4 + (3 + Cos[v]) Sin[u], 4 + (3 + Cos[v]) Cos[u], 4 + Sin[v]} and {8 + (3 + Cos[v]) Cos[u], 3 + Sin[v], 4 + (3 + Cos[v]) Sin[u]}.

2.      Give the range for u and v i. e, {u,0, 2 Pi},{v,0, 2 Pi}.

3.   Use PlotStyle {Red,Green} to Show the different colure of the given Surface.

Example:

ParametricPlot3D[{{4 + (3 + Cos[v]) Sin[u], 4 + (3 + Cos[v]) Cos[u], 4 + Sin[v]}, {8 + (3 + Cos[v]) Cos[u], 3 + Sin[v], 4 + (3 + Cos[v]) Sin[u]}}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle  {Red, Green}]

Q.No.25: Continuous Probability Distribution Template

Construction Steps:

1.      Give the input 'a' =1; b=4

Note: Where 'a' means initial value of the distribution and 'b' means final value of the distribution.

2.      Enter the function , f[x] = x^2 / 21

3.   Input  = ProbabilityDistribution[x^2 / 21, {x, a, b}];

4.   Input PDF[ , x] then Shift+ Enter

5.    Check total probability by using input

Then, you will get 1 as a result.

1.   Give the input mean; Mean [ ] // N for finding the mean of the distribution.

2.   Give the input the Variance Variance[ ] // N for finding the variance of the distribution.

3.      For finding the Probability within the certain interval firstly Enter the range e.g. c=2; d=4.5

4.   Input ;

5.   Give the command Plot[ PDF{ , x}]

Q.No.26: Discrete Probability Distribution Template.

Construction Steps:

1.      Give the input a=1; b=3

2.      Note: Where 'a' means initial value of the distribution and 'b' means final value of the distribution.

3.   Give the input  =ProbablityDistributation[x^2/14,{x, a, b, 1}] ;

4.   Input the PDF[ , x]  then press Shift+ Enter.

5.   Give the input Mean  for finding the mean of the distribution.

6.    Give the input Variance[ ] // N for finding the variance of the distribution.

Q.No.27: The Explicit representation of the space curve.

Construction Steps:

1.      Use Manipulate command to manipulate the parametric curve

2.      Use Parametric Plot3D command for ploting the graph [{t+1, t^3,t^2}] under the show command.

3.      Give the range  of the parameter 't' i.e. {t,-2,2}

4.      Decorate the graph using the box ratio, axis level, Graphics3D command.

Example:

Manipulate[Show[ParametricPlot3D[{t + 1, t^3, t^2}, {t, -2, 2}, BoxRatios →{1, 1, 1}, AxesLabel → {X - Axis, Y - Axis, Z - Axis}], Graphics3D[{PointSize[0.03], Red, Point[{t + 1, t^3, t^2}]} /. {t → parameter}]], {parameter, -2, 2}].

Q.No.28: The Implicit representation of the Viviani curve.

Construction Steps:

1.      Give input h=x^2 + y^2 + z^2 - 4 ; and g = (x - 1)^2 + y^2 - 1;

2.      Use CounterPlot3D command to plot {h == 0,   g == 0}.

3.      Give the rang for x,y and z i.e. {x, -2, 2}, {y, -2, 2}, {z, -2, 2}

4.      Decorate the curve by using BoxRatio, AxesLable and PlotRange command.

Example:

h = x^2 + y^2 + z^2 - 4 ;

g = (x - 1)^2 + y^2 - 1;

ContourPlot3D[{h== 0, g== 0}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},BoxRatios → {1, 1, 1}, AxesLabel →{X - Axis, Y - Axis, Z - Axis}, PlotRange → Full]

Q.No.29: Unit vector along tangent to a space curve.

Construction Steps:

1.      Inter the space curve r[t]= {t^3, t^3 + 7 t + 5 t^2, t^2} press Shift+ enter

2.      Inter r'[t] for finding the derivative of the space curve.

3.      Then give the command tangentvector = FullSimplify[r'[t], t ∈ Reals]

4.      Give magnitude = FullSimplify[Norm[r'[t]], t ∈ Reals] for finding the magnitude of the space curve.

5.      Give the command unittangentvector = tangentvector / magnitude for finding unit tangent vector.

Q.No.30. Plot a Helix (a Sin[u], a Cos[u], u/10) using ParametricPlot3D.And Manipulate the parameter a from 1 to 5.

