Calculus II   
Mathematica Project 2
Due Date: November 15, 2012

1.	Find  the following limits
a)	 (lim_(x->∞))((x^2)/(e^x))				b)	(lim_(x->0))((arcsin x)/x)	 c)	(lim_(x->∞))(x^((1/x)))

2.	Integrate 
a) 	(∫_1^∞) (1/(arctan x))dx			b)	(∫_0^2)((4-x)/(Sqrt[4-(x^2)]))dx 	c)	∫(((Cos[(x/2)]))^4)dx

3.	The Gamma function is defined by Γ(x)=(∫_0^∞)(t^(x-1))(e^(-t)) d t. This function has many applications in several branches of mathematics.  Mathematica's in-built gamma function is denoted by Gamma[x]. )

a)	Use Mathematica to evaluate the integral of (t^(x-1))(e^(-t))  on the interval [0,∞) (Hint: Evaluate h[x_, t_]:=(t^(x-1))(e^(-t))  and G[x_]:=Integrate[h[x,t],{t,0,Infinity}].
b)	Find G(10) and compare it with 9!
c)	Find G((1/2))  and comapre it with Gamma[1/2].

4.	The following Mathematica program can be used to verify the convergence or divergence of an improper integral. (∫_a^∞)f(x) d x  where a and f(x)  are given. For example let a=0 and f(x)= (e^(-x))

Clear[ x, f, g ]
f[ x_ ] :=(E^(-x))
g[b_]: =Integrate [f[x], {x,0,b}]
Limit[g[x], x ->∞] 
Plot[  g[x] , { x, 0 , 70 } ];   

The last command plots the graph of the integral over the interval [0,70] and shows graphically why the given improper integral coverges or diverges.

a)	Modify the above program  for the function f(x)=(sin^3) x  
b)	Explain  Mathematica's output of the limit.
c)	What does the graph tell us about the convergence of the improper integral? Expalin.
d)	Evaluate N[ g [ {10,100,1000} ] , 10]
e)	What, if any, do the values in (d) suggest about the  limit?
f)	Redo (a) to (e) for the function f(x)=(1/((x^3)+2x+1))