MULTIVARIABLE CALCULUS, HOMEWORK 4
(1) Find the limit, if it exists, or show that the limit does not exist
x4 − y 4
(x,y)→(0,0) x2 + y 2
lim
(2) Verify that the function
u = e−α
2 k2 t
sin kx
is a solution of the heat equation ut = α2 uxx .
(3) Find the linear approximation of the function
p
f (x, y, z) = x2 + y 2 + z 2
at (3, 2, 6) and use it to approximate the number
√
3.022 + 1.972 + 5.992 .
(4) The wind-chill index is modelled by the function
W = 13.12 + 0.6215T − 11.37v 0.16 + 0.3965T v 0.16
where T is the temperature in ◦ C and v is the wind speed in km/h. The
wind speed is measured as 26 ± 2 km/h and the temperature is measured as
−11±1 ◦ C. Use differentials to estimate the maximum error in the calculated
value of W due to the measurement errors in T and v.
(5) For the implicitly defined function z = z(x, y) find δz/δx and δz/δy
x2 − y 2 + z 2 − 2z = 4
(6) A manufacturer has modeled its yearly production function P as a CobbDouglas function
P (L, K) = 1.47L0.65 K 0.35
where L is the labor hours (in thousands) and K is the invested capital (in
millions of $). Suppose that when L = 30 and K = 8, the labour force is
decreasing at the rate of 2 thousand labour hours per year and capital is
increasing at a rate of half a million dollars per year. Find the rate of change
of production.
2
MULTIVARIABLE CALCULUS, HOMEWORK 4
(7) The temperature at a point (x, y, z) is given by
T (x, y, z) = 200e−x
2 −3y 2 −9z 2
where T is measured in ◦ C and x, y, z in meters.
a) Find the rate of change of temperature at the point P = (2, −1, 2) and the
direction toward the point (3, −3, 3).
b) In which direction does the temperature increase the fastest at P ?
c) Find the maximum rate of increase at P .
(8) Find the equation of the tangent plane and the normal line to the surface
xy + yz + zx = 5
at the point (1, 2, 1).