Construction Steps:

1.      Use manipulate command for manipulating the helix.

2.      Use ParamatricPlot3D[{a u Sin[v],a u Cos[v],u v}] command for plotting the helix

3.      Give the range for 'u' and 'v' i.e. {u, 0, 10}, {v, 0, 10}.

4.      Give the range of the manipulation for a. i.e. {a, 1, 5}

Example:

Manipulate[ParametricPlot3D[{a u Sin[v], a u Cos[v], u v}, {u, 0, 10}, {v, 0, 10}].

Q.No.31: Limit of the function.

Construction Steps:

1.      Give the limit command for the function Limit[(x^2 - 2 x - 8) / (x - 4)].

2.      Give the limit point for x. i.e. x→4

3.      Use Direction → -1 command: Limit[(x^2 - 2 x - 8) / (x - 4), x → 4, Direction →1]

Example: 

Limit[(x^2 - 2 x - 8) / (x - 4), x → 4, Direction →-1]

Then, Limit[x^a, x → ∞, Assumptions → a < 0]

Q.No.32: Limit of a function in Plot.

Construction Steps:

1.      Use the Manipulate command for manipulation of the given function.

2.      Use Plot Plot[{a x Sin[1 / (b x)], Abs[a x], - Abs[a x]}] command

3.      Give the range for x. i.e. {x, -1, 1}

4.      Use PlotRange{-5, 5}

5.      Give the range of manipulation{{a, 1, "Amplitude"}, 1, 5}, {{b, 1, "Periodicity"}, 1, 5}

Example:

Manipulate[Plot[{a x Sin[1 / (b x)], Abs[a x], - Abs[a x]}, {x, -1, 1},PlotRange → {-5, 5}], {{a, 1, "Amplitude"}, 1, 5}, {{b, 1, "Periodicity"}, 1, 5}]

Q.No.33: Test of Left Hand and Right hand limit.

Construction Steps:

1.      Give command limit = Limit[Abs[x] / x, x → 0]

2.       Give the command limitright = Limit[Abs[x] / x, x → 0, Direction → -1] for finding right hand limit

3.      Similarly, limitleft = Limit[Abs[x] / x, x → 0, Direction →1] for finding left hand limit.

4.      Use the limit == limitright == limitleft command whether the limit is exist or not.

Q.No.34: Derivative of function by first principle.

Construction Steps:

1.      Input the function f[x_] := Tan[x]

2.      Give the command; Limit[(f[x + h] - f[x]) / h, h → 0, Analytic → True] press Shift+ Enter

3.      Example: f[x_] := Tan[x]

4.      Limit[(f[x + h] - f[x]) / h, h →0, Analytic → True]

Q.No.35: Application of Integration.

Construction Steps:

1.      Give the Manipulate command for manipulation of h

2.      Use [ContourPlot[{x^2 + y^2== 16, y==h}] under manipulation.

3.      Give the range of x and y respectively i.e.{x, -4.1, 4.1}, {y, 0, 4.1}

4.      Use AspectRatio→Automatic command

5.      Give the range of manipulation i.e. {h, 0, 4}

Example:  Manipulate[ContourPlot[{x^2 + y^2==16, y==h},

{x, -4.1, 4.1}, {y, 0, 4.1}, AspectRatio → Automatic], {h, 0, 4}]

6.      Use the following input

Part II: Mathematica

Construction on Mathematica:

Q.No.1: Plot the parabola from –3 to 3.

Construction Steps:

1.      Use Plot command for plotting the parabola .

2.      Give the range for x, e.g. {x -3, 3}.

3.      Use command PlotStyle to show the color of the graph. e.g. PlotStyle Red.

4.      To show the name of the graph, use the command PlotLegends. e. g. PlotLegends

5.      For example:

Plot[ , {x, -3, 3}, PlotStyle {Red}, PlotLegends { }]

[Note: Use the command Shift + Enter for the result]

Q.No.2: Plot the parabola and from –3 to 3.

