Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 1
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 28/29 January 2016
Name:
1. A curve C in the xy-plane is represented by the equation
Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 .
(1)
In the x0 y 0 -plane obtained by rotating the xy-plane through an angle φ
x0 = x cos φ + y sin φ ,
y 0 = −x sin φ + y cos φ ,
(2)
the curve C is represented by a similar equation
A0 x02 + B 0 x0 y 0 + C 0 y 02 + D0 x0 + E 0 y 0 + F 0 = 0 .
(3)
(a) Express A0 , B 0 , C 0 , D0 , E 0 , F 0 in terms of A, B, C, D, E, F and φ.
(b) Prove that if the angle φ satisfies
cot 2φ =
A−C
,
B
(4)
then the curve C is represented by the equation
A0 x02 + C 0 y 02 + D0 x0 + E 0 y 0 + F 0 = 0 ,
(5)
i.e. B 0 = 0.
2. Use Mathematica, and the result of the previous question to identify the curve. Find a
parametric representation and plot the curve in the xy-plane. The Mathematica function
ParametricPlot can be used to plot parametric curves in the xy-plane.
√
√
(a) 3x2 + y 2 − 2 3xy − 8x − 8 3y = 0 .
√
√
(b) 57x2 + 14 3xy + 36x + 43y 2 − 36 3y − 540 = 0 .
√
√
(c) 2x2 + 5xy + 9 2x + 2y 2 + 9 2y + 36 = 0 .
3. A curve C is the intersection of the cone
z 2 = x2 + y 2 ,
(6)
with a plane.
Identify the curve, find a parametric representation and plot the curve in the xyz-space
for the planes below. The Mathematica function ParametricPlot3D can be used to plot
parametric curves in the xyz-space.
1
(a) z = 2 .
(b) y = 0 .
(c) x = 1 .
(d) x + y = 1 .
(e) x + z = 1 .
(f) x + y + z = 1 .
4. Consider the vector-valued function (with values in R3 )
r(t) = ln(3 −
√
t) i + (1 +
√
√
(3 − t)2
t) j +
k
4
(7)
(a) Find the domain D(r) of the vector-valued function r(t).
(b) Find the derivative dr/dt.
(c) Find the norm |dr/dt|.
Simplify the expressions obtained.
(d) Find the unit tangent vector T for all values of t in D(r).
(e) Find the vector equation of the line tangent to the graph of r(t) at the point
P0 (0, 3, 41 ) on the curve.
Bonus questions (each bonus question is worth extra 25 marks)
1. The parametrisation
x = α + a cosh t ,
y = β + b sinh t ,
−∞ < t < ∞
(8)
of the hyperbola
(x − α)2 (y − β)2
−
= 1,
a2
b2
represents only one branch of the hyperbola.
a > 0,
b > 0,
(9)
Find a parametrisation which represents both branches of the hyperbola.
2. A curve C is the intersection of the cone
x2 + y 2 − z 2 = 0 ,
(10)
ax + by + cz + d = 0 .
(11)
with the plane
Prove that the curve C is one of these 1) a circle; 2) an ellipse; 3) a parabola; 4) a
hyperbola; 5) a pair of intersecting lines; 6) a single line; 7) a point.
2