Calculus II   
Mathematica Project 1
Due Date: October 18, 2012

1.	Plot the graphs of the following functions using at least one plot option discussed in this section.

	(a)	  f(x)=(x^4)+2(x^3)+1	 for   -3≤x≤3		(b)	f(x)= x ln x      for   0≤x≤4

2.	Plot the graphs of the following  pairs of functions on the same axes. Use a plot style to identify the graphs.

	(a)	f(x)= ((2x)/(x-5)) and g(x)=((x-5)/(2x))   

	(b)	f(x)=   (x^2)-sin x   and   g(x)=(Sqrt[(x^4)+1])- (Sqrt[(x^2)+1])

3.	Evaluate each of the following limits:

	(a) 	({{lim        (((2x-1)/(4-3x)))}, {x→2                     }})  			(b)	({{lim            (((1-ln x)/(e^((1/x) ))) )      }, {x→(SuperPlus[0])                           }})
	
	(c)	({{lim             ((1/x)-ln x) }, {x→(SuperPlus[0])                              }}) 			(d)	({{lim           ((( sin  x))^(1/x))}, {x→(SuperPlus[0])                       }})

4	 Differentiate the following
		(a)	y=((3x-5)^(1/4))			(b)	y=(1/(Sqrt[cos ((x^2)) ])) 		
	
		(c)	y=(e^(x ln x))			(d)	y=ln(cos((e^x)))	
	
		(e)	y=(e^x)arcsec x			(f)	y=(sec^(-1))x+(csc^(-1))x

5.	  Let f(x)=(e^(sin x)) for -2π≤x≤2π 
		(a)	Define f in Mathematica 
		(b)	Find the solutions of f ' (x)=0. (Hint: Use FindRoot )
		(c)	Identify the relative extrema of f. 
		(d)	Find the inflection points of f.
		(e)	Verify your answers in (b), (c), and (d) by drawing the graph of f.

6.  	Integrate each of the following. Simplify your answers.

	(a)	∫(((x^5)+3(x^4)-2(x^2)+1)/(x^3))d x			(b)	∫(( 1)/(1+ sin x))d x		
	
	(c)	 ∫(sec^2)5 x (tan^3)5x  d x			(d)	∫(e^(3x))cos 2x  d x

7.	 Evaluate the following integrals.
	
	(a)	 ((∫_0)^3)((x^3)-4(x^2)+x)d x			(b)	((∫_1)^4)((( 1)/(Sqrt[x]))+2(Sqrt[x]))d x		
	
	(c)	 ((∫_0)^(π/4)) sec x  d x				(d) 	(∫_0^((Sqrt[2])/4))(2/(Sqrt[1-4(x^2)])) d x