this tutorial is a brief slide-show introduction to

« ≈ 1 of 23

This tutorial is a brief slide-show introduction to elementary mathematics by Mathemat-ica. It covers the following subjects:

Basic Operations; Solving Equations; Sequences & Series

Basic Plotting; Curves and Surfaces

May 26, 2010

Hsin-Yun Hu, Department of Mathematics, Tunghai University

東海大學數學系胡馨云

Printed from the Mathematica Help Browser 1

©1988-2002 Wolfram Research, Inc. All rights reserved.

« ≈ 2 of 23

Basic OperationsDo arithmetic

In[1]:= 2.3 + 5.36 ∗ 2 − 8.9^2 + 4.5ê1.2

Out[1]= −62.44

Handle numbers of any size

2 Printed from the Mathematica Help Browser


In[2]:= 100!

N@Pi, 100D

Out[2]= 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

Out[3]= 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

Find the prime factors of an integer

In[4]:= FactorInteger@45DFactorInteger@242D

Out[4]= 883, 2<, 85, 1<<

Out[5]= 882, 1<, 811, 2<<

Find the greatest common divisor & least common multiple

In[6]:= GCD@12, 16DLCM@12, 16D

Out[6]= 4

Out[7]= 48

« ≈ 3 of 23

Basic OperationsFactor the polynomials



In[8]:= Factor@x2 − y2DFactor@x9 − y9D

Out[8]= Hx − yL Hx + yL

Out[9]= Hx − yL Hx2 + x y + y2L Hx6 + x3 y3 + y6L

Multiply out products and power

In[10]:= Expand@H1 + xL5DExpandAHx2 − y2 + 10L3E

Out[10]= 1 + 5 x + 10 x2 + 10 x3 + 5 x4 + x5

Out[11]= 1000 + 300 x2 + 30 x4 + x6 − 300 y2 − 60 x2 y2 − 3 x4 y2 + 30 y4 + 3 x2 y4 − y6

Simplify factors the polynomials

In[12]:= Simplify@x2 + 2 x + 1DSimplify@−108 + 108 x + 45 x2 − 40 x3 − 10 x4 + 4 x5 + x6D

Out[12]= H1 + xL2

Out[13]= H−2 + xL2 H−1 + xL H3 + xL3

« ≈ 4 of 23

Basic OperationsPut all terms over a common denominator

In[14]:= TogetherA 5

x − 3−

2

x + 1E

Out[14]=11 + 3 x

H−3 + xL H1 + xL



Separate into terms with simple denominators

In[15]:= ApartA x3 − 3 x + 2

Hx + 1L Hx − 3L3E

Out[15]=5

H−3 + xL3 +19

4 H−3 + xL2 +17

16 H−3 + xL −1

16 H1 + xL

Use some mathematical functions

In[16]:= a = 10.0;Sqrt@aDExp@aDLog@aDSin@aDClear@aDb = 1;4 ∗ ArcTan@bDClear@bD

Out[17]= 3.16228

Out[18]= 22026.5

Out[19]= 2.30259

Out[20]= −0.544021

Out[23]= π

« ≈ 5 of 23

Advance OperationsLimit



In[27]:= Clear@nDLimit@xn, x → 2D

Out[28]= 2n

Derivatives

In[29]:= Clear@nDD@xn, xD

Out[30]= n x−1+n

Anti-derivatives

In[31]:= Clear@nD

‡ xn x

Out[32]=x1+n

1 + n

« ≈ 6 of 23

Solving EquationsSolve a polynomial equation in one variable

In[33]:= Solve@a x2 + b x + c == 0, xD

Out[33]= 99x →−b −

è!!!!!!!!!!!!!!!!!!!!b2 − 4 a c2 a

=, 9x →−b +

è!!!!!!!!!!!!!!!!!!!!b2 − 4 a c2 a

==

In[34]:= Solve@x2 + 5 x + 3 == 0, xD

Out[34]= 99x →12I−5 −è!!!!!!13 M=, 9x →

12I−5 +è!!!!!!13 M==



In[35]:= Solve@x2 + x + 3 == 0, xD

Out[35]= 99x →12I−1 − è!!!!!!11 M=, 9x →

12I−1 + è!!!!!!11 M==

Solve a simple equation of higher degree

In[36]:= Solve@3 x3 + 2 x2 − 2 x − 1 0, xD

Out[36]= 98x → −1<, 9x →16I1 −è!!!!!!13 M=, 9x →

16I1 +è!!!!!!13 M==

In[37]:= Solve@x6 1, xD

Out[37]= 88x → −1<, 8x → 1<, 8x → −H−1L1ê3<, 8x → H−1L1ê3<, 8x → −H−1L2ê3<, 8x → H−1L2ê3<<

