Below are the commands we had Mathematica execute in class on Friday.  I've deleted all of the output to keep the file size small.  I've also edited a few things.  In particular, I named the characteristic polynomial as charpoly so I could refer to it by name in later commands rather than by the output number.  This is preferable since the output numbers will change each time we run the commands in this file.

2 + 2

Plot[Sin[x], {x, 0, 2  Pi}]

Plot[{BesselJ[0, x], Sin[x]}, {x, 0, 
        50}, 
PlotStyle -> {RGBColor[1, 0, 0], 
          RGBColor[0, 1, 0]}]

Integrate[Sin[x], {x, 0, Pi}]

∫Sin[x] \[DifferentialD]x

(∫_0^π) Sin[x] \[DifferentialD]x

a = {{1, -1}, {1, 3}}

MatrixForm[a]

(({{1, -1}, {1, 3}}))

a

Eigenvalues[a]

Eigenvectors[a]

b=(({{3, 5}, {1, 2}}))

Eigenvalues[b]

Eigenvectors[b]

Eigensystem[b]

a - λ IdentityMatrix[2]

MatrixForm[%]

charpoly = Det[a - λ IdentityMatrix[2]]

Factor[charpoly]

Solve[charpoly == 0, λ]

RowReduce[a - 2  IdentityMatrix[2]]

MatrixForm[%]

a

a*a

a . a

a . b

b . a

Inverse[a]

DSolve[{y''[t] + t y'[t] + y[t] == 0}, y[t], 
      t]

NDSolve[{y''[t] + t y'[t] + y[t] == 0, 
        y[0] == 1, y'[0] == 0}, y[t], {t, 0, 2}]

Plot[y[t] /. %, {t, 0, 2}]

(∑_(n = 1)^∞) (1/(n^2))

(∑_(n = 1)^∞) c[n] (t^n)

D[%, t]

Table[n^2, {n, 1, 4}]

Table[
Plot[Sin[x - n], {x, -10, 10}], {n, 0, 
        10, 0.2}]