Key to the Practice Final

Problem 1

1. False a_n = 1/n² (Converges by PST, b_n = 1/n (Diverges by HST) then a_n/b_n = 1/n which diverges by HST.

2. True.   ||uxv|| = ||u|| ||v|| sinq = ||v|| sinq is maximized at p/2.

3. True, since y = x is perpendicular to i - j and i - j is perpendicular to the level curve.

Problem 2
A.  Diverges by IT

B.  Converges conditionally first by AST then by LCT with 1/sqrt(n) which diverges by PST.

Problem 3
0.80 with n = 87

Problem 4

  [4/3,8/3)

Problem 5
x + 9y + 3z = 28

Problem 6
r² = arsinq + brcosq

x² + y² = ax + by

(x - b/2)² + (y - a/2)² = (a² + b²)/4

Problem 7
If the cube has side length r, then the diagonal is v = <r,r,r> and the edge is w = <r,0,0>.

cosq = (v ^. w)/(||v|| ||w||) = r²/[(rsqrt3)(r)] = 1/sqrt3

So that 

q = cos^-1(1/sqrt3)

Problem 8
Let x = 0 then the lim is 0 and let y = x², then the lim is 1.  Since they tend towards two different limits, the limit does not exist.

Problem 9
Relative minimum at (1,1/2)

Problem 10

T = k/[x² + y² + z²]

gradT = <-2kx/[x² + y² + z²],-2ky/[x² + y² + z²],-2kz/[x² + y² + z²]>

= -2k/[x² + y² + z²]<x,y,z> which is a negative multiple of <x,y,z>, the vector pointing from (x,y,z) towards the origin.