Math2018/2019 Engineering Mathematics 2D/2E
Engineering Mathematics 2D/2E
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Math2018/2019 Engineering Mathematics 2D/2E
Written Test Practice Version
Time allowed: 45 minutes 7 questions Attempt all questions
Leibniz Rule for Differentiation of Integrals
d
dx
∫ v
u
f(x, t)dt =
∫ v
u
?f
?x
dt + f(x, v)
dv
dx
? f(x, u)du
dx
.
Multivariable Taylor Series
f(x, y) = f(a, b) + (x? a)?f
?x
(a, b) + (y ? b)?f
?y
(a, b)
+
1
2!
(
(x? a)2?
2f
?x2
(a, b) + 2(x? a)(y ? b) ?
2f
?x?y
(a, b) + (y ? b)2?
2f
?y2
(a, b)
)
+ · · ·
1. [5 marks]
Consider the function
z = f(x, y) = x2y3 + e4x sin(y).
(a) Find
?f
?x
and
?f
?y
.
(b) Verify that
?2f
?y?x
=
?2f
?x?y
.
2. [5 marks]
Determine the Taylor series expansion of f(x, y) = e?x sin(y) about (0, pi
2
)
up to and including quadratic terms.
3. [5 marks]
Find and classify the critical points of
f(x, y) = 2x2 ? 20x + 3y2 + 24y ? 10.
Also give the function values at the critical points.
4. [5 marks]
Let u = 3i + 4j? k and v = 2i? j + 2k.
Evaluate (if possible):
(a) v × u
(b) u · v
(c) v × u + u · v
5. [5 marks]
A particle moves along a curve with parametric equations
x(t) = e?t, y(t) = cos(t), z(t) = 3t,
where t is time.
Determine the magnitude of its acceleration vector at t = 0.
6. [5 marks]
For the vector field
f = xyzi + y2j + xk
calculate if possible:
(a) ?× f
(b) ?× (? · f)
(c) ? · (?× f)
7. [5 marks]
Calculate
∫
C F · dr where
F(x, y) = (1? 3y)i + 3xj
and C is the path in R2 from (0, 2) to (2, 0) clockwise along the circle x2+y2 = 4.