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Calculus II

• These questions are mainly based on the material in Sections 2.3 to 2.5 of the lecture notes.
• The question marked with (x) you will solve together with your tutor in the tutorials.
• The feedback question marked with (*) should be solved in groups of 2 or 3 students. Your joint solution
must be submitted by 5pm on Monday 29th February (via the yellow box on the ground floor of the
Maths building) and clearly display the following information:
– The student number, full name (with surname underlined) and signature of all team members;
– The surname of the class tutor, and the time and location of the tutorial.
• Model solutions will be available on QMplus after 6pm on Monday 29th February and marked submissions
returned in the tutorials of Friday 4th March.
(x) 1. (a) Use the chain rule of partial differentiation to express ∂w/∂u and ∂w/∂v as functions of u
and v for
w = xy + yz + xz, x = u + v, y = u − v, z = uv.
How can you check your answers?
(b) Find the directional derivative of the function f(x, y, z) = xy +yz +xz at the point (1, −1, 2),
in the direction of the vector 3i + 6j − 2k.
(*) 2. (a) Find all first-order and second-order partial derivatives of the function
f(x, y) = e3y cos x + ln(2y) − ex sin y. [2013 Q1(b)]
(b) Find the directional derivative of the function f(x, y) = exy at the point (−2, 0) in the direction
of the unit vector that makes an angle of π/3 with the positive x-axis. [2010 Q1(d)]
3. Use the chain rule of partial differentiation to express dw/dt as a function of t for
w =
x 2z
+
y 2z
, x = cos2 t, y = sin2 t, z = 1
t
.
Show that one obtains the same result by expressing w in terms of t directly. [2013 Q2]
4. Find all the first-order and second-order partial derivatives of the following functions:
(a)
f(x, y) = sin(x2y) − ex cos y, [2011 Q1(c)]
(b)
f(x, y) = ey cos x − ex sin y + 4x2y3 − 3 ln x, [2009 Q1(f)]
(c)
f(x, y) = yex − x sin y + x2 − y2. [2008 Q1(f)]

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