« ≈ 7 of 23

Solving EquationsSolve two simultaneous algebraic equations

In[38]:= sys1 = 8y 1 + 2 x, x 5 − y<;Solve@sys1, 8x, y<D

Out[39]= 99x →43

, y →113

==

In[40]:= sys2 = 9 1 + x

−1 + y== 0, 2 − 3 y + y2 == 0=;

Solve@sys2, 8x, y<D

Out[41]= 88x → −1, y → 2<<

Solve an equations involving exponentials, logarithms and trigonometric functions



In[44]:= eqn1 = x−2 8;Solve@eqn1, xD

Out[45]= 88x → 2 + Log@8D<<

In[46]:= eqn2 = Log@x + 2D 5;Solve@eqn2, xD

Out[47]= 88x → −2 + 5<<

In[48]:= eqn3 = Sin@xD 1ê2;Solve@eqn3, xD

Out[49]= 99x →π6==

« ≈ 8 of 23

Sequences & SeriesList the terms of a sequence

In[50]:= Table@i2, 8i, 10<DTable@j3, 8j, −10, 10<D

TableA 1

k, 8k, 10<E

TableA H−1Lk−1

k2, 8k, 10<E

Out[50]= 81, 4, 9, 16, 25, 36, 49, 64, 81, 100<

Out[51]= 8−1000, −729, −512, −343, −216, −125, −64, −27, −8, −1, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000<

Out[52]= 91,12

,13

,14

,15

,16

,17

,18

,19

,1

10=

Out[53]= 91, −14

, 19

, −1

16, 1

25, −

136

, 149

, −1

64, 1

81, −

1100

=



The telescoping sequence

In[54]:= TableA 1

n−

1

n + 1, 8n, 1, 10<E

SumA 1

n−

1

n + 1, 8n, 1, 10<E

Out[54]= 9 12

,16

,1

12,

120

,1

30,

142

,1

56,

172

,1

90,

1110

=

Out[55]=1011

« ≈ 9 of 23

Sequences & SeriesThe arithmetic sequence

In[56]:= a = 2; d = 2;Table@a + Hn − 1L ∗ d, 8n, 1, 10<DSum@a + Hn − 1L∗ d, 8n, 1, 10<D

Out[57]= 82, 4, 6, 8, 10, 12, 14, 16, 18, 20<

Out[58]= 110

by formula

In[59]:= n = 10; dsum =Ha + a + Hn − 1L∗ dL

2∗ n

Out[59]= 110

The geometric sequence



In[60]:= b = 1; r = 1 ê3;Table@b ∗ rn−1, 8n, 1, 10<DSum@b ∗ rn−1, 8n, 1, 10<D

Out[61]= 91, 13

, 19

, 127

, 181

, 1243

, 1729

, 12187

, 16561

, 119683

=

Out[62]=2952419683

by formula

In[63]:= n = 10; rsum =b ∗H1 − rnL

1 − r

Out[63]=2952419683

« ≈ 10 of 23

Basic PlottingThe two-dimensional functions plots



In[64]:= Plot@x2 − 2, 8x, −4, 4<, Frame → TrueD;

-4 -2 0 2 4

0

2.5

5

7.5

10

12.5

In[65]:= Plot@Exp@xD, 8x, −4, 4<, Frame → TrueD;

-4 -2 0 2 40

5

10

15

20

25

30

35



In[66]:= Plot@Log@xD, 8x, 0, 4<, Frame → TrueD;

0 1 2 3 4-10

-8

-6

-4

-2

0

« ≈ 11 of 23

Basic PlottingCombining several plots



In[67]:= Plot@82x, 3x, 5x<, 8x, −5, 5<, Frame → True, PlotStyle → 8RGBColor@1, 0, 0D, RGBColor@0, 1, 0D, RGBColor@0, 0, 1D<D;

