Math 225: Calculus III     Solution 1

Initialization

<<Calculus`Math225`

Problem 1

Find the value of  (2*Pi + Sqrt[3])^(5/2) to 20 significant digits. (Pi is a constant built into Mathematica  that represents the exact mathematical value of π. Other important constants to be aware of like this are E == 2.71828... for the base of the natural logarithms and I == Sqrt[-1]). Click in the line BELOW the word "Solution" to enter the expression. Then hit ENTER (not RETURN).

Solution

N[(2*Pi + Sqrt[3])^(5/2),20]

Problem 2

Find the first and second derivatives of the function Sin[Log[Sqrt[1-x^2]]]
using the function D[] and using the prime notation f'[x], f''[x] as above.

Solution

D[Sin[Log[Sqrt[1-x^2]]], x]

D[%,x]

f[x_] = Sin[Log[Sqrt[1-x^2]]]

f'[x]

f''[x]

Problem 3

Find an anti-derivative of 2/(1-x^2). Integrate 2/(1-x^2) from 2 to 3. Find an approximation to the integral of 1/(2 + E^Sqrt[x]) from 0 to 1.

Solution

Integrate[2/(1-x^2), x]

Integrate[2/(1-x^2), {x,2,3}]

NIntegrate[1/(2 + E^Sqrt[x]), {x,0,1}]

Problem 4

Solve the system of equations x^2 y - x y + x y^2 == 0,
x^2 + y - y^2 == 10.

Solution

Solve[{x^2 y - x y + x y^2 == 0,x^2 + y - y^2 == 10}]

N[%]

NSolve[{x^2 y - x y + x y^2 == 0,x^2 + y - y^2 == 10}]

The real soutions are x == 10, y == -9 and x == Sqrt[10], y == 0, and
x == -Sqrt[10], y == 0

Problem 5

Plot the graph of the function Cos[x]/(9+x^2).
Plot the graph of the function  Sin[2*x] - Cos[y/2].
Plot the graph of the curve {3*Cos[t], 2*Sin[t]}.
Plot the graph of the curve {t, Cos[2*Pi*t], Sin[2*Pi*t]}.
Plot the graph of the curve {t, Cos[2*Pi*t], Sin[3*Pi*t]}.

Solution

Plot[Cos[x]/(9+x^2),{x,-4Pi,4Pi}]

Plot3D[Sin[2*x] - Cos[y/2],{x,-4Pi,4Pi},{y,-4Pi,4Pi}, PlotPoints->50]

ParametricPlot[{3*Cos[t],2*Sin[t]},{t,0,2Pi}]

ParametricPlot3D[{t, Cos[2*Pi*t], Sin[2*Pi*t]},{t,0,4}]

ParametricPlot3D[{t, Cos[2*Pi*t], Sin[3*Pi*t]},{t,0,4}]