mathematica quickstart math101c - ohlone librarycosinecurve = plot@5 cos@td, 8t, 0, 4 p

Mathematica QuickStart for Calculus 101C

Algebra

Solving Equations

Exact Solutions to single equation:

In[88]:= Solve@x^3 + 5 x - 6 ã 0, xD

Out[88]= :8x Ø 1<, :x Ø1

2I-1 - Â 23 M>, :x Ø

1

2I-1 + Â 23 M>>

Exact Solution to a system of equations:

In[89]:= SolveA9x2 + y2 ã 4, x2 + 4 y2 ã 9=, 8x, y<E

Out[89]= ::x Ø -7

3, y Ø -

5

3>, :x Ø -

7

3, y Ø

5

3>,

:x Ø7

3, y Ø -

5

3>, :x Ø

7

3, y Ø

5

3>>

Approximate Solutions to a single equation:

In[90]:= NSolve@x^3 + 5 x - 6 ã 0, xD

Out[90]= 88x Ø -0.5 - 2.39792 Â<, 8x Ø -0.5 + 2.39792 Â<, 8x Ø 1.<<

Factoring

In[91]:= FactorAx2 y3 + 6 x y2 - 7 yE

Out[91]= y H-1 + x yL H7 + x yL

Expansions

In[92]:= ExpandAHx + 2 yL5E

Out[92]= x5 + 10 x4 y + 40 x3 y2 + 80 x2 y3 + 80 x y4 + 32 y5

Partial Fractions

In[93]:= ApartB4 x - 3

x2 - 7 x + 12F

Out[93]=13

-4 + x-

9

-3 + x

Functions

Functions in Mathematica use the notation f[x_] when declaring the function. Notice the underscore character after the variable.

In[94]:= f@x_D = x^2 - 5 x

Out[94]= -5 x + x2

When using the function, you do not need the underscore character.

In[95]:= fAx2E

Out[95]= -5 x2 + x4

In[96]:= f@x + hD - f@xD

h

Out[96]=5 x - x2 - 5 Hh + xL + Hh + xL2

h

In[97]:= ExpandBf@x + hD - f@xD

hF

Out[97]= -5 + h + 2 x

Calculus Operations

2 mathematica_quickstart_math101c.nb

Calculus OperationsMathematica has all the standard calculus operations, including left and right side limits.First create a function.

In[98]:= f@x_D =Sin@xD

x

Out[98]=Sin@xD

x

In[99]:= g@x_D = PiecewiseA99x2, x > 0=, 94 - x2, x < 0==E

Out[99]=

x2 x > 0

4 - x2 x < 00 True

Limits

In[100]:= Limit@f@xD, x Ø 0D

Out[100]= 1

In[101]:= Limit@g@xD, x Ø 0, Direction Ø 1D H*Note this means heading to the right.*L

Out[101]= 4

In[102]:= Limit@g@xD, x Ø 0, Direction Ø -1D H*Note this means heading to the right.*L

Out[102]= 0

Derivatives

The notation is simple for functions.

In[103]:= f'@xD

Out[103]=Cos@xD

x-Sin@xD

x2

mathematica_quickstart_math101c.nb 3

In[104]:= f'Bp

2F

Out[104]= -4

p2

Indefinite Integrals

In[105]:= IntegrateAx3, xE

Out[105]=x4

4

Definite Integrals

In[106]:= IntegrateAx3, 8x, 1, a<E

Out[106]= -1

4+a4

4

Series

To find ⁄k=1

5k2

In[107]:= SumAx2, 8x, 1, 5<E

Out[107]= 55

To find 5 terms of the Taylor series expansion of x7 at c = 1,

In[108]:= SeriesAx7, 8x, 1, 5<E

Out[108]= 1 + 7 Hx - 1L + 21 Hx - 1L2 + 35 Hx - 1L3 + 35 Hx - 1L4 + 21 Hx - 1L5 + O@x - 1D6

To eliminate the “O” notation at the end, use Normal at the end of the command.

