Calc III
Homework 5
1. Express dw/dt as a function of t by using Chain Rule
(a) w =
x
z
+ yz , with x = cos t + sin t, y = cos t − sin t and z = 1/t
(b) w = z − sin xy, with x = t, y = ln t and z = et−1
∂z
∂z
2. Compute ∂u
and ∂v
where z = arctan (x/y) with x = u cos v and y =
∂z
∂z
u sin v. Evaluate ∂u and ∂v
when (u, v) = (−2, 0) .
3. Let w = x2 e2y cos 3z. Find the value of dw/dt at the point (1, ln 2, 0) on
the curve
c(t) = cos t, ln t2 + 2 , t , t ∈ R.
4. Let T = g(x, y) = xy − 2 be the temperature at the point (x, y) on the
ellipse. Find the maximum and minimum values of T if
√
√
x = 2 2 cos t, y = 2 sin t, 0 ≤ t ≤ 2π
5. Find ∇f at the given point f (x, y, z) = x2 + y 2 − 2z 2 + z ln x at (1, 1, 1)
6. Find the derivative of the function g(x, y) =
→
−
→
−
→
−
u = 12 i + 5 j .
x−y
xy+2
in the direction of
7. Find the derivative of the function h(x, y) = cos(xyz) + ln(zx) in the
→
−
→
− →
−
−
direction of →
u =2 i + j −2k.
8. Find the points on the ellipsoid x2 + 2y 2 + 3z 2 = 1 where the tangent
plane is parallel to the plane 3x − y + 3z = 1. Hint: think about normal
vectors
9. Find the points on the surface z 2 = xy + 1 which are closest to the origin.
Hint: you want to minimize the distance away from the origin of points
in R3 with the constraint function g(x, y, z) = z 2 − xy − 1
10. Find the dimensions of the rectangular box with largest volume if the total
surface area is given as 100cm2 .
11. Use Lagrange multiplier to minimize the function f (x, y, z) = yz + xy
subjected to the conditions xy = 1 and y 2 + z 2 = 1.
12. Find the local max, local min and saddle points to the functions if they
exist
(a) f (x, y) = x2 − y 2
(b) f (x, y) = ex cos y
1