University of Aarhus
Department of Mathematical Sciences
Calculus Quiz
week 38.2
Legend: A 4 indicates a correct response; a 8, indicates an incorrect response, in this case, the correct answer is marked with a l. Correct and follow
a 4,l for a solution. Return through n.
Fall 2004
2
The Chain Rule
1. If f (x, y) = x2 + y 2 , x = sin t, y = cos t, then by the chain rule
f 0 (t) = 1
Yes
No
2. If z = x + y, x = u(s, t), y = v(s, t), then by the chain rule
zs = u s
zs = vs
zs = us + vs
zt = vt
3. Mark the Jacobian matrix of
fx
gx
fy
gy
(x, y) 7→ (f (x, y), g(x, y))
fxx fyy
fx
gxx gyy
fy
gx
gy
3
4. If z = f (x, y) = x2 − xy, x = s + t, y = st then the differential is
dz = (s + t)ds + stdt
dz = (2s + 2t − 2st − t2 )ds + (2s + 2t − 2st − s2 )dt
5. If f, g are composable differential maps and the Jacobian matrix
du (g) = 0, then the Jacobian matrix of the composite f ◦ g map
du (f ◦ g) = 0
Yes
No
6. Let F (x, y) = Arctan(x + y). The equation F (x, y) = 0 defines a
function y(x) for x ≈ 0. Then
y 0 (0) = 0
y 0 (0) = 1
y 0 (0) = −1
Solutions to Quizzes
4
Solutions to Quizzes
Solution to Quiz:
No, z = f (x, y) = x2 + y 2 , x = sin t, y = cos t gives
zx = 2x, zy = 2y, x0 = cos t, y 0 = − sin t
By [S] 11.5 Theorem 2
z 0 = zx x0 + zy y 0
I.e.
f 0 (t) = 2 sin t cos t − 2 cos t sin t = 0
Solutions to Quizzes
5
Solution to Quiz:
z = x + y, x = u(s, t), y = v(s, t) gives
zx = zy = 1, xs = us , ys = vs
By [S] 11.5 Theorem 2
zs = zx xs + zy ys
I.e.
zs = us + vs
Solutions to Quizzes
6
Solution to Quiz:
By [LA] Section 2.2 the Jacobian matrix of
(x, y) 7→ (f (x, y), g(x, y))
is given
d(x,y) =
fx
gx
fy
gy
Solutions to Quizzes
7
Solution to Quiz:
z = f (x, y) = x2 − xy, x = s + t, y = st gives
dz = (2x − y)dx − xdy, dx = ds + dt, dy = tds + sdt
I.e.
dz = (2(s + t) − st)(ds + dt) − (s + t)(tds + sdt)
= (2s + 2t − st)ds + (2s + 2t − st)dt − (st + t2 )ds − (s2 + st)dt
= (2s + 2t − 2st − t2 )ds + (2s + 2t − 2st − s2 )dt
Solutions to Quizzes
8
Solution to Quiz:
Yes, if du (g) = 0 then by [LA] Section 2.2
du (f ◦ g) = dg(u) (f )du (g) = 0
Solutions to Quizzes
9
Solution to Quiz:
Given F (x, y) = Arctan(x + y) and calculate
1
1
Fx =
, Fy =
1 + (x + y)2
1 + (x + y)2
By [S] 11.5 6 the derivative is
y 0 (x) = −
Fx
Fy
I.e.
y 0 (0) = −1