MA1132 2015-2016 Problem Set 1 PDF

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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 x′ y ′ -plane obtained by rotating the xy-plane through an angle φ x′ = x cos φ + y sin φ ,

y ′ = −x sin φ + y cos φ ,

(2)

the curve C is represented by a similar equation A′ x′2 + B ′ x′ y ′ + C ′ y ′2 + D′ x′ + E ′ y ′ + F ′ = 0 .

(3)

(a) Express A′ , B ′ , C ′ , D ′ , E ′ , F ′ 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 A′ x′2 + C ′ y ′2 + D′ x′ + E ′ y ′ + F ′ = 0 ,

(5)

i.e. B ′ = 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 k t) j + 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, − b2 a2 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...