Construction Steps:

1.      Input the two functions  and

2.      Use Plot command for plotting the two parabolas' f(x) and g(x).

3.      Give the range for the x i. e. {x, -3, 3}.

4.   Use command PlotStyle to show the color of the graph. e. g. .

For example.

f(x_) : =  

g(x_) : = 9-

Plot [{f[x], g[x], {x, -3, 3}, ]

Q.No.3: Plot a function whose axes origin is (1, 6).

Construction Steps:

1.      Use plot the command for plotting the function .

2.      Give the range for x, i. e. {x, 1, 2}.

3.   Use the AxesOrigin {1, 6}. To change the origin of axes.

4.      For example:

Plot [ , {x, 1, 2}, AxesOrigin {1, 6}]

Q.No.4: Plot the graph of from x = 0 to x = 5.

Construction Steps:

1.   Use the Plot command for plotting the function

2.      Use palettes and go to Basic Math Assistant for the exponential function [ ].

3.      Give the range for x. from 0 to 5.

For example:

Plot[ -x, {x, 0, 5}]

Q.No.5: Plot the graph of on the interval [-3, 3].

Construction Steps:

1.      Use the Plot command for plotting the function y = Abs[1-Abs[x]].

2.      Give the range for x, -3 to 3.

For example:

Plot[Abs[1-Abs[x]], {x, -3, 3}.

Q.No.6: Plot the graph of standard normal curve form x = -3, to x = 3.

Construction Steps:

1.      Use the Plot command for plotting the function for standard normal curve eq.

2.      Use Palettes and go to Basic Math assistant for standard normal equation.

3.      Give the range for x. -3 to 3.

For example:

Plot[

Q.No.7: Plot the graph of , - and Sin[10x] on the interval [-2 Pi, 2 Pi].

Construction Steps:

1.      Use the Plot command for three functions. , - and Sin[10 x].

2.   Give the range for x. i. e. { }.

Example:

Plot[{ , - 2, Sin[10 x]}, { }].

Q.No.8: Plot a function ranged from (-3,  3) in 3D.

Construction Steps:

1.    Use 3D Plot command for plotting the graph

Note: This function is associated with three variable so, we need 3D Plot command.

2.      Give the different range for x and y. i.e. {x, -3, 3} and {y,  -3,  3}.

Example:

Plot3D [Sin [x-y], {x, -3, 3}, {y, -3, 3}]

Q.No.9: Plot a parametric curve the value of t ranged from (0 to 13) in 3D.

Construction Steps:

1.      Since, t is a parameter. Use parametric Plot 3D for plotting the function (Sin (t), Cos (t), t/3).

Note: This function is associated with three co-ordinates so, use the 3D plot command.

2.      Give the range for t. i.e. {t, 0, 13}.

Example:

ParametricPlot3D [{Sin[t], Cos[t], t/3}, {t, 0, 13}]

Q.No.10: Sketch the surface of revolution generated when the curve   from x=, 0 to x = 4, is rotated about the z-axis.

Construction Steps:

For, plotting generating function.

1.    Use plot command for the function .

2.      Give the range for x. i.e. { x,  0,  4}.

3.   Use AspectRatio 1.

Note: We use AspectRatio for ratio between height and wide.

4.   Use the AxesLabel command to show the axes name. i. e. AxesLabel  {"x", "z"}

For, Plotting Revolution function 3D.

1.    Use the RevolutionPlot3D command for generating the revolution solid. i.e. .

2.      Give the range for x. i.e. {x, 0, 4}

3.   Use the command BoxRatio 1.

4.   Use the command ViewPoint {1, -5, 1}.

5.   Use the command AxesLabel {"x", "y", "z"}. To show the name of axes.

Example:

Plot[√x, {x, 0, 4}, AspectRatio 1, AxeLable {"x", "z"}]

RevolutionPlot3D[√x, {x, 0, 4}, BoxRatios 1, ViewPoint {1, -5, 1}, AxesLabel {"x", "y", "z"}]

Q.No.11: Use Manipulate to control the graph of , 0 ≤ x < 2 π, with controls for a, b, and c varying between 1 and 10. Move the sliders and observe the affect upon the graph.

