A Little Calculus with Mathematica

Reminders

• Press Shift-Enter to evaluate each command.
• Mathematica is case-sensitive; upper and lower case letters matter.
• If you want to make changes or add options, you can edit previous input and re-evaluate it.
• To get the arrow character ->type a hyphen - followed by the greater than sign >.
• Your cursor is the horizontal bar between cells.

Limits

Limit[(x-3)/(x^2-9),x->4]

Limit[(x-3)/(x^2-9),x->3]

Limit[(4x^3-5)/(6x^3+2x^2-8),x->Infinity]

HoldForm[Limit[(4x^3-5)/(6x^3+2x^2-8),x->Infinity]]

Limit[1/x,x->0]

Limit[E^x,x->Infinity]

Derivatives

D[3x^4+2x+4,x]

TraditionalForm[D[3x^4+2x+4,x]]

HoldForm[D[3x^4+2x+4,x]]

f[x_]:=x^3Cos[x^2]

f[3]

Plot[f[x],{x,-2Pi,2Pi}]

D[f[x],x]

D[f[x],x]/.x->3

D[f[x],x]/.x->3//N

Plot[Evaluate[D[f[x],x]],{x,-2Pi,2Pi}]

Plot[{f[x],Evaluate[D[f[x],x]]},{x,Pi/2,5Pi/4},PlotRange->All]

Plot[{f[x],Evaluate[D[f[x],x]]},{x,Pi/2,5Pi/4},PlotRange->All,PlotStyle->{Blue,Red}]

g[x_]:=x^3+2x^2-8x

D[g[x],x]

D[D[g[x],x],x]

D[g[x],{x,2}]

D[g[x],{x,3}]

Try this one below and hover your mouse over each curve if you use Tooltip.

Plot[Tooltip@{g[x],Evaluate@D[g[x],x],Evaluate@D[g[x],{x,2}]},{x,-5,5}]

Integrals

Integrate[x^3+2x^2-8x,x]

Integrate[x^3+2x^2-8x,{x,-4,0}]

NIntegrate[x^3+2x^2-8x,{x,-4,0}]

HoldForm[Integrate[x^3+2x^2-8x,{x,-4,0}]]

Plot[x^3+2x^2-8x,{x,-5,5}]

Plot[x^3+2x^2-8x,{x,-5,5},Filling->Axis]

Show[
Plot[x^3+2x^2-8x,{x,-5,5}],
Plot[x^3+2x^2-8x,{x,-4,0},Filling->Axis]
]

Row[{HoldForm[Integrate[x^3+2x^2-8x,{x,-4,0}]]," = ",Integrate[x^3+2x^2-8x,{x,-4,0}]}]

Abby Brown
www.abbymath.com
September 2020