2S1: Assignment 2
28 November 2008
Due date: Tuesday 6th January
1. Given the equation w = x2 +xy 2 +z 2 where x = rs+1, y = ers and z = 2r+3s,
use the Chain Rule to find ∂w
and ∂w
.
∂r
∂s
2. Given the equation yx + ln(y + z) = 1, use implicit differentiation to find
∂z
and ∂y
.
∂z
∂x
3. A company manufactures cylindrical storage containers of radius 3m and
height 4m. Suppose a maximum error of 0.03m can occur in the radius and
height of a storage container. Using the total differential of the volume function, approximate the maximum error in the volume of a storage container.
p
4. Let T (x, y) = 9 − x2 − y 2 be a function which describes temperature at a
point (x, y) in the plane.
(i) Sketch the graph of T and sketch the level curves for k = 0, 1, 2.
(ii) What is the rate of change of the temperature at the point P (2, 1) in a
north-easterly direction?
(iii) In which direction is the temperature increasing/decreasing most rapidly
at P ? In which directions is the temperature not increasing or decreasing?
5.
(i) Find all critical points of the function f (x, y) = x3 + 3xy + y 3 and classify
each critical point as a local maximum, local minimum or saddle point.
(ii) Use the method of Lagrange multipliers to find the maximum volume of
a rectangular box with surface area 18m2 .
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