Construction Steps:

1.      Use the Manipulate command to control the, b, c.

2.      Use Plot command under manipulation for the plotting function f(x) = a Sin (b x+ c).

3.   Give the range x. i.e. {x, 0, 2 }

4.      Give the range of manipulation for the a, b and c. i.e. {a, 1, 10}, {b, 1, 10} and {c, 1, 10}

Example:

Manipulate[Plot[a Sin[b x+ c], {x, 0, 2 }], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}]

Q.No.12: Show that the function satisfies Rolle's Theorem on the interval [0, 1] and find the value of c referred to in the theorem.

Construction Steps:

1.      Input the function f[x_] = ( +2 +15x +2)Sin[π x] Check the value for the initial point of the interval. i. e. f [0] and final point of the interval. i. e. f [1].

2.      Check, whether f[0] = f[1].

3.      Give the command FindRoot for f'[c] = = 0. Where, C Î (0, 1).

4.      Note: We use double equal to sign (= =) for the command finding the root.

5.      It gives the value of c.

6.      Use the plot command for plotting the function f[x] and f [0.640241]. To show the Rolle's theorem.  

7.      Give the range for x. i.e. {x, 0, 1}.

Example:

f[x_] = ( +2 +15x +2)Sin[π x] Press Shift+ Enter.

f[0] Press Shift+ Enter.

f[1] Press Shift+ Enter.

FindRoot[f'[c] = = 0, {x, 0, 1} ] Press Shift+ Enter

Plot[{f[x], f[0.640241]}, {x, 0, 1}].

Q.No.13: Sketch the graph of f and its derivative, on the set of axes, for -10<x<10.

Construction Steps:

1.    Input ;  press Shift+ Enter

2.      Give the range of x i.e. {x, -10, 10},

3.   Decorate the plot by using

Example:

           

Use Manipulate to show that the tangent line at various points of the curve  for 0<x<2pi.

Construction Steps:

1.    Input ;  press Shift+ Enter

2.      Give the input t[x_, u_] := f[u] + f'[u] (x - u) /; u - 0.5 < x < u + 0.5 for a the tangent at a point u

3.      Use the command manipulate for manipulation of tangent

4.      Use plot command to plot graph and its tangent

5.      Give the range of x i.e. {x, 0, 6},

6.      Decorate the plot by using PlotStyle and Graphic command

Example :

Manipulate[Show[Plot[{f[x], t[x, u]}, {x, 0, 6}, PlotStyle → {Red, {Green, Thickness[0.01]}}],Graphics[{PointSize[0.02], Point[{u, f[u]}]}]], {u, 0, 6}]

Q.No.14: Area enclosed by two curve.

Construction Steps:

1.       In put the function f[x_] = 1-  and g[x_] = -3

2.       Use Plot the command for plotting the function f[x] and g[x].

3.       Give the range for x. i.e. {x, -2, 2}.

4.    Use the PlotStyle Command for coloring curve line. i. e. PlotStyle  {Red, Blue}.

5.    Use the Filling command for filling the intersection of two curves. i.e. Filling  {1  {{2}, {None, Yellow}}}

Example:

f[x_]= 1-  Press Shift+ Enter

g[x_] =   -3 Press Shift + Enter

Plot[{f[x], g[x]}, {x, -2, 2}, PlotStyle  {Red, Blue}, Filling{1   {{2}, {None, Yellow}}}] Press Shift + Enter.

For Area of enclosed by two curve i.e. Yellow area.

1.      Input point=Solve[f[x]= =g[x]] Press Shift + Enter.

2.      Input {a, b, c, d}=x/.point [Press Shift+ Enter]

3.   Use palettes and go to basic math assistant and input the symbol //N [Press Shift + Enter]

Q.No.15: Binomial Probability Distribution Template.

Construction Steps:

1.      Define the binomial parameters n, p, q for example: n = 2; p = 0.5; q = 1 - p;

2.   Define  = ProbabilityDistribution[Binomial[n, x] p^x q^(n - x), {x, 0, n, 1}] then, Shift+ Enter

3.   Give the input PDF[ , x] Press Shift+ Enter

4.   Give the input Mean[ ] for finding mean of the distribution

5.   Give the input Variance[ ] for finding variance of the distribution

6.   Write a command TableForm[{Table[PDF[ , x], {x, 0, n, 1}]}, TableHeadings →{{"f(x)"}, Table[x, {x, 0, n, 1}]}] to show distribution in table form.

