QUIZ #1 -­ SOLUTIONS 1) 2) Find ∂u
∂u
and for u ( r,θ ) = sin ( r cosθ ) . ∂r
∂θ
∂u
= cos ( r cosθ ) ⋅ cosθ = cosθ cos ( r cosθ ) ∂r
∂u
= cos ( r cosθ ) ⋅ ( −r sin θ ) = −r sin θ cos ( r cosθ ) ∂θ
5y 4 cos 2 x
Find lim
( x,y )→( 0,0 ) x 4 + y 4
5y 4 cos 2 x 0
= indeterminate. ( x,y )→( 0,0 ) x 4 + y 4
0
lim
5y 4 cos 2 x 0
= 4 =0
( x,y )→( 0,0 ) x 4 + y 4
x
Along x-­‐axis: y = 0 and lim
3) 5y 4 cos 2 x 5y 4
Along y-­‐axis: x = 0 and lim
= 4 = 5 ≠ 0. ( x,y )→( 0,0 ) x 4 + y 4
y
Hence, the limit does not exist. Let f ( x, y ) = 1+ 4 − y 2 . a)
Find the domain of f. The domain of f = {( x, y ) : −2 ≤ y ≤ 2} . b)
Find the range of f. The range of f is [1, 3].