-4 -2 0 2 40

20

40

60

80

100

120

In[68]:= g1 = Plot@Sin@xD, 8x, −2 π, 4 π<, PlotRange → 8−2, 2<, Frame → True, PlotStyle → RGBColor@1, 0, 1DD;g2 = Plot@Cos@xD, 8x, −2 π, 4 π<, PlotRange → 8−2, 2<, Frame → True, PlotStyle → RGBColor@0, 1, 1DD;Show@g1, g2D;



-5 -2.5 0 2.5 5 7.5 10 12.5

-1.5

-1

-0.5

0

0.5

1

1.5

2

-5 -2.5 0 2.5 5 7.5 10 12.5

-1.5

-1

-0.5

0

0.5

1

1.5

2



-5 -2.5 0 2.5 5 7.5 10 12.5

-1.5

-1

-0.5

0

0.5

1

1.5

2

« ≈ 12 of 23

Basic PlottingThe three-dimensional surfaces plots



In[71]:= g1 = Plot3D@Sin@x yD, 8x, 0, 3<, 8y, 0, 3<D;

0

1

2

3 0

1

2

3

-1-0.5

00.5

1

0

1

2



In[72]:= g2 = Plot3D@Exp@−Hx2 + y2LD, 8x, −3, 3<, 8y, −3, 3<, PlotRange → AllD;

-2

0

2-2

0

2

00.250.5

0.751

-2

0

2

« ≈ 13 of 23

Basic PlottingConti.



In[73]:= Show@g1, Mesh → FalseD;

0

1

2

3 0

1

2

3

-1-0.5

00.5

1

0

1

2



In[74]:= Show@g2, Shading → FalseD;

-2

0

2-2

0

2

00.250.5

0.751

-2

0

2

« ≈ 14 of 23

Basic PlottingConti.



In[75]:= Show@ContourGraphics@g1DD;

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

In[76]:= Show@g2, ViewPoint → 8−1, −3, 1<D;

-20

2

-20

2

0

0.25

0.5

0.75

1

02

-20



In[77]:= Show@g2, Boxed → FalseD;

-2

0

2

-2

0

2

00.250.5

0.75

1

-2

0

2

« ≈ 15 of 23

Curves In PlaneTeardrop



In[78]:= a = 1.5; b = 3;ParametricPlot@8a ∗ Cos@tD − aê2 ∗ Sin@2 ∗ tD, b ∗ Sin@tD<, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-2 -1 1 2

-3

-2

-1

1

2

3

Ellipse

In[80]:= a = 4; b = 2;ParametricPlot@8a ∗ Cos@tD, b ∗ Sin@tD<, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-4 -2 2 4

-2

-1

1

2



Epicycloid

In[82]:= a = 32; b = 4;

ParametricPlotA9Ha + bL ∗ Cos@tD − b ∗ CosA a + b

b∗ tE, Ha + bL∗ Sin@tD − b ∗ SinA a + b

b∗ tE=, 8t, 0, 2 ∗ Pi<, AspectRatio → AutomaticE;

-30 -20 -10 10 20 30

-30

-20

-10

10

20

30

« ≈ 16 of 23

Involute of a circle



In[84]:= ParametricPlot@8Sin@θD − θ ∗ Cos@θD, Cos@θD + θ ∗ Sin@θD<, 8θ, 0, 6 ∗ Pi<, AspectRatio → AutomaticD;

-15 -10 -5 5 10 15

-15

-10

-5

5

10

Rose curve (odd petal)



In[85]:= r = 2 ∗ Cos@5 ∗ θD;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, Pi<, AspectRatio → AutomaticD;

-1.5 -1 -0.5 0.5 1 1.5 2

-2

-1

1

2

Rose curve (even petal)



In[87]:= r = 2 ∗ Cos@2 ∗ θD;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-2 -1 1 2

-2

-1

1

2

« ≈ 17 of 23

Cardioid / Heart-shaped



In[89]:= r = 2 ∗H1 − Cos@θDL;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-4 -3 -2 -1