7.   Give  DiscretePlot[PDF[ , x], {x, -1, n + 1}, PlotStyle →Black, FillingStyle→Directive[Opacity[1], Red],AxesOrigin → {-1, 0}, ExtentSize → 0.2, AxesLabel → {"Value of X", "Probability f(x)"}] command for ploting the graph of the probability distribution.

Q.No.16: Plot the straight line in Mathematica using Contour plot. Show Axis and AxesLable. Take the range from - 2 to 2.

Construction Steps:

1.      Use ContourPlot command for the Polting the Straightline 2x+3y == 1.

Note: Use = = instead of = for ContourPlot command.

2.      Give the range of x and y i.e. {x , -2, 2}, {y,-2, 2}.

3.   Use Axes True to show Axes and AxesLabel {"X","Y"} to give the name of the Axes.

Example:

ContourPlot[2 x + 3 y == 1, {x, -2, 2}, {y, -2, 2},Axes True, AxesLabel  {X - Axis, Y - Axis}]

Q.No.17: Solve and Plot two linear equation x+ y=7 and x-y=1 in Mathematica using solve and contourplot command. Show Axis and intersection point. Take the range from -6 to 6.

Construction Steps:

1.      Give the input for solving Solve[ x + y == 7 && x - y == 1, {x, y}].

2.      Use the show command to show the CountourPlot {x + y == 7, x - y == 1}.

3.      Give the range of x and y i.e. {x, -6, 6}, {y, - 6, 6}.

4.   Use Axes  True and AxesLabel  {"X","Y"} to show and name for the Axes.

5.   Use AxesStyle   Directive[Orange, 12] for coluring and Width of the Axes.

6.      Use Graphics [{PointSize[0.03], Point[{4, 3}]}] for point size and intersection point.

Example:

Solve[ x + y == 7 && x - y == 1, {x, y}] press Shift + enter.

Show[ContourPlot[{x + y == 7, x - y == 1}, {x, -6, 6}, {y, -6, 6}, Axes   True,

AxesLabel   {X - Axis, Y - Axis}, AxesStyle   Directive[Orange, 12]],

Graphics[{PointSize[0.03], Point[{4, 3}]}]].

Q.No.18: Plot the hyperboloid using Contourplot3D. Take the range of plot from -2 to 2. ShowAxesLable and PlotLabel.

Construction Steps:

1.    Use the ContourPlot3D command for Plotting hyperboloid .

2.      [Note: This equation is related with three variables so we use 3D Plot 

3.      Use = = instead of = for ContourPlot.]

4.      Give the range of x, y and z. i.e, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}.

5.   Give the command AxesLabel l  {x,y,z} to Labled the Axes.

6.   Give the command PlotLabel  l   x^2 + y ^ 2 - z^ 2 == 1 to name the graph.

Example:

ContourPlot3D[ , {x, -2, 2}, {y, -2, 2}, {z, -2, 2},

AxesLabel l   {x, y, z}, PlotLabel l   ].

Q.No.19: Plot a helix  using ParametricPlot3D.  And Manipulate the parameter a from 1 to 5.

Construction Steps:

1.      Since this function is determined by parameter, u and having 3 co-ordinates show we use parametricPlot3D.

2.      Use Manipulate command for manipulating a.

3.      Use ParametricPlot3D for [{a Sin[u], a Cos[u], u / 10}, {u, 0, 20}].

4.      Give the manipulation rang for a i. e {a, 1, 5}.

Example:

Manipulate[ParametricPlot3D[{a Sin[u], a Cos[u], u / 10}, {u, 0, 20}], {a, 1, 5}]

Q.No.20: Surface plot

Construction Steps:

1.      Use the command ParametricPlot3D[{u + v, u - v, u^2 + v^2}].

2.      Give the range for u and v i. e {u,-3, 3},{v,-3, 3}.

3.   Use the PlotStyle Yellow command for coloring the surface.

4.   Use the AxesLabel {X, Y, Z} to name the Axes.

5.   Use the command ImageSize Large for enlarging the surface.

Example:

ParametricPlot3D[{u + v, u - v, u^2 + v^2}, {u, -3, 3},{v, -3, 3}, BoxRatios  {1, 1, 1}, PlotStyle  Yellow, AxesLabel  {X - Axis, Y - Axis, Z - Axis}, ImageSize  Large].