In[109]:= SeriesAx7, 8x, 1, 5<E êê Normal

Out[109]= 1 + 7 H-1 + xL + 21 H-1 + xL2 + 35 H-1 + xL3 + 35 H-1 + xL4 + 21 H-1 + xL5

Simple Graphing


Simple Graphing

Graphing functions

Standard functions are graphed with the Plot function.

In[110]:= Plot@3 Sin@tD, 8t, 0, 4 p<D

Out[110]=2 4 6 8 10 12

-3

-2

-1

1

2

3

To change the style, use the PlotStyle option.

In[111]:= Plot@3 Sin@tD, 8t, 0, 4 p<,PlotStyle Ø 8Red, [email protected]<D

Out[111]=2 4 6 8 10 12

-3

-2

-1

1

2

3

To combine plots, create named plots separately and use the Show command to combine them.


In[112]:= SineCurve = Plot@3 Sin@tD, 8t, 0, 4 p<, PlotStyle Ø 8Red, [email protected]<D;CosineCurve = Plot@5 Cos@tD, 8t, 0, 4 p<, PlotStyle Ø 8Blue, [email protected]<D;Show@SineCurve, CosineCurve, PlotRange -> AllD

Out[114]=2 4 6 8 10 12

-4

-2

2

4

Fancy Graphing

Polar Curves

Polar curves can be graphed using the PolarPlot command. For example, to graph r = 3cos 2q, use


In[115]:= PolarPlot@3 Cos@3 thetaD, 8theta, 0, 2 p<D

Out[115]=-1 1 2 3

-2

-1

1

2

Parametric Curves

To graph a set of parametric functions x(t) = cos 3t and y(t) = sin 2t, use the ParametricPlot com-mand.


In[116]:= ParametricPlot@8Cos@3 tD, Sin@2 tD<, 8t, 0, 2 p<D

Out[116]=-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

In three dimensions, use ParametricPlot3D


In[117]:= ParametricPlot3D@8Cos@tD, Sin@tD, Sin@4 tD<, 8t, 0, 2 p<D

Out[117]=

-1.0

-0.5

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

Surfaces

To plot the surface f(x, y) = x3 + y3 + 4 sin HxyL, use the Plot3D command.


In[118]:= Plot3DAx3 + y3 + 4 Sin@x yD, 8x, -2, 2<, 8y, -2, 2<E

Out[118]=

Implicit surfaces such as x2 + y2 + z2 = 4 are graphed using the ContourPlot3D command.

In[119]:= ContourPlot3DAx2 + y2 + z2 ã 4, 8x, -2, 2<, 8y, -2, 2<, 8z, -2, 2<E

Out[119]=

Parametric surfaces are graphed using the ParametricPlot3D command. For example, given


x Hu, vL = cos u cos vy Hu, vL = cos u sin vz Hu, vL = u

In[120]:= ParametricPlot3D@8Cos@uD Cos@vD, Cos@uD Sin@vD, u<, 8u, 0, 3 p<, 8v, 0, 2 p<D

Out[120]=

To add nice level cures to any of the above, use the MeshFunctions option.


In[121]:= Plot3DAx2 + y2, 8x, -2, 2<, 8y, -2, 2<,

MeshFunctions Ø 8Function@8x, y, z<, zD<E

Out[121]=

To add nice coloring to any of the above, use the ColorFunction option.

In[122]:= Plot3DAx2 + y2, 8x, -2, 2<, 8y, -2, 2<,MeshFunctions Ø 8Function@8x, y, z<, zD<,ColorFunction Ø Function@8x, y, z<, Hue@zDDE

Out[122]=

To eliminate the corners from sticking up, use the RegionFunction option.


In[123]:= Plot3DAx2 + y2, 8x, -2, 2<, 8y, -2, 2<,MeshFunctions Ø 8Function@8x, y, z<, zD<,ColorFunction Ø Function@8x, y, z<, Hue@zDD,RegionFunction Ø Function@8x, y, z<, x^2 + y^2 < 4DE

Out[123]=


mathematica quickstart math101c - ohlone librarycosinecurve = plot@5 cos@td, 8t, 0, 4 p

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