Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 1
To be solved during the tutorial session Thursday/Friday, 21/22 January 2016
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
where
A0 x02 + B 0 x0 y 0 + C 0 y 02 + D0 x0 + E 0 y 0 + F 0 = 0 ,
(3)
A0
B0
C0
D0
E0
F0
(4)
= A cos2 (φ) + B sin(φ) cos(φ) + C sin2 (φ) ,
= −A sin(2φ) + B cos(2φ) + C sin(2φ) ,
= A sin2 (φ) − B sin(φ) cos(φ) + C cos2 (φ) ,
= D cos(φ) + E sin(φ) ,
= E cos(φ) − D sin(φ) ,
=F.
If the angle φ satisfies
A−C
,
B
then the curve C is represented by the equation
cot 2φ =
A0 x02 + C 0 y 02 + D0 x0 + E 0 y 0 + F 0 = 0 ,
(5)
B0 = 0 .
(6)
1. 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) x2 − 2xy + y 2 − 4 2x − 4 2y = 0 .
√
√
(b) 31x2 − 10 3xy − 32x + 21y 2 − 32 3y − 80 = 0 .
√
√
(c) 32x2 − 7y 2 − 52xy − 144 5x + 72 5y + 900 = 0 .
2. A curve C is the intersection of the cone
z 2 = x2 + y 2 ,
(7)
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 = 2.5 .
3. Consider the vector-valued function (with values in R3 )
r(t) = ln(−t) i − t j +
t2
k
4
(8)
(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, 1, 41 ) on the curve.
2