Q.No.21: Sphere plot

Construction Steps:

1.      Use the ParametricPlot3D[{Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]}]

2.      Give the range for u and v i. e {u, 0, Pi}, {v,-Pi, Pi + 3 Pi / 4}.

3.   Use the PlotStyle Opacity[0.5]

Example:

ParametricPlot3D[{Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]},{u, 0, Pi}, {v, -Pi, Pi + 3 Pi / 4}, PlotStyle  Opacity[0.5]].

Q.No.22: Cylinder plot.

Construction Steps:

1.      Use the ContourPlot3D[(x - 1)^2 + y^2 == 1].

2.      Give the range for x, y and z i. e, {x,-2,2},{y,-2, 2},{z,-2, 2}.

3.   Use AxesLabel {X, Y, Z} for name the axes for naming for the Axes.

4.   Use PlotRnge Full for complet appear of cylinder inside the axes.

Example:

ContourPlot3D[(x - 1)^2 + y^2 == 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},BoxRatios  {1, 1, 1}, AxesLabel  {X - Axis, Y - Axis, Z - Axis}, PlotRange Full].

Q.No.23: Tangent Plane plot

Construction Steps:

1.      Use Manipulate command to Manipulate to Tangent Plane.

2.      Use ParametricPlot3D[{u + v, u - v, u^2 + v^2}] under Show command.

3.      Give BoxRatios, PlotStyle, AxesLabel, ImageSize as you’re interested.

For example:

BoxRatios {1, 1, 1},PlotStyle Yellow, AxesLabel  {X -Axis, Y-Axis, Z-Axis}, ImageSize  Large.

Use Graphics3D command for decorating the tangent Plane.

Example:

Manipulate[Show[ParametricPlot3D[{u + v, u - v, u^2 + v^2},{u, -3, 3}, {v, -3, 3}, BoxRatios  {1, 1, 1}, PlotStyle  Yellow,AxesLabel  {X - Axis, Y - Axis, Z - Axis}, ImageSize  Large],Graphics3D[{Red, PointSize[0.02], Point[{u + v, u - v, u^2 + v^2}]} /. {u  p, v q}],Graphics3D[Arrow[{{u + v, u - v, u² + v²}, {1 + u + v, 1 + u - v, 2 u + u² + v²}} /.{u  p, v  q}]], Graphics3D[Arrow[{{u + v, u - v, u² + v²}, {1 + u + v, -1 + u - v, u² + 2 v + v²}}/. {u  p, v ¦ q}]], ParametricPlot3D[({u + v, u - v, u^2 + v^2} + a {1, 1, 2 u} + b {1, -1, 2 v}) /.{u  p, v  q}, {a, -2, 2}, {b, -2, 2}, PlotStyle  Blue]], {p, -3, 3}, {q, -3, 3}].

Q.No.24: Surface plot

Construction Steps:

1.      Use ParametericPlot3D command for Ploting {4 + (3 + Cos[v]) Sin[u], 4 + (3 + Cos[v]) Cos[u], 4 + Sin[v]} and {8 + (3 + Cos[v]) Cos[u], 3 + Sin[v], 4 + (3 + Cos[v]) Sin[u]}.

2.      Give the range for u and v i. e, {u,0, 2 Pi},{v,0, 2 Pi}.

3.   Use PlotStyle {Red,Green} to Show the different colure of the given Surface.

Example:

ParametricPlot3D[{{4 + (3 + Cos[v]) Sin[u], 4 + (3 + Cos[v]) Cos[u], 4 + Sin[v]}, {8 + (3 + Cos[v]) Cos[u], 3 + Sin[v], 4 + (3 + Cos[v]) Sin[u]}}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle  {Red, Green}]

Q.No.25: Continuous Probability Distribution Template

Construction Steps:

1.      Give the input 'a' =1; b=4

Note: Where 'a' means initial value of the distribution and 'b' means final value of the distribution.

