Key to the Practice Final

Problem 1

A.  False the gradient vector must also be zero

B.  False, the harmonic series 1/n is a counter-example

C.  True, since the cos(theta) must therefore equal zero.

Problem 2

A.  Diverges by the integral test

B.  Converges conditionally by the AST and the LCT with 1/n.

Problem 3

1-1/18+1/600 = .9461 (since when n = 2, |a_n+1| < .0001 )

Problem 4

R = 2

Problem 5

4pi/3

Problem 6

u = <-1,-2,2>, v = <1,2,0>, w = <2,4,-2>

u@v = -5, u@w = -14, and v@w = 10

since none of the above is 0, the triangle is not a right triangle

Problem 7

2xu - (2x - 2y)(1 - u)

Problem 8

(0,0) is a saddle, and (2/3,2/3) is a local min