Calculus Answers

1.   Let  {Un} and {Vn}  be the sequences defined by

   And{U0=3

{Un+1=Un/1+Un and Vn=1/Un

i.            Find U1 and U2                                                                                                

ii.          Prove that  {Vn} is an A.P                                                                               

iii.         Express Un and Vn in terms of n                                                                           

iv.         Find the limits of the two sequences.                                                              

v.            Are Vn, Un  convergent or divergent ?  

Determine the dimensions of a rectangular box, open at the top, having volume V, and requiring the least amount of material for its construction. Use: (i). The constraint to eliminate a variable (Second Partials Test (SPT)). [Verify using Mathematica] (ii). Lagrange multipliers. [Verify using Mathematica]

Provide all detailed steps to find the limit of the following functions

A) lim (tan x)^(cos x) as x approaches pi/2

B)lim x^(ln2/1+ln x) as x approaches infinity

C) lim (e^(4x)-1-4x/x^2) as x approaches infinity

D) lim e^(2x)-e^(-2x)/ln (x+1) as x approaches infinity to

E)lim( tan 4x)/(x+sin 2x) as x approaches 0

Suppose f is differentiable on R(real number) and has two roots. Show that f' has at least one root