2.      Enter the function , f[x] = x^2 / 21

3.   Input  = ProbabilityDistribution[x^2 / 21, {x, a, b}];

4.   Input PDF[ , x] then Shift+ Enter

5.    Check total probability by using input

Then, you will get 1 as a result.

1.   Give the input mean; Mean [ ] // N for finding the mean of the distribution.

2.   Give the input the Variance Variance[ ] // N for finding the variance of the distribution.

3.      For finding the Probability within the certain interval firstly Enter the range e.g. c=2; d=4.5

4.   Input ;

5.   Give the command Plot[ PDF{ , x}]

Q.No.26: Discrete Probability Distribution Template.

Construction Steps:

1.      Give the input a=1; b=3

2.      Note: Where 'a' means initial value of the distribution and 'b' means final value of the distribution.

3.   Give the input  =ProbablityDistributation[x^2/14,{x, a, b, 1}] ;

4.   Input the PDF[ , x]  then press Shift+ Enter.

5.   Give the input Mean  for finding the mean of the distribution.

6.    Give the input Variance[ ] // N for finding the variance of the distribution.

Q.No.27: The Explicit representation of the space curve.

Construction Steps:

1.      Use Manipulate command to manipulate the parametric curve

2.      Use Parametric Plot3D command for ploting the graph [{t+1, t^3,t^2}] under the show command.

3.      Give the range  of the parameter 't' i.e. {t,-2,2}

4.      Decorate the graph using the box ratio, axis level, Graphics3D command.

Example:

Manipulate[Show[ParametricPlot3D[{t + 1, t^3, t^2}, {t, -2, 2}, BoxRatios →{1, 1, 1}, AxesLabel → {X - Axis, Y - Axis, Z - Axis}], Graphics3D[{PointSize[0.03], Red, Point[{t + 1, t^3, t^2}]} /. {t → parameter}]], {parameter, -2, 2}].

Q.No.28: The Implicit representation of the Viviani curve.

Construction Steps:

1.      Give input h=x^2 + y^2 + z^2 - 4 ; and g = (x - 1)^2 + y^2 - 1;

2.      Use CounterPlot3D command to plot {h == 0,   g == 0}.

3.      Give the rang for x,y and z i.e. {x, -2, 2}, {y, -2, 2}, {z, -2, 2}

4.      Decorate the curve by using BoxRatio, AxesLable and PlotRange command.

Example:

h = x^2 + y^2 + z^2 - 4 ;

g = (x - 1)^2 + y^2 - 1;

ContourPlot3D[{h== 0, g== 0}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},BoxRatios → {1, 1, 1}, AxesLabel →{X - Axis, Y - Axis, Z - Axis}, PlotRange → Full]

Q.No.29: Unit vector along tangent to a space curve.

Construction Steps:

1.      Inter the space curve r[t]= {t^3, t^3 + 7 t + 5 t^2, t^2} press Shift+ enter

2.      Inter r'[t] for finding the derivative of the space curve.

3.      Then give the command tangentvector = FullSimplify[r'[t], t ∈ Reals]

4.      Give magnitude = FullSimplify[Norm[r'[t]], t ∈ Reals] for finding the magnitude of the space curve.

5.      Give the command unittangentvector = tangentvector / magnitude for finding unit tangent vector.

Q.No.30. Plot a Helix (a Sin[u], a Cos[u], u/10) using ParametricPlot3D.And Manipulate the parameter a from 1 to 5.

Construction Steps:

1.      Use manipulate command for manipulating the helix.

2.      Use ParamatricPlot3D[{a u Sin[v],a u Cos[v],u v}] command for plotting the helix

3.      Give the range for 'u' and 'v' i.e. {u, 0, 10}, {v, 0, 10}.

4.      Give the range of the manipulation for a. i.e. {a, 1, 5}

Example:

Manipulate[ParametricPlot3D[{a u Sin[v], a u Cos[v], u v}, {u, 0, 10}, {v, 0, 10}].

Q.No.31: Limit of the function.