-2

-1

1

2

Star-shaped



In[91]:= r = 50 ∗ H1 + 0.2 ∗ Cos@6 ∗ θDL;ParametricPlot@8r ∗ Cos@θD, r ∗ Sin@θD<, 8θ, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-60 -40 -20 20 40 60

-40

-20

20

40

« ≈ 18 of 23

Curves In SpaceHelix



In[93]:= ParametricPlot3D@83 ∗ Cos@3 tD, 3 ∗ Sin@3 tD, t<, 8t, 0, 4 ∗ Pi<, AspectRatio → AutomaticD;

-20

2

-20

2

0

5

10

-20

2

Perodic curve



In[94]:= ParametricPlot3D@82 ∗ t, Sin@5 ∗ tD + Sin@tD, t<, 8t, 0, 4 ∗ Pi<, AspectRatio → AutomaticD;

0

10

20

-2-10120

5

10

0

5

« ≈ 19 of 23

Surfaces In Space Cylindrical surface



In[95]:= ParametricPlot3D@8Cos@uD, Sin@uD, v<, 8u, 0, 2 ∗ Pi<, 8v, 0, 1<, AspectRatio → AutomaticD;

-1-0.5

00.5

1 -1

-0.5

0

0.5

1

00.250.5

0.751

-1-0.5

00.5

Riemann surface



In[96]:= ParametricPlot3D@8u ∗ Cos@vD, u ∗ Sin@vD, 2 ∗ v<, 8u, 0, 5<, 8v, 0, 4 ∗ Pi<, BoxRatios → 81.5, 1.5, 2<D;

-5 -2.5 0 2.55

-5-2.5

02.5

5

0

10

20

5-2.5

02.5

Sphere



In[97]:= ρ = 1;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 12 ∗ Piê6<, AspectRatio → AutomaticD;

-1-0.5

00.5

1

-1

-0.50

0.51

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-0.50

0.5

« ≈ 20 of 23

Ellipsoid



In[99]:= a = 1; b = 1; c = 1.5;ParametricPlot3D@8a ∗ Cos@vD∗ Cos@uD, b ∗ Cos@vD∗ Sin@uD, c ∗ Sin@vD<, 8u, 0, 2 ∗ Pi<,8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-1 -0.5 0 0.51

-1-0.5

00.5

1

-1

0

1

1-0.5

00.5

Paraboloid



In[101]:=

ParametricPlot3D@82 ∗ u ∗ Cos@vD, 2 ∗ u ∗ Sin@vD, u2<, 8u, 0, 4<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-50

5

-5

0

5

0

5

10

15

-50

5

-5

0

5

« ≈ 21 of 23

Tours



In[102]:=

a = 2;ParametricPlot3D@8H4 + a ∗ Cos@vDL∗ Cos@uD, H4 + a ∗ Cos@vDL∗ Sin@uD, a ∗ Sin@vD<, 8u, 0, 2 ∗ Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-5

0

5-5

0

5

-2-1012

-5

0

5

Wrinkled sphere



In[104]:=

ρ = 1 + 0.2 ∗ Sin@uD∗ Sin@5 ∗ vD;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

« ≈ 22 of 23

Bumpy sphere



In[106]:=

ρ = 1 + 0.2 ∗ Sin@8 ∗ uD ∗ Sin@8 ∗ vD;ParametricPlot3D@8ρ ∗ Sin@uD∗ Cos@vD, ρ ∗ Sin@uD∗ Sin@vD, ρ ∗ Cos@uD<, 8u, 0, Pi<, 8v, 0, 2 ∗ Pi<, AspectRatio → AutomaticD;

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

Mobius strip



In[108]:=

ParametricPlot3D@8H4 − v ∗ Sin@uDL∗ Cos@2 ∗ uD, H4 − v ∗ Sin@uDL∗ Sin@2 ∗ uD, v ∗ Cos@uD<,8u, 0, Pi<, 8v, −2, 2<, ViewPoint → 81.2, −2.5, 1.2<, AspectRatio → AutomaticD;

-5-2.5

02.5

-5-2.5

02.5

5

-2

-1

0

1

2

-5-2.5

02 5

-5-2.5

02.5

« ≈ 23 of 23

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