Algebra. Mathematica QuickStart for Calculus 101C. Solving Equations. Factoring. Exact Solutions to single equation:

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1 Mathematica QuickStart for Calculus 101C Algebra Solving Equations Exact Solutions to single equation: In[88]:= + 5 x - 6 ã 0, xd Out[88]= :8x Ø 1<, :x Ø 1 2 I-1 - Â 23 M>, :x Ø 1 2 I-1 + Â 23 M>> Exact Solution to a system of equations: In[89]:= SolveA9x 2 + y 2 ã 4, x y 2 ã 9=, 8x, y<e Out[89]= ::x Ø - 7 3, y Ø >, :x Ø - 7 3, y Ø 5 3 >, :x Ø 7 3, y Ø >, :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 Â<, 8x Ø Â<, 8x Ø 1.<< Factoring In[91]:= FactorAx 2 y x y 2-7 ye Out[91]= y H-1 + x yl H7 + x yl

2 2 mathematica_quickstart_math101c.nb Expansions In[92]:= ExpandAHx + 2 yl 5 E Out[92]= x x 4 y + 40 x 3 y x 2 y x y y 5 Partial Fractions In[93]:= ApartB 4 x - 3 F x 2-7 x + 12 Out[93]= x 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 + x 2 When using the function, you do not need the underscore character. In[95]:= fax 2 E Out[95]= -5 x 2 + x 4 In[96]:= f@x + hd - f@xd h Out[96]= 5 x - x 2-5 Hh + xl + Hh + xl 2 h In[97]:= f@x + hd - f@xd ExpandB F h Out[97]= -5 + h + 2 x

3 mathematica_quickstart_math101c.nb 3 Calculus Operations Mathematica 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 = PiecewiseA99x 2, x > 0=, 94 - x 2, x < 0==E Out[99]= x 2 x > x 2 x < 0 0 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 x 2

4 4 mathematica_quickstart_math101c.nb In[104]:= f'b p 2 F Out[104]= - 4 p 2 Indefinite Integrals In[105]:= IntegrateAx 3, xe Out[105]= x 4 4 Definite Integrals In[106]:= IntegrateAx 3, 8x, 1, a<e Out[106]= a4 4 Series 5 To find k 2 k=1 In[107]:= SumAx 2, 8x, 1, 5<E Out[107]= 55 To find 5 terms of the Taylor series expansion of x 7 at c = 1, In[108]:= SeriesAx 7, 8x, 1, 5<E Out[108]= Hx - 1L + 21 Hx - 1L Hx - 1L Hx - 1L Hx - 1L 5 + O@x - 1D 6 To eliminate the O notation at the end, use Normal at the end of the command. In[109]:= SeriesAx 7, 8x, 1, 5<E êê Normal Out[109]= H-1 + xl + 21 H-1 + xl H-1 + xl H-1 + xl H-1 + xl 5

5 mathematica_quickstart_math101c.nb 5 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]= To change the style, use the PlotStyle option. In[111]:= Plot@3 Sin@tD, 8t, 0, 4 p<, PlotStyle Ø 8Red, Thickness@0.01D<D Out[111]= To combine plots, create named plots separately and use the Show command to combine them.

6 6 mathematica_quickstart_math101c.nb In[112]:= SineCurve = Plot@3 Sin@tD, 8t, 0, 4 p<, PlotStyle Ø 8Red, Thickness@0.01D<D; CosineCurve = Plot@5 Cos@tD, 8t, 0, 4 p<, PlotStyle Ø 8Blue, Thickness@0.01D<D; Show@SineCurve, CosineCurve, PlotRange -> AllD 4 2 Out[114]= Fancy Graphing Polar Curves Polar curves can be graphed using the PolarPlot command. For example, to graph r = 3cos 2q, use

7 mathematica_quickstart_math101c.nb 7 In[115]:= PolarPlot@3 Cos@3 thetad, 8theta, 0, 2 p<d 2 1 Out[115]= Parametric Curves To graph a set of parametric functions x(t) = cos 3t and y(t) = sin 2t, use the ParametricPlot command.

8 8 mathematica_quickstart_math101c.nb In[116]:= td, td<, 8t, 0, 2 p<d Out[116]= In three dimensions, use ParametricPlot3D

9 mathematica_quickstart_math101c.nb 9 In[117]:= ParametricPlot3D@8Cos@tD, Sin@tD, Sin@4 td<, 8t, 0, 2 p<d Out[117]= Surfaces To plot the surface f(x, y) = x 3 + y sin HxyL, use the Plot3D command.

10 10 mathematica_quickstart_math101c.nb In[118]:= Plot3DAx3 + y3 + 4 Sin@x yd, 8x, - 2, 2<, 8y, - 2, 2<E Out[118]= Implicit surfaces such as x 2 + y 2 + 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

11 mathematica_quickstart_math101c.nb 11 x Hu, vl = cos u cos v y Hu, vl = cos u sin v z 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.

12 12 mathematica_quickstart_math101c.nb 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.

13 mathematica_quickstart_math101c.nb 13 In[123]:= Plot3DAx 2 + y 2, 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]=