Construction Steps:

1.      Give the limit command for the function Limit[(x^2 - 2 x - 8) / (x - 4)].

2.      Give the limit point for x. i.e. x→4

3.      Use Direction → -1 command: Limit[(x^2 - 2 x - 8) / (x - 4), x → 4, Direction →1]

Example: 

Limit[(x^2 - 2 x - 8) / (x - 4), x → 4, Direction →-1]

Then, Limit[x^a, x → ∞, Assumptions → a < 0]

Q.No.32: Limit of a function in Plot.

Construction Steps:

1.      Use the Manipulate command for manipulation of the given function.

2.      Use Plot Plot[{a x Sin[1 / (b x)], Abs[a x], - Abs[a x]}] command

3.      Give the range for x. i.e. {x, -1, 1}

4.      Use PlotRange{-5, 5}

5.      Give the range of manipulation{{a, 1, "Amplitude"}, 1, 5}, {{b, 1, "Periodicity"}, 1, 5}

Example:

Manipulate[Plot[{a x Sin[1 / (b x)], Abs[a x], - Abs[a x]}, {x, -1, 1},PlotRange → {-5, 5}], {{a, 1, "Amplitude"}, 1, 5}, {{b, 1, "Periodicity"}, 1, 5}]

Q.No.33: Test of Left Hand and Right hand limit.

Construction Steps:

1.      Give command limit = Limit[Abs[x] / x, x → 0]

2.       Give the command limitright = Limit[Abs[x] / x, x → 0, Direction → -1] for finding right hand limit

3.      Similarly, limitleft = Limit[Abs[x] / x, x → 0, Direction →1] for finding left hand limit.

4.      Use the limit == limitright == limitleft command whether the limit is exist or not.

Q.No.34: Derivative of function by first principle.

Construction Steps:

1.      Input the function f[x_] := Tan[x]

2.      Give the command; Limit[(f[x + h] - f[x]) / h, h → 0, Analytic → True] press Shift+ Enter

3.      Example: f[x_] := Tan[x]

4.      Limit[(f[x + h] - f[x]) / h, h →0, Analytic → True]

Q.No.35: Application of Integration.

Construction Steps:

1.      Give the Manipulate command for manipulation of h

2.      Use [ContourPlot[{x^2 + y^2== 16, y==h}] under manipulation.

3.      Give the range of x and y respectively i.e.{x, -4.1, 4.1}, {y, 0, 4.1}

4.      Use AspectRatio→Automatic command

5.      Give the range of manipulation i.e. {h, 0, 4}

Example:  Manipulate[ContourPlot[{x^2 + y^2==16, y==h},

{x, -4.1, 4.1}, {y, 0, 4.1}, AspectRatio → Automatic], {h, 0, 4}]

6.      Use the following input

Input[1]; Solve[x^2 + y^2== 16, y]

f[x_] :=  ; 2 * Integrate[f[x], {x, 0, h}, Assumptions →0 < h < 4]

A[h_] := 2 [1/2h √(16-h^2) + 8 ArcSin[h/4]

A[4]

Plot[A[h], {h, 0, 4}]

FindRoot[A[h] == 2 Pi, {h, 1}]

Q.No.36: Formula Expansion.

Construction Steps:

1.      Use manipulation command

2.      Give the command Expand[(a + b)^n]

3.      Give the range of manipulating {{n, 1, "Power of (a+ b)"}, 1, 10, 1}

Example:

Manipulate[Expand[(a + b)^n], {{n, 1, "Power of (a+b)"}, 1, 10, 1}]

Input[1]; Solve[x^2 + y^2== 16, y]

f[x_] :=  ; 2 * Integrate[f[x], {x, 0, h}, Assumptions →0 < h < 4]

A[h_] := 2 [1/2h √(16-h^2) + 8 ArcSin[h/4]

A[4]

Plot[A[h], {h, 0, 4}]

FindRoot[A[h] == 2 Pi, {h, 1}]

Q.No.36: Formula Expansion.

Construction Steps:

1.      Use manipulation command

2.      Give the command Expand[(a + b)^n]

3.      Give the range of manipulating {{n, 1, "Power of (a+ b)"}, 1, 10, 1}

Example:

Manipulate[Expand[(a + b)^n], {{n, 1, "Power of (a+b)"}, 1, 